stats ch 7-12 vocab
level of education for residents
( )#1 the other one is #2
Find the z-scores that separate the middle 72% of the distribution from the area in the tails of the standard normal distribution.
(1-.72)/2=0.14 0.14=-1.08 z scores are (-1.08, 1.08)
If the consequences of making a Type I error are severe, would you choose the level of significance, α, to equal 0.01, 0.05, or 0.10?
0.01
Determine the point estimate of the population proportion, the margin of error for the following confidence interval, and the number of individuals in the sample with the specified characteristic, x, for the sample size provided. Lower bound=0.541, upper bound=0.879, n=1000
0.879+.541=1.42 1.42/2=0.71(a) 0.71-.541=0.169(b) .71*1000
Compute the critical value zα/2 that corresponds to a 81% level of confidence.
1-.81=.19 .19/2=0.095 1-.095=.905 st norm= 1.31
Find the value of zα. z0.28
1-0.28=0.72 0.72=0.58
The area under the normal curve to the right of μ equals _______.
1/2
A researcher wants to show the mean from population 1 is less than the mean from population 2 in matched-pairs data. If the observations from sample 1 are Xi and the observations from sample 2 are Yi, and di=Xi−Yi, then the null hypothesis is H0: μd=0 and the alternative hypothesis is H1: μd ___ 0.
<
Two researchers, Jaime and Mariya, are each constructing confidence intervals for the proportion of a population who is left-handed. They find the point estimate is 0.26. Each independently constructed a confidence interval based on the point estimate, but Jaime's interval has a lower bound of 0.247 and an upper bound of 0.273, while Mariya's interval has a lower bound of 0.194 and an upper bound of 0.288. Which interval is wrong? Why?
A. Mariya's interval is wrong because it is not centered on the point estimate
Some have argued that throwing darts at the stock pages to decide which companies to invest in could be a successful stock-picking strategy. Suppose a researcher decides to test this theory and randomly chooses 200 companies to invest in. After 1 year, 104 of the companies were considered winners; that is, they outperformed other companies in the same investment class. To assess whether the dart-picking strategy resulted in a majority of winners, the researcher tested H0: p=0.5 versus H1: p>0.5 and obtained a P-value of 0.2858.
About 29 in 100 samples will give a sample proportion as high or higher than the one obtained if the population proportion really is 0.5. Because the P-value is large, do not reject the null hypothesis. There is not sufficient evidence to conclude that the dart-picking strategy resulted in a majority of winners.
The headline reporting the results of a poll stated, "Majority of Adults at Personal Best in the Morning." The results indicated that a survey of 1000 adults resulted in 56% stating they were at their personal best in the morning. The poll's results were reported with a margin of error of 4%. Explain why the poll's headline is accurate.
All the values within the margin of error are greater than 50%.
Explain what a P-value is. What is the criterion for rejecting the null hypothesis using the P-value approach?
A P-value is the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the statement in the null hypothesis is true. f P-value<α, reject the null hypothesis.
Twenty years ago, 46% of parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 185 of 750 parents of children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did twenty years ago? Use the α=0.05 level of significance.
Because np01−p0=186.3>10,the sample size is LESS than 5% of the population size, and the sample can be reasonably assumed to be random, the requirements for testing the hypothesis are satisfied.
In a survey conducted by the Gallup Organization, 1100 adult Americans were asked how many hours they worked in the previous week. Based on the results, a 95% confidence interval for the mean number of hours worked had a lower bound of 42.7 and an upper bound of 44.5. Provide two recommendations for decreasing the margin of error of the interval.
Decrease the confidence level. Increase the sample size.
(d) We are 95% confident that the mean number of hours worked by adults in a particular area of this country in the previous week was between 41.9 hours and 45.4 hours.
Flawed. The interpretation should be about the mean number of hours worked by adults in the whole country, not about adults in the particular area.
In a survey, 1100 adults in a certain country were asked how many hours they worked in the previous week. Based on the results, a 95% confidence interval for mean number of hours worked was lower bound: 41.9 and upper bound: 45.4. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw. Complete parts (a) through (d) below. (a) There is a 95% chance the mean number of hours worked by adults in this country in the previous week was between 41.9 hours and 45.4 hours.
Flawed. This interpretation implies that the population mean varies rather than the interval.
The following data represent the level of health and the level of education for a random sample of 1608 residents.
H0: Level of education and health are independent. H1: Level of education and health are dependent. test stat: contingency table w summary conditional dist: ( )#1
A manufacturer of colored candies states that 13% of the candies in a bag should be brown, 14% yellow, 13% red, 24% blue, 20% orange, and 16% green. A student randomly selected a bag of colored candies. He counted the number of candies of each color and obtained the results shown in the table.
H0: The distribution of colors is the same as stated by the manufacturer. H1: The distribution of colors is not the same as stated by the manufacturer. expected counts: stats>goodness of fit> chi square
Explain what "90% confidence" means in a 90% confidence interval
If 100 different confidence intervals are constructed, each based on a different sample of size n from the same population, then we expect 90 of the intervals to include the parameter and 1 to not include the parameter.
Suppose a researcher is testing the hypothesis H0: p=0.4 versus H1: p<0.4 and she finds the P-value to be 0.19. Explain what this means. Would she reject the null hypothesis? Why?
If the P-value for a particular test statistic is 0.19, she expects results at least as extreme as the test statistic in about 19 of 100 samples if the null hypothesis is true. Since this event is not unusual, she will not reject the null hypothesis.
Katrina wants to estimate the proportion of adults who read at least 10 books last year. To do so, she obtains a simple random sample of 100 adults and constructs a 95% confidence interval. Matthew also wants to estimate the proportion of adults who read at least 10 books last year. He obtains a simple random sample of 400 adults and constructs a 99% confidence interval. Assuming both Katrina and Matthew obtained the same point estimate, whose estimate will have the smaller margin of error? Justify your answer.
Matthew's estimate will have the smaller margin of error because the larger sample size more than compensates for the higher level of confidence.
A simple random sample of size n=71 is obtained from a population with μ=65 and σ=6. Does the population need to be normally distributed for the sampling distribution of x to be approximately normally distributed? Why? What is the sampling distribution of x?
No because the Central Limit Theorem states that regardless of the shape of the underlying population, the sampling distribution of x becomes approximately normal as the sample size, n, increases.
Explain what "statistical significance" means.
Statistical significance means that the result observed in a sample is unusual when