stats final

Suppose that the probability density function of the random variable X is given by the graph below. Determine the following probabilities. Round your answers to 3 digits after the decimal point. (a) 𝑃(𝑋=1)= (b) 𝑃(𝑋≤4)= (c) 𝑃(4≤𝑋≤6)= (d) 𝑃(𝑋>4)=

(a) 0 (b) 0.333 (c) 0.333 (d) 0.667

A random sample of 30 male students and a random sample of 30 female students of STAT2510 was selected. The female students scored significantly higher on the final than the male students. (a) We may conclude that the difference between the mean scores of the samples is due to the gender of the students. (b) We can be confident that the female students of STAT2510 scored in average higher than the male students of STAT2510.

(a) false (b) true

Find the area under the standard normal curve to the left of −1.724−1.724 using the function pnorm(). Round your answer to 4 digits after the decimal point.

0.0424

You toss a coin 200 times. Find the probability that exactly half of the coin tosses will be heads. Round your answer to 4 digits after the decimal point.

0.0563

Find the area under the standard normal curve to the right of 1.1731.173 using the function pnorm(). Round your answer to 4 digits after the decimal point.

0.1204

There are two containers. One contains the letters ALLOSAURUS the other the letters CEPHALOPOD. You choose a container by coin toss and then randomly select a letter. Find the probability of selecting the letter O.

0.15

Suppose X is a discrete random variable whose distribution is given by the table below. X0123456p(x)0.10?0.150.100.200.150.05 Find the probability P(X = {2, 4, 6})

0.4

The manager of an ice cream store finds that 42 of the 97 people who tried the new flavor are buying it. Use the data to estimate the probability that a person who tries the new flavor will buy it. Round your answer to 3 digits after the decimal point.

0.433

Suppose that E and F are independent events, P(E) = 0.3 and P(F) = 0.2. Find P(E or F).

0.44

Suppose X is a discrete random variable whose distribution is given by the table below. X0123456p(x)0.10?0.150.100.200.150.05 Find the probability P(X ≥ 2)

0.65

You roll a die 7 times. What is the probability of rolling at least one 6? Round your answer to 3 digits after the decimal point

0.721

Suppose that a 95% confidence interval for 𝜇μ is (3, 5). What is the margin of error?

1

Suppose that the distribution of systolic blood pressure among patients has mean 135 mmHg and standard deviation 13 mmHg. Consider a random sample of 11 patients. What is the standard deviation of the sample mean?

3.92

Use the function mean() to calculate the mean of the values 2.42, 9.62, 1.47, 1.92, 5.72, 6.4, 4.67. Round your answer to 4 digits after the decimal point.

4.6029

Suppose that a biologist measures the time intervals between male mating calls (measured in seconds) of a certain species of tree frogs at different temperatures (measured in Celsius). She finds a linear relationship between the two variables and calculates the following prediction equation: time interval = 8.4 - 0.2 temperature Which of the following interpretations of the slope is correct?

An increase in temperature by 1 degree decreases the time interval by 0.2 seconds

The quality control inspector of the company claims that the average weight is between 0.45 and 0.55 grams. What would you recommend to provide evidence for this claim.

Calculate a two-sided confidence interval.

A 95% two-sided confidence interval for 𝜇μ which has been calculated using R turns out to be (0, 1). A 90% two-sided confidence interval based on the same data will contain the value 0.9.

Cannot tell

Suppose that you roll a fair die 1200 times. What is the approximate distribution of the sample proportion 𝑝̂ p^ of the number of sixes that you roll?

N(u = 1/6, theta = 0.0108)

A 95% confidence interval for 𝜇μ turns out to be (45.6,56.7)(45.6,56.7). Is it correct to say that 𝜇μ falls between 45.6 and 56.7 with probability 0.95? Yes, that is correct.

No, but we can be 95% confident about that

The following graph is a plot of some data and their regression line. (a) Choose the most appropriate value of the response variable when the explanatory variable is 1. (b) Choose the most appropriate value of the correlation coefficient r.

(a) 4 (b) -0.9

A regression analysis of the length of a Western Hognose snake (in mm) and its age (in days) produced the prediction equation Length = 278 + 0.44 · Age. All measurements of the length were performed for ages between 200 and 700 days. (a) Predict the length of a Western Hognose snake after 450 days. (b) A scatter plot reveals a linear relationship between Length and Age, with a correlation coecient of 0.93 and a coecient of determination of 0.86. Which of the following conclusions would be most appropriate?

(a) 476 mm (b) About 86% of the variation in the length of the snake is explained by its age.

According to the American Social Health Organization, one out of four teens in the United States becomes infected with an STD each year. What is the probability that in a random selection of 4 teens at least one becomes infected with an STD in a given year? Round your answer to 3 digits after the decimal point.

0.684

A club with 31 members is voting on a proposal. A group of 4 has decided to vote in favor of the proposal while each of the rest of the members vote independently with a chance of 50% in favor of the proposal. What are the chances that the proposal will get the majority vote. Round your answer to 4 digits after the decimal point.

0.779

Let X be the litter size for a certain species of animals. Suppose that X has the following distribution. X012f(x)0.40.20.4 The expected value of X is 1. Find the variance of X. Round your answer to 3 digits after the decimal point.

0.8

Find the area under the standard normal curve between −2.135−2.135 and 1.2781.278 using the function pnorm(). Round your answer to 4 digits after the decimal point.

0.883

Decide whether or not you should reject the null hypothesis 𝐻0:𝜇=2H0:μ=2 in favor of the alternative hypothesis 𝐻𝐴:𝜇>2HA:μ>2 at significance level 𝛼=0.05α=0.05 when 𝑇=−2.5T=−2.5 and 𝑛=42n=42.

fail to reject the null hypothesis

In a game, a spinner with five equal-sized spaces is labeled from A to E. If a player spins an A the player wins 15 points. If any other letter is spun the player loses 4 points. What is the expected gain or loss from playing 40 games?

loss of 8 points

A 95% two-sided confidence interval for 𝜇μ, which has been calculated using R, turns out to be (0, 1). A 99% confidence interval based on the same data ______ contain the value 0.9.

will

Consider the following scatterplot. (a) Identify the form of the plot. (b) Identify the direction of the plot. positive association (c) Is the correlation coefficient an appropriate measure of the strength of the relationship between the two variables.

(a) curved relationship (b) positive association (c) no

Which of the following numbers are possible descriptions of probabilities. (a) 0 (b) 12.5% (c) 1.3 (d) 200% (e) -0.2

(a) possible (b) possible (c) impossible (d) impossible (e) impossible

Let A and B events with P(A) = 1/3, P(B) = 1/4, and P(A and B) = 1/8. Find the probability P(B | A).

0.375

There are 125 students in a certain class. Suppose that 75 of these students are female, 44 are sophomores, and 28 are both, that is sophomore and female. A student from this class is randomly selected. Denote the event that the selected student is a sophomore with SO, and the event that the student is female with F. Based on these numbers calculate the probability 𝑃(𝑆𝑂𝐶)P(SOC). Round your answer to 3 digits after the decimal point.

0.648

Two coins are tossed. Find the probability that at least one of the coins shows head.

0.75

Decreasing the sample size decreases the margin of error.

false

A typical russet potato weighs about 173 g. Farmer Joe believes that his potatoes are bigger and needs your help with providing the evidence. Let 𝜇μ denote the true mean weight of a potato from Farmer

𝐻0:𝜇=173 H0:μ=173 versus 𝐻𝐴:𝜇>173

Echinacea has been widely used as an herbal remedy for the common cold, but previous studies evaluating its efficacy as a remedy have produced conflicting results. In a new study, researchers randomly assigned 437 volunteers to receive either a placebo or echinacea treatment before being infected with rhinovirus. Healthy young adult volunteers were recruited for the study from the University of Virginia community. (a) Suppose that the population of interest were healthy young adults from the University of Virginia. The sample in this study was then a (b) Can the results be generalized to the population? (c) Can the findings of the study be used to establish causal relationships?

(a) convenience sample (b) No, because the sample was self selected. (c) Yes, because the treatment was randomly assigned.

Use the function qnorm() to find the value z such that the area to the left of z under the standard normal curve is 0.25. Round your answer to 4 digits after the decimal point. Correct!

-0.6745

Suppose that X is a continuous random variable whose probability density function is given by the graph below. Find the probability P(X = 0.5)

0

Suppose that the distribution of scores in a statistics exam is approximately normal with a mean of 64 and a standard deviation of 12. Use the Empirical Rule to find approximately the proportion of students with a score greater than 76.

0.16

In Monopoly one rolls a pair of dice. Find the probability of getting a sum of 7. Round your answer to 3 digits after the decimal point.

0.167

You write each letter of the word CEPHALOPOD on a piece of paper and put all ten pieces in a hat. You then randomly draw a piece of paper from the hat. Find the probability of drawing the letter O.

0.2

Suppose X is a discrete random variable whose distribution is given by the table below. X0123456p(x)0.10?0.150.100.200.150.05 Find the probability P(X = 1)

0.25

There are two types of sneetches, the star-bellied sneetch (S. astra) and the plain sneetch (S. simplex). Professor O. B. Servant classified each sneetch of a population of 125 sneetches by type (S. astra, S. simplex) and by habitat (ground, tree, lake). GroundTreeLakeS. Astra12249S. Simplex183626 Find the probability that a randomly selected sneetch is of the type S. astra. Round your answer to 3 digits after the decimal point.

0.36

Suppose that X is a continuous random variable whose probability density function is given by the graph below. Find the probability P(0 < X < 1)

0.375

Suppose that E and F are disjoint events, P(E) = 0.1 and P(F) = 0.4. Find P(E or F).

0.5

A typical russet potato weighs about 173 g. Farmer Joe believes that his potatoes are bigger and needs your help with providing the evidence. The average weight of 33 potatoes from Joe is 175 grams and the sample standard deviation is 22 grams. Find the value of the appropriate test statistic.

0.522

There are two types of sneetches, the star-bellied sneetch (S. astra) and the plain sneetch (S. simplex). Professor O. B. Servant classified each sneetch of a population of 125 sneetches by type (S. astra, S. simplex) and by habitat (ground, tree, lake). GroundTreeLakeS. Astra12249S. Simplex183626 Find the probability that a randomly selected sneetch is of the type S. simplex given that its habitat is a tree. Round your answer to 3 digits after the decimal point.

0.6

In the United States about 7% of the male population and about 0.4% of the female population is red-green color blind (that is they cannot distinguish red from green, or see red and green differently from how others do). The population in the United States consist of about 49% males and 51% females. A person is randomly selected. Let M be the event that the person is male, let F be the event that the person is female, let CB be the event that the person is red-green color blind, and let NC be the event that the person is not red-green color blind. Calculate the probability that the selected person is male given that the person is red-green color blind. Round your answer to 3 digits after the decimal point.

0.944

Find the value z such that the area to the right of z under the standard normal curve is 0.1. Round your answer to 4 digits after the decimal point.

1.2816

Let X be the litter size for a certain species of animals. Suppose that X has the following distribution. X0123f(x)0.20.30.40.1 Find the expected value of X.

1.4

Use the function sd() to calculate the standard deviation s of the values 1.196, 2.372, 4.754, 7.203, 6.873, 4.223, 9.253. Round your answer to 3 digits after the decimal point.

2.84

The distribution of a discrete random variable X is given by the table below. X-271027p(x)0.40.20.4 Calculate the standard deviation of X. Round your answer to 3 digits after the decimal point.

24.479

Suppose X is a random variable for which 𝐸[𝑋]=E[X]= 4. Find E[9 - X].

5

An insurance for an appliance costs $50 and will pay $526 if the insured item breaks plus the cost of the insurance. The insurance company estimates that the proportion 0.07 of the insured items will break. Let X be the random variable that assigns to each outcome (item breaks, item does not break) the profit for the company. A negative value is a loss. The distribution of X is given by the following table. X50-526p(x)?0.07 Complete the table and calculate the expected profit for the company. In other words, find the expected value of X.

9.68

Decide whether or not to reject the null hypothesis in favor of the alternative hypothesis at significance level 𝛼=0.05 based on the following information: 𝑛=32, 𝐻𝐴:𝜇<200, 𝑇=2.1. I recommend that you sketch the area that represents the p-value.

Fail to reject the null hypothesis.

Denote the average weight loss of diet 1 with 𝜇1μ1 and the average weight loss of diet 2 with 𝜇2μ2 (measured in pounds). A 95% confidence interval for 𝜇1−𝜇2μ1−μ2 based on a random sample of size n = 10 from an approximately normal population turns out to be (0.01, 10.48). Decide if the following statement is true or false. We cannot trust the result because the sample size is too small.

False

A company claims that less than 0.5% of its items are defective. You suspect that this claim is not true. Denote by p the true proportion of defective items produced by the company. Which of the following is the correct alternative hypothesis?

Ha: P (greater than) 0.005

In 2013, the Pew Research Foundation reported that "45% of U.S. adults report that they live with one or more chronic conditions". However, this value was based on a sample, and therefore is only an estimate of the proportion of all U.S. adults who live with one or more chronic conditions. The study reported a standard error of about 1.2%, and a normal model may reasonably be used in this setting. Calculate a 95% confidence interval for the proportion of U.S. adults who live with one or more chronic conditions and interpret this interval. One can be 95% confident that between 42.6 and 47.4 percent of U.S. adults live with one or more chronic conditions.

One can be 95% confident that between 42.6 and 47.4 percent of U.S. adults live with one or more chronic conditions.

A 95% confidence interval for 𝜇μ turns out to be (45.6,56.7)(45.6,56.7). Is the mean 𝜇μ included in the 95% confidence interval? Perhaps, but we cannot be sure about that.

Perhaps, but we cannot be sure about that

A 95% two-sided confidence interval for 𝜇μ which has been calculated using R turns out to be (0, 1). One can be 95% confident that 𝜇μ is between 0 and 1.

True

Denote the average weight loss of diet 1 with 𝜇1μ1 and the average weight loss of diet 2 with μ2 (measured in pounds). A 95% confidence interval for μ1−μ2 based on a random sample of size n = 10 from an approximately normal population turns out to be (0.01, 10.48). Decide if the following statement is true or false. One can be 95% confident that μ1 is at least 0.01 pounds larger and at most 10.48 pounds larger than μ2.

True

Denote the average weight loss of diet 1 with 𝜇1μ1 and the average weight loss of diet 2 with 𝜇2μ2 (measured in pounds). A 95% confidence interval for 𝜇1−𝜇2μ1−μ2 based on a random sample of size n = 10 from an approximately normal population turns out to be (0.01, 10.48). Decide if the following statement is true or false. 𝜇1μ1 is significantly different from 𝜇2μ2 at significance level 0.05.

True

Researchers randomly divided 868 Navajo Indian children into 2 groups. Half of the children received a placebo while the rest received 1,000 mg of vitamin C daily. The number of colds and the length of duration of each cold was recorded. The researchers found that the recovery time in the group that received vitamin C was significantly shorter than those who received the placebo.

We can be confident that Vitamin C caused the shorter recovery time.

Suppose we fit a regression line to predict the shelf life of an apple based on its weight. For a particular apple, we predict the shelf life to be 4.6 days. The apple's residual is -0.6 days. Did we over or under estimate the shelf-life of the apple?

We overestimated the shelf-life of the apple.

Let X be the number of girls in a random sample of 12 newborn babies.

X is binomially distributed

Let X be the number of suicides in Atlanta during December.

X is not binomially distributed

Let X be the weight of a baby at birth at the local hospital.

X is not binomially distributed

Denote by X the number of children in a randomly selected family. Describe the event that the selected family has at least 5 children using the random variable X.

X ≥ 5

Suppose that the lower-bound 98% confidence interval for the parameter 𝜇μ consists of all values greater than 4.5. Can you reject 𝐻0:𝜇≤4H0:μ≤4 in favor of 𝐻𝐴:𝜇>4HA:μ>4 at significance level 𝛼=0.02α=0.02?

Yes

Read the following brief report of statistical research and identify whether it is an observational study or an experiment. Athletes who had suffered hamstring injuries were randomly assigned to one of two exercise programs. Those who engaged in static stretching returned to sports activity in a mean of 37.4 days. Those assigned to a program of agility and trunk stabilization exercises returned to sports in a mean of 22.2 days.

experiment

Read the following brief report of statistical research and identify whether it is an observational study or an experiment. Brain scans revealed that older people walking between six and nine miles a week appeared to have more brain tissue in key areas. The Pittsburgh University study of 299 people suggested they had less "brain shrinkage", which is linked to memory problems.

observational study

A company sells saffron in packets advertised to contain 0.5 grams for $4.99. Let 𝜇μ be the average weight of saffron in each packet. A consumer advocate group is concerned that the amount of saffron in each packet is in average at most 0.4. Which test should be done to provide evidence?

𝐻0:𝜇=0.4 versus 𝐻𝐴:𝜇<0.4

A company sells saffron in packets advertised to contain 0.5 grams for $4.99. Let 𝜇μ be the average weight of saffron in each packet. The company is loosing money and the management suspects that the amount of saffron in each packet is in average at least 0.6. What test could provide evidence?

𝐻0:𝜇=0.6 versus 𝐻𝐴:𝜇>0.6

Suppose X and Y are random variables for which Var(X) = 7 and Var(Y) = 2. Find Var(X - Y + 5).

9

Suppose that 32 residential home sales in a city are used to fit a least squares regression line relating the sales price, y, to the square feet of living space, x. The sampled homes range from 1,500 square feet to 4,000 square feet. The resulting prediction equation is y = −25,000+80x. Furthermore, the coefficient of determination is 0.85. (a) Predict the price of a house that has 3000 square feet. $215,000 (b) Which of the following statements is true? 85% of the variation in the sales price of the house can be explained by the amount of living space of the house.

(a) $215,000 (b) 85% of the variation in the sales price of the house can be explained by the amount of living space of the house.

. Lipitor is a drug used to control cholesterol. In clinical trials of Lipitor, 94 subjects were treated with Lipitor and 270 subjects were given a placebo. Among those treated with Lipitor, 7 developed infections. Among those given a placebo, 27 developed infections. Denote by p1 the true proportion of patients that develope an infection when treated with Lipitor and denote by p2 the true proportion of patients that develope an infection when treated with a placebo. (a) Calculate a confidence interval for p2. Use z? = 1.96. (b) Calculate the test statistic for the test H0 : p1 = p2 versus Ha : p1 6= p2.

(a) (0.0643, 0.1358) (b) Z = -0.7326

The table below is based on records of accidents in 1988 in the State of Florida. Safety Equipment Injury Total Fatal Nonfatal None 1,601 162,527 164,128 Seat belt 510 412,368 412,878Total2,111574,895577,006 (a) Calculate the proportion of fatal accidents. Round your answer to 4 digits after the decimal point. (b) What percent of the fatal accidents used a seat belt? Round your answer to 1 digit after the decimal point. (c) What is the relative risk of a fatal accident comparing those not wearing a seat belt with those wearing a seat belt. Round your answer to 1 digit after the decimal point.

(a) 0.0037 (b) 24.2% (c) 7.9

A store has an alarm system. If there is a break in then the alarm will go off with probability 0.99. With no break in during a particular night, the alarm will go o↵ with probability 0.005 (perhaps triggered by a mouse or inadvertently activated by the owner). The probability of a break in during a particular night is 0.001. (a) What is the probability that the alarm goes o↵ during a particular night? (b)What is the probability that someone broke into the store given that the alarm just went o↵.

(a) 0.005985 (b) 0.1654

The popular statistical software package SAS reports that the p-value for the one-sample t-test of H0 : µ = 1 versus Ha : µ 6= 1 turns out to be 0.06. The test statistic is 1.7. Hint: Sketch the density curve of the test statistic and indicate the test statistic and the corresponding p-value for each test. (a) The p-value for the test H0 : µ (less than or equal to) 1 versus Ha : µ > 1 is: (b) The p-value for the test H0 : µ (greater than or equal to) 1 versus Ha : µ < 1 is:

(a) 0.03 (b) 0.97

Download the file assignment10data2.csv download(comma separated text file) and read the data into R. Your data set consists of 100 measurements in Celsius of body temperatures from women and men. Use the function t.test() to answer the following questions. Do not assume that the variances are equal. Denote the mean body temperature of females and males by 𝜇𝐹μF and 𝜇𝑀μM respectively. (a) Find the p-value for the test 𝐻0:𝜇𝐹=𝜇𝑀H0:μF=μM versus 𝐻𝐴:𝜇𝐹≠𝜇𝑀HA:μF≠μM. (b) Are the body temperatures for men and women significantly different? Use significance level 𝛼=0.1α=0.1. (c) Denote the sample mean of the females by 𝑥¯𝐹x¯F and the sample mean of the males by 𝑥¯𝑀x¯M . Then 𝑥¯𝐹=x¯F= _________________________ and 𝑥¯𝑀x¯M = _____________. (d) The 90% two-sided confidence interval for the difference 𝜇𝐹−𝜇𝑀μF−μM ranges from ________________ to ______________.

(a) 0.0369 (b) Yes (c) 36.9131, 36.7349 (d) 0.0384, 0.3181

In Monopoly one rolls a pair of dice. (a) Find the probability of getting a sum of 10

(a) 0.0833

Suppose that the frequencies of blood phenotypes in the population are as follows: A B AB O 0.42 ? 0.04 0.44 (a) What is the probability that a randomly chosen person has blood type B? (b) What is the probability that two randomly chosen people both have blood type A? (c) What is the probability that neither of two randomly chosen people has blood type O?

(a) 0.1 (b) 0.1764 (c) 0.3136

Let X be the lifetime of an electronic device. It is known that the average lifetime of the device is 777 days and the standard deviation is 135 days. Let 𝑥¯x¯ be the sample mean of the lifetimes of 171 devices. The distribution of X is unknown, however, the distribution of 𝑥¯x¯ should be approximately normal according to the Central Limit Theorem. Calculate the following probabilities using the normal approximation. (a) 𝑃(𝑥≤766)= (b) 𝑃(𝑥≥790)= (c) 𝑃(766≤𝑥≤796)=

(a) 0.1433 (b) 0.104 (c) 0.8238

Suppose that the random variable X has the probability mass function given by the table below. X12345678p(x)0.180.070.2?0.050.030.220.1 (a) 𝑃(𝑋=4)= (b) 𝑃(𝑋≤2)= (c) 𝑃(𝑋>2)= (d) 𝑃(𝑋∈{2,6,7})=

(a) 0.15 (b) 0.25 (c) 0.75 (d) 0.32

A large class is composed as follows. Freshman -Sophomore- Junior -Senior Male 0 23 35 12 Female 3 53 80 24 You randomly select a student from this class. (a) Find the probability that the selected student is a junior and female. (b) Find the probability that the selected student is not a junior given that the student is male

(a) 0.3478 (b) 0.5

Consider the following relative frequency histogram. What proportion of the data is at most 4? What proportion of the data is greater than 6?

(a) 0.35 (b) 0.55

Control of the disease caused by the Eastern Equine Encephalomyelitis (EEE) virus requires a good understanding of which vectors are most important in transmitting the EEE virus to the different hosts. There are six species of mosquitoes in Alabama carrying this virus. A DNA analysis of a sample of blood-engorged female mosquitoes of the species Ae. vexans shows that 75 of the 𝑛=135n=135 mosquitoes have fed on mammals. Use the R function binom.test() to find the following 93% confidence intervals for the proportion of blood meals taken from mammals by the species Ae. vexans. (a) The 93% two-sided confidence interval ranges from __________ to _________ (b) The 93% lower-bound confidence interval ranges from _________ to 1. (c) The 93% upper-bound confidence interval ranges from 0 to _____________ Now calculate the following confidence intervals using the R function prop.test(). (d) The 93% two-sided confidence interval ranges from 0.474 to ____________ (e) The 93% lower-bound confidence interval ranges from ___________ to 1. (f) The 93% upper-bound confidence interval ranges from 0 to _______

(a) 0.4739, 0.635 (b) 0.4884 (c) 0.6212 (d) 0.474, 0.6344 (e) 0.4884 (f) 0.6209

In some state the mean ACT score in Mathematics is 𝜇ACT=μACT= 22 and the standard deviation is 𝜎ACT=σACT= 6, while the mean SAT score in Mathematics is 𝜇SAT=μSAT= 470 and the standard deviation is 𝜎SAT=σSAT= 110. Suppose that the ACT scores and the SAT scores have a normal distribution. Terence gets a score of 19 on the Mathematics portion of the ACT and Philip gets a score of 449 on the Mathematics portion of the SAT. Answer questions (a)-(d) using the Empirical Rule. You should not use R here since the point of this exercise is to practice the Empirical Rule. (a) What proportion of ACT scores are less than 22? (b) What proportion of SAT scores are greater than 690? (c) What proportion of ACT scores are between 16 and 34? (d) What proportion of SAT scores are between 250 and 470? (e) What is the z-score of Terence? (f) What is the z-score of Philip? (g) Who did better in Mathematics, Terence or Philip?

(a) 0.5 (b) 0.025 (c) 0.815 (d) 0.475 (e) -0.5 (f) -0.1909 (g) Philip

A class has 32 students of which 16 are juniors, 13 are male, and 9 are both, that is, junior and male. You randomly select a student. Denote the event that the student is a junior by J and the event that the student is male by M. Find the following probabilities rounded to 4 digits after the decimal point. (a) 𝑃(𝐽)P(J) = (b) 𝑃(𝐽𝐶)P(JC) = (c) 𝑃(𝐽 and 𝑀)P(J and M)= (d) 𝑃(𝐽 or 𝑀)P(J or M) =

(a) 0.5 (b) 0.5 (c) 0.2813 (d) 0.625

Suppose there is a school with 60% boys and 40% girls as its students. The female students wear trousers or skirts in equal numbers; the boys all wear trousers. An observer sees a (random) student from a distance, and what the observer can see is that this student is wearing trousers. Let G be the event that the student is a girl, let B the event that the student is a boy, let T be the event that the student wears trousers, and let S be the event that the student wears a skirt. (a) Find P(T): (b) Find P(G!T):

(a) 0.8 (b) 0.25

Peter and Paul visit Grandpa on the farm. To keep them busy he asks Peter to find the biggest apple and Paul to find the biggest potato. Joe knows that his apples have an average weight of 200 grams and a standard deviation of 40 grams and his potatoes have an average weight of 160 grams and a standard deviation of 30 grams. Peter brings an apple that weighs 264 grams and Paul brings a potato that weighs 205 grams. (a) Find the z-score of the weight of Peter's apple. (b) Who did better?

(a) 1.6 (b) Peter

Insurance for an appliance costs $30 and will pay $330 if the insured item breaks within 2 years. The insurance company estimates that 6% of the insured items will break within 2 years. Let X be the random variable that assigns to each outcome (item breaks, item does not break) the profit to the company. A negative value of X is a loss. Then the distribution of X is given by the following table. Value of X 30 -300 Probability 0.94 0.06 16. (a) Calculate the mean value of X. (b) Calculate the standard deviation of X

(a) 10.2 (b) 78.3707

The boxplot below is based on 60 measurements. (a) About 75% of the measurements are less than 130 . (b) About 30 measurements of the data are greater than 90 . (c) The interquartile range is approximately 60 .

(a) 130 (b) 90 (c) 60

The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days. (a) What percentage of patients will recover in less than 3.2 days? Use the Empirical Rule (b) What percentage of patients will take longer than 9.5 days? Use the Empirical Rule. (c) What percentage of patients will take between 7.4 and 11.6 days? Use the Empirical Rule (d) Suppose a patient has a recovery time of 5 days. What is the z-score of that time? (e) Suppose a patient has a recovery time of 7 days. Is this time shorter or longer than the 90th percentile of the recovery time?

(a) 16% (b) 2.5% (c) 15.85 (d) -0.1429 (e) shorter

Consider the following frequency histogram of scores obtained in a statistics exam. (a) How many scores are in the data set? (b) What proportion of scores is at most 80? (c) What proportion of scores is greater than 60?

(a) 20 (b) 0.7 (c) 0.75

The National Survey of Family Growth conducted by the Centers for Disease Control gathers information on family life, marriage and divorce, pregnancy, infertility, use of contraception, and men's and women's health. One of the variables collected on this survey is the age at first marriage. The histogram below shows the distribution of ages at first marriage of 5,534 randomly sampled women between 2006 and 2010. The average age at first marriage among these women is 23.44 with a standard deviation of 4.72. (a) Calculate a 95% lower-bound confidence interval for the average age at first marriage. Round your answer to two digits after the decimal point. The lower bound is (b) Suppose that the 95% upper-bound confidence interval for the average age at first marriage is the interval (−∞,23.54)(−∞,23.54). Interpret this interval. (c) Is the distribution of the variable "average age at first marriage" normally distributed? (d) Is it necessary that the distribution of the variable "average age at first marriage" is normal for the method you used in (a) to be valid

(a) 23.34 (b) One can be 95% confident that average age at first marriage is at most 23.54 years. (c) No (d) No

Suppose 𝑦=3.7−1.2𝑥y=3.7−1.2x is the equation of a least squares regression line. (a) What is the value of the intercept? (b) What is the value of the slope? (c) What is the change in y if x increases by 1 unit? (d) Use the given equation to predict y for x = 3.

(a) 3.7 (b) -1.2 (c) -1.2 (d) 0.1

Trait anger is defined as a relatively stable personality trait that is manifested in the frequency, intensity, and duration of feelings associated with anger. People with high trait anger have rage and fury more often, more intensely, and with longer lasting episodes than people with low trait anger. It is thought that people with high trait anger might be particularly susceptible to coronary heart disease; 12,986 participants were recruited for a study examining this hypothesis. Participants were followed for five years. The following table shows data for the participants identified as having normal blood pressure (normotensives). Trait Anger Score Low Moderate High Total CHD Event Yes 53 110 27 190 No 3057 4704 606 8367 Total 3110 4814 633 8557 All answers of the questions below are rounded to one digit after the decimal point. (a) What percentage of participants have moderate anger scores? (b) What percentage of individuals who experienced a CHD event have moderate anger scores? (c) What percentage of participants with high trait anger scores experienced a CHD event (i.e., heart attack)? (d) What percentage of participants with low trait anger scores experienced a CHD event? (e) Are individuals with high trait anger more likely to experience a CHD event than individuals with low trait anger? (f) Calculate the relative risk of a CHD event for individuals with high trait anger compared to low trait anger.

(a) 56.3 (b) 57.9 (c) 4.3 (d) 1.7 (e) Yes (f) 2.5

In an exam students earned these scores: 57, 48, 67, 62, 78, 67, 86, 95, 79, 73, 100, 65, 93, 63, 82 Calculate the following measures using R. Round your answers to 4 digits after the decimal point: (a) mean = (b) median = (c) interquartile range = (d) first quartile =

(a) 74.3333 (b) 73 (c) 20 (d) 64

What is the life span of a lab mouse? You measured the following life spans (in days) for a certain standard inbred laboratory strain. 781, 928, 584, 557, 598, 1017, 1003, 891, 920, 749, 616, 1065, 990, 736, 724, 817, 625, 791, 841, 1118, 765, 936 You may assume for the following questions that the distribution of life span is normal. (a) Calculate the sample mean 𝑥¯.x¯. 𝑥¯=x¯= (b) Calculate the sample standard deviation 𝑠.s. 𝑠=s= (c) Calculate the critical value 𝑡⋆t⋆ for a 83 percent two-sided confidence interval. 𝑡⋆=t⋆= (d) Calculate the margin of error 𝑚m for a 83 percent two-sided confidence interval. 𝑚=m= (e) The lower bound of the 83 percent two-sided confidence interval is ______________ and the upper bound of the two-sided confidence interval is _________________- (f) Calculate the critical value 𝑡⋆t⋆ for the 83 percent lower-bound confidence interval. 𝑡⋆=t⋆= (g) Calculate the lower bound of the 83 percent lower-bound confidence interval. We can be 83 percent confident that the mean life span of the lab mouse is more than (h) Calculate the upper bound of the 83 percent upper-bound confidence interval. We can be 83 percent confident that the mean life span of the lab mouse is less than 956.06 days. Please note that the confidence intervals in (g) and (h) are not simultaneously valid.

(a) 820.55 (b) 164.91 (c) 1.42 (d) 49.96 (e) 770.58, 870.51 (f) 0.98 (g) 786.22 (h) 854.87

A hospital administrator hoping to improve wait times decides to estimate the average emergency room waiting time at her hospital. She collects a simple random sample of 64 patients and determines the time (in minutes) between when they checked in to the ER until they were first seen by a doctor. A 95% confidence interval based on this sample is (128 minutes, 147 minutes), which is based on the normal model for the mean. Determine whether the following statements are true or false. (a) This confidence interval is not valid since we do not know if the population distribution of the ER wait times is nearly Normal. (b) We are 95% confident that the average waiting time of these 64 emergency room patients is between 128 and 147 minutes. (c) We are 95% confident that the average waiting time of all patients at this hospital's emergency room is between 128 and 147 minutes. (d) A 99% confidence interval would be narrower than the 95% confidence interval since we need to be more sure of our estimate. (e) The margin of error is 9.5 and the sample mean is 137.5. (f) Halving the margin of error of a 95% confidence interval requires doubling the sample size.

(a) False (b) False (c) True (d) False (e) True (f) False

In 2013, the Pew Research Foundation reported that "45% of U.S. adults report that they live with one or more chronic conditions" and the standard error for this estimate is 1.2%. Identify each of the following statements as true or false. (a) We can say with certainty that the confidence interval calculated in the previous problem contains the true percentage of U.S. adults who suffer from a chronic illness. (b) If we repeated this study 1,000 times and constructed a 95% confidence interval for each study, then approximately 950 of those confidence intervals would contain the true fraction of U.S. adults who suffer from chronic illnesses. (c) Since the standard error is 1.2%, only 1.2% of people in the study communicated uncertainty about their answer.

(a) False (b) True (c) False

A company claims that less than 0.5% of its items are defective. You suspect that this claim is not true. Denote by p the true proportion of defective items produced by the company. (a) Which test should you perform to support your suspicion

(a) H0 : p = 0.005 versus Ha : p > 0.005

Denote by pA the true proportion of patients that would be cured by drug A and by pB the true proportion of patients that would be cured by drug B. You would like to show that drug A helps more patients than drug B. (a) Which of the following is the correct null and alternative hypothesis?

(a) H0 : pA = pB versus Ha : pA > pB

Denote by µ the true weight loss (weight before the program minus the weight after the program) of someone following Dr. Quack's Diet Program for 4 weeks. Dr. Quack wants to claim that the weight loss from the 4 week program is statistically significant. Note: µ > 0 means that there is a weight loss. (a) Which of the following tests is most appropriate for supporting Dr. Quack's claim?

(a) H0 : µ = 0 versus Ha : µ > 0

Denote the true average yield (kg/ha) of Sundance winter wheat by µsun and the true average yield of Manitou spring wheat by µman. You want to show that the yield of Sundance winter wheat is significantly di↵erent form the Manitou spring wheat yield (per unit area). (a) Which of the following tests is most appropriate? (b) A 95% confidence interval of µsun µman turns out to be (1, 634). Based on this information there is

(a) H0 : µsun = µman versus Ha : µsun 6= µman (b) a statistically significant difference of yield between the two types of wheat (theta = 5%)

You are interested in the weight gain of the Auburn Freshman. You think that the average weight gain of the Auburn Freshman student is not 15 pounds. You collect a random sample of measurements. The weight gains are (a negative value indicates a weight loss): 16, 11, 10, 12, 8, 12, 13, 18, 11, 15, 8, 18, 13, 16, 12, 12, 15, 7 You may assume for the following questions that the distribution of weight gain is normal. (a) State the appropriate hypothesis test. (This was done before collecting the data) Null Hypothesis: ____________________ Alternative Hypothesis: The weight gain is not equal to 15 pounds. (b) Calculate the sample mean 𝑥¯x¯. 𝑥¯=x¯= (c) Calculate the sample standard deviation 𝑠s. 𝑠=s= (d) Evaluate the test statistic 𝑇=𝑥¯−𝜇0𝑠/𝑛√T=x¯−μ0s/n. 𝑇=T= (e) The test statistic is t-distributed with ________ degrees of freedom. (f) The p-value for the test is (g) Should you reject the null hypothesis in favor of the alternative at significance level 𝛼=0.01α=0.01? (h) Is the weight gain significantly different from 15 lbs?

(a) The weight gain is equal to 15 pounds. (b) 12.6111 (c) 3.2565 (d) -3.1123 (e) 17 (f) 0.0063 (g) Yes (h) Yes

Suppose that there is a linear relationship between the variables x and y. Determine if the following statement are TRUE or FALSE. (a) A correlation of -0.9 indicates a stronger linear relationship than a correlation of 0.8. (b) If the correlation is negative then there is a negative association between x and y.

(a) True (b) True

Let X be the number of rooms in a randomly selected house in Auburn. Describe the following events using the random variable X. (a) The house contains at least three rooms. (b) The house contains at most three rooms. (c) The house contains more than three rooms. (d) The house contains less than three rooms.

(a) X ≥ 3 (b) X ≤ 3 (c) X > 3 (d) X < 3

Suppose that measurements of your systolic blood pressure are normally distributed with some unknown mean 𝜇μ and the known standard deviation 𝜎=7.3.σ=7.3. (a) You suspect that your average systolic blood pressure is high, that is 𝜇>140μ>140 mmHg. Which is the most appropriate alternative hypothesis to support your suspicion with a hypothesis test? (b) In order to perform the test you are taking 𝑛=6n=6 independent measurements over several days. The sample mean 𝑥¯x¯ is 153. Your test statistic is the standardized sample mean 𝑥¯x¯, that is, 𝑍=𝑥¯−140𝜎/𝑛√Z=x¯−140σ/n. Find the value of this test statistic. 𝑍=Z= 3.1221 (c) The p-value for the appropriate test in (a) is the area under the standard normal distribution to the right of the observed test statistic. Find the p-value. (d) Should you reject the null hypothesis at significance level 𝛼=0.05α=0.05? (e). Draw the appropriate conclusion at significance level 𝛼=0.05α=0.05. Your blood pressure ________ significantly higher than 140.

(a) Your true mean blood pressure is higher than 140. (b) 4.3621 (c) less than 0.0001 (d) Reject the null hypothesis. (e) is

Let X be the number of rooms in a randomly selected house in Auburn. Select an appropriate description of the event in words. (a) Event: X ≥ 3 In words: The house contains ___________ rooms. (b) Event: X ≤ 3 In words: The house contains ___________ rooms. (c) Event: X ≥ 4 In words: The house contains ____________ rooms. (d) Event: X ≤ 2 In words: The house contains ____________ rooms.

(a) at least three (b) at most three (c) more than three (d) less than three

A study of fish in the Tennessee River and its three tributary creeks (Flint Creek, Limestone Creek, and Spring Creek) recorded the following variables for each fish: river/creek where each fish was captured, species (channel catfish, largemouth bass, or smallmouth buffalofish), weight (grams), and DDT concentration (ppm). Identify each of these variables as numerical or categorical variable. (a) River/Creek (b) Species (c) Weight (d) DDT

(a) categorical (b) categorical (c) numerical (d) numerical

A researcher is interested in the effects of exercise on mental health and he proposes the following study: Use stratified random sampling to recruit 18-30, 31-40 and 41-55 year olds from the population. Next, randomly assign half the subjects from each age group to exercise twice a week, and instruct the rest not to exercise. Conduct a mental health exam at the beginning and at the end of the study, and compare the results. (a) What type of study is this? (b) What are the treatment groups in this study? (b, 2) What are the control groups in this study? (c) Does this study make use of blocking? (c, 2) If so, what is the blocking variable? (d) Can the results of the study be used to establish a causal relationship between exercise and mental health? (d, 2) Can the conclusions can be generalized to the population at large?

(a) experiment (b) Groups that exercise twice a week. (b, 2) Groups that don't exercise. (c) Yes. (c, 2) Age (d) Yes, because the study was an experiment. (d, 2) Yes the conclusions can be generalized because the sample is likely to be representative of the population.

Download the file icecore1.csv download(comma separated text file) and read the data into R. Your data set consists of 1000 cases that were randomly selected from the Vostok Ice Core Data. It contains the values for the variables age, (age of trapped air in the ice measured in years before present) temperature, (temperature difference with respect to the mean recent time value measured in Celsius; a positive difference means that the observed temperature was warmer than the reference value, while a negative difference means that the observed value was cooler than the reference value) and carbonDioxide (carbon dioxide concentration measured in ppm). (a) Plot the variable temperature against the variable age. What do you observe? Remark: A time plot would be more suitable here, although it is not required. If y contains the temperature values and x contains the age values then the option type='l' will create the appropriate time plot. (b) Plot the variable carbonDioxide against the variable age. What do you observe? (c) Plot the variable temperature against the variable carbonDioxide. What do you observe? The remaining questions refer to the relationship between temperature and carbonDioxide. The variable temperature is the response variable or dependent variable and the variable carbonDioxide is the explanatory variable or independent variable. (d) Calculate the least squares regression line. The slope is (e) What percent of the variation in the variable temperature is explained by the linear regression on the variable carbonDioxide? (f) The carbon dioxide value on July 5 2020 was 415.13ppm. Use the least squares prediction equation that you calculated to predict the variable temperature for that day. Round your answer to two digits after the decimal point. (g) Typically we do not know if the observed pattern persists outside the range of the collected data. Is the carbon dioxide level of 415.13ppm415 outside the range of your data set?

(a) no direction (b) no direction (c) positive association (d) 0.0908 (e) 77 (f) 12.46 (g) Yes, it is.

Identify which of the following variables can be described by a binomial random variable. (a) Number of phone calls during an hour. (b) Number of coin tosses until heads occurs. (c) Number of correct answers on a multiple choice test with 20 questions when guessing and there are 5 choices for each question. (d) Number of people that are red-green color blind in a random selection of 100 people.

(a) not a binomial random variable (b) not a binomial random variable (c) binomial random variable (d) binomial random variable

A study is to be conducted to determine the proportion p of patients with ulcers healed by a new drug.You want to find a 95% confidence interval of p with a margin of error that is at most 0.02. 24. What sample size do you need? Use p = 0.5 and the critical value z? = 1.96.

2401

A study that surveyed a random sample of otherwise healthy high school students found that they are more likely to get muscle cramps when they are stressed. The study also noted that students drink more coffee and sleep less when they are stressed. (a) What type of study is this? (b) Can this study be used to conclude a causal relationship between increased stress and muscle cramps? (c) State possible confounding variables that might explain the observed relationship between increased stress and muscle cramps.

(a) observational study (b) The study cannot be used to conclude a causal relationship because this is an observational study. (c) Coffee consumption, Duration of sleep.

Read each brief report of statistical research and identify whether it is an observational study or an experiment. (a) Researchers wanted to know if there is a link between proximity to high-tension wires and the rate of leukemia in children. To conduct the study, researchers compared the incidence rate of leukemia for children who lived within 1/2 mile of high-tension wires to the incidence rate for children who did not live within 1/2 mile of high-tension wires. (b) Rats with cancer are divided randomly into two groups. One group receives 5 mg of medication that is thought to fight cancer, and the other receives 10 mg. After 2 months, the spread of the cancer is measured.

(a) observational study (b) experiment

Indicate which of the plots show a positive association, negative association, no association. Also determine if the positive and negative associations are linear or nonlinear. Plot 1: __________ , ________ (dots scattered, going upwards) Plot 2: _________ (dots scattered, all over the place) Plot 3: _________ , _________ (dots condensed, going upwards) Plot 4: _________ , _________ (dots scattered, going downwards)

(a) postive association, linear (b) no association (c) positive association, nonlinear (d) negative association, linear

The Buteyko method is a shallow breathing technique developed by Konstantin Buteyko, a Russian doctor, in 1952. Anecdotal evidence suggests that the Buteyko method can reduce asthma symptoms and improve quality of life. In a scientific study to determine the effectiveness of this method, researchers recruited 600 asthma patients aged 18- 69 who relied on medication for asthma treatment. These patients were randomly split into two research groups: one practiced the Buteyko method and the other did not. Afterwards, patients were scored on quality of life, activity, asthma symptoms, and medication reduction on a scale from 0 to 10. On average, the participants in the Buteyko group experienced a significant reduction in asthma symptoms and an improvement in quality of life. (a) The main research question was if the Buteyko method _________ (b) The population of interest here are the _____________ (c) The sample in this study was a convenience sample . ________ (d) This was an ____________ (e) Can one be confident that the Buteyko method caused a reduction in asthma symptoms and an improvement in quality of life for the population of interest? __________ (f) Can one be confident that the Buteyko method caused a reduction in asthma symptoms and an improvement in quality of life for the selected sample? _________ (g) The variable quality of life, which was scored on a scale from 0 to 10, is a __________

(a) reduced asthma symptoms and improved quality of life (b) asthma patients aged 18-69 who relied on medication for asthma treatment (c) convenience sample (d) experimental design (e) No. (f) Yes (g) ordinal categorical variable

Create a histogram and a boxplot for the following data: 7.65, 12.85, 9.77, 7.62, 6.68, 13.8, 8.85, 11.78, 11.88, 5.88, 12.81, 8.89, 7.8, 9.72, 8.65, 5.77, 10.73, 10.82, 10.8, 12.63, 10.83, 12.86, 9.86, 6.65, 9.69, 13.72, 13.7, 4.7, 10.73, 12.87, 11.71, 11.87, 11.67, 11.63, 7.66, 8.6, 8.82, 9.81, 10.89, 9.73, 6.61, 10.84, 7.65, 12.85, 9.77, 7.62, 6.68, 13.8, 8.85, 11.78, 11.88, 5.88, 12.81, 8.89, 7.8, 9.72, 8.65, 5.77, 10.73, 10.82, 10.8, 12.63, 10.83, 12.86, 9.86, 6.65, 9.69, 13.72, 13.7, 4.7, 10.73, 12.87, 11.71, 11.87, 11.67, 11.63, 7.66, 8.6, 8.82, 9.81, 10.89, 9.73, 6.61, 10.84, 0.9Hint: To enter the data into R type "x = c()" (without the quotation marks) into the R console and then copy and paste the data between the parentheses. Select all correct descriptions of the histogram. ______________. _______________ there is an outlier ________________

(a) unimodal (b) skewed left (c) there is an outlier

In a 2003 survey examining weights and body image concerns among young Korean women, researchers administered a questionnaire to 264 female college students in Seoul, South Korea. The survey was designed to assess excessive concern with weight and dieting, consisting of questions such as "If I gain a pound, I worry that I will keep gaining." Questionnaires were given numerical scores on the Drive for Thinness Scale. Roughly speaking, a score of 15 is typical of Western women with eating disorders, but unusually high (90th) percentile for other Western women. (a) The distribution of the dietary concern score is (b) The distribution of the dietary concern score is (c) Which measure of center will be robust?

(a) unimodal (b) skewed right (c) median

Identify all the correct descriptions of the data based on the histogram and boxplot below. (a) The shape of the distribution is (b) The shape of the distribution is (c) There are

(a) unimodal (b)skewed right (c) outliers

Let pN be the proportion of patients cured with a new drug and pR be the proportion of patients cured by a reference drug. A clinical trial was performed and now a confidence interval for pN pR needs to be constructed. Which of the following statements is true? (a) The 95% confidence interval will be narrower than the 90% confidence interval. (b) The 95% confidence interval will be narrower than the 90% confidence interval. (C) Both intervals will be of the same length, because they are computed from the same sample. (D) The length of the confidence interval will depend on the sample size, not on the confidence level. (E) The length of the confidence interval will depend on the sample standard deviation, not on the confidence level.

(b) The 95% confidence interval will be narrower than the 90% confidence interval.

There are 150 students in a certain class. Suppose that 69 of these students are female, 44 are sophomores, and 21 are both, that is sophomore and female. A student from this class is randomly selected. Denote the event that the selected student is a sophomore with SO, and the event that the student is female with F. Based on these numbers calculate the probability P(F). Round your answer to 3 digits after the decimal point.

0.46

Suppose that X is a continuous random variable whose probability density function is given by the graph below. Find the probability P(X < 0)

0.5

There are 146 students in a certain class. Suppose that 74 of these students are female, 43 are sophomores, and 24 are both, that is sophomore and female. A student from this class is randomly selected. Denote the event that the selected student is a sophomore with SO, and the event that the student is female with F. Based on these numbers calculate the probability P(F or SO). Round your answer to 3 digits after the decimal point.

0.637

The blood group of 200 people is distributed as follows: 50 have type A blood, 65 have type B blood, 70 have type O blood and 15 have type AB blood. A person from this group is randomly selected. What is the probability that this person does not have blood type B?

0.675

Suppose X and Y are random variables for which 𝐸[𝑋]=E[X]= 8 and 𝐸[𝑌]=E[Y]=6. Find E[6X - 8 Y]. Correct!

0

Suppose a certain drug test is 96% sensitive, that is, the test will correctly identify a drug user as testing positive 96% of the time, and 80% specific, that is, the test will correctly identify a non-user as testing negative 80% of the time. Suppose a corporation decides to test its employees for drug use, and that only 0.4% of the employees actually use the drug. What is the probability that, given a positive drug test, an employee is actually a drug user? Round your answer to 3 digits after the decimal point.

0.019

According to the American Social Health Organization, one out of four teens in the United States becomes infected with an STD each year. What is the probability that in a random selection of 7 teens no one becomes infected with an STD in a given year? Round your answer to 3 digits after the decimal point.

0.133

There are 137 students in a certain class. Suppose that 63 of these students are female, 47 are sophomores, and 29 are both, that is sophomore and female. A student from this class is randomly selected. Denote the event that the selected student is a sophomore with SO, and the event that the student is female with F. Based on these numbers calculate the probability P(F and SO). Round your answer to 3 digits after the decimal point.

0.212

The blood group of 200 people is distributed as follows: 50 have type A blood, 65 have type B blood, 70 have type O blood and 15 have type AB blood. A person from this group is randomly selected. What is the probability that this person has blood type O?

0.35

Suppose X and Y are random variables for which Var(X) = 3 and Var(Y) = 9. Find Var(4X - 3Y)

129

A study is to be conducted to determine the proportion p of patients with ulcers healed by a new drug. You want to find a 95% confidence interval of p with a margin of error that is at most 0.02. What sample size do you need? Use 𝑝=0.5p=0.5 and the critical value 𝑧∗=1.96z∗=1.96.

2401

Suppose you have a box with a very large number of orange and blue beads. You want to estimate the proportion 𝑝p of orange beads in the box and you want to be 96% confident that your point estimate, which is the sample proportion 𝑝̂ p^, is within 5% of the true proportion 𝑝p. Find the appropriate sample size.

422

A typical russet potato weighs about 173 g. Farmer Joe would like to estimate the average weight of his potatoes. The average weight of 43 potatoes from Joe is 181 grams and the sample standard deviation is 20 grams. Use the critical value 𝑡⋆=2.0141t⋆=2.0141 to find the margin of error for a two-sided 95% confidence interval for 𝜇μ.

6.143

A statistics exam consists of 30 multiple choice questions with 4 choices to each question. Suppose that a student just guesses the answer to each question. (a) What is the expected number of correct answers.

7.5

Suppose that the distribution of scores in a statistics exam is approximately normal with a mean of 64 and a standard deviation of 12. Use the Empirical Rule to find approximately the percentage of students with a score less than 88.

97.5%

Decide whether or not you should reject the null hypothesis 𝐻0:𝜇=2H0:μ=2 in favor of the alternative hypothesis 𝐻𝐴:𝜇>2HA:μ>2 at significance level 𝛼=0.05α=0.05 when 𝑇=0.0001T=0.0001 and 𝑛=42

fail to reject the null hypothesis

Decide whether or not to reject the null hypothesis in favor of the alternative hypothesis at significance level 𝛼=0.05α=0.05. The purpose of this exercise is to visualize p-values and to improve your intuition for the t-distribution. Sketch the area that represents the p-value in each case. The sample size is denoted by 𝑛n and the test statistic from the one-sample t-test is denoted by 𝑇T. You should do this exercise without using R. Hint: Recall that for sufficiently large sample size the t-distribution is approximated by the standard normal distribution. Use the Empirical Rule. 𝑛=67n=67, 𝐻𝐴:𝜇≠3HA:μ≠3, 𝑇=2.3T=2.3

Reject the null hypothesis

Decide whether or not to reject the null hypothesis in favor of the alternative hypothesis at significance level 𝛼=0.05α=0.05. The purpose of this exercise is to visualize p-values and to improve your intuition for the t-distribution. Sketch the area that represents the p-value in each case. The sample size is denoted by 𝑛n and the test statistic from the one-sample t-test is denoted by 𝑇T. You should do this exercise without using R. Hint: Recall that for sufficiently large sample size the t-distribution is approximated by the standard normal distribution. Use the Empirical Rule. 𝑛=45n=45, 𝐻𝐴:𝜇>3HA:μ>3, 𝑇=4.2T=4.2

Reject the null hypothesis

Decide whether or not to reject the null hypothesis in favor of the alternative hypothesis at significance level 𝛼=0.05α=0.05 based on the following information: 𝑛=32n=32, 𝐻𝐴:𝜇<200HA:μ<200, 𝑇=2.1T=2.1. I recommend that you sketch the area that represents the p-value.

fail to reject the null hypothesis

Decide whether or not to reject the null hypothesis in favor of the alternative hypothesis at significance level 𝛼=0.05α=0.05 based on the following information: 𝑛=32n=32, 𝐻𝐴:𝜇≠200HA:μ≠200, 𝑇=0.001T=0.001. I recommend that you sketch the area that represents the p-value. Correct!

fail to reject the null hypothesis

Suppose that (23.2, 34.5) is a 95% confidence interval for a parameter 𝜇μ. Decide whether the following statement is true or false: The parameter 𝜇μ is significantly different from 20 at significance level 0.05.

True

The General Social Survey asked a random sample of 1,390 Americans thefollowing question: "On the whole, do you think it should or should not be the government's responsibility to promote equality between men and women?" 82% of the respondents said it "should be". At a 95% confidence level, this sample has 2% margin of error. Based on this information, determine if the following statement is true or false. We are 95% confident that between 80% and 84% of all Americans think it's the government's responsibility to promote equality between men and women.

True

Healthy Tigers, a Wellness Program at Auburn University, measures the cholesterol levels of the employees of Auburn University. Suppose that a researcher takes the results and randomly selects 100 subjects from those employees who exercise regularly and another 100 subjects from those employees who do not exercise regularly. The researcher then compares the cholesterol levels between the two groups and finds a significant difference. Check all the answers that apply

We can be confident that there is a significant difference in cholesterol levels between those AU employees who exercise regularly and those who don't.

Suppose that you toss a fair coin 25 times. 25. What is the approximate distribution of the sample proportion pˆ of the number of heads that you toss?

pˆ-N(µ = 0.5, theta = 0.1)

Decide whether or not to reject the null hypothesis in favor of the alternative hypothesis at significance level 𝛼=0.05α=0.05 based on the following information: 𝑛=32n=32, 𝐻𝐴:𝜇>200HA:μ>200, 𝑇=2.1T=2.1. I recommend that you sketch the area that represents the p-value.

reject the null hypothesis

Decide whether or not to reject the null hypothesis in favor of the alternative hypothesis at significance level 𝛼=0.05α=0.05 based on the following information: 𝑛=32n=32, 𝐻𝐴:𝜇≠200HA:μ≠200, 𝑇=−3.5T=−3.5. I recommend that you sketch the area that represents the p-value.

reject the null hypothesis

Decide whether or not you should reject the null hypothesis 𝐻0:𝜇=2H0:μ=2 in favor of the alternative hypothesis 𝐻𝐴:𝜇>2HA:μ>2 at significance level 𝛼=0.05α=0.05 when the p-value of the test is 0.0001.

reject the null hypothesis

Complete the following sentence. A type I error occurs when

the null hypothesis is rejected and the null hypothesis is true

Decreasing the confidence level decreases the margin of error.

true

Blood tests are used to screen possible donors for HIV. You perform the following test. Null Hypothesis: Donor does not have HIV. Alternative Hypothesis: Donor has HIV. The test indicates that the donor has HIV even though the donor does not have HIV. What type of error is this?

type 1

The police inspects the tires of your car. H0 : The tires of your car do not violate the safety regulations in Alabama. HA : The tires of your car violate the safety regulations in Alabama. The police does not find anything wrong with your tires even though the tread depth of your left front tire is below the legal limit of 1/16 inch. The police committed:

type II error

You want to visit a friend who lives 10 miles away. H0 : There is enough gasoline in your car to drive another 10 miles. HA : There is not enough gasoline in your car to drive another 10 miles. Based on the fuel gauge you decide that there is enough gas in the car to drive to your friend, however, after 8 miles of driving your car runs out of gas. This is an example of:

type II error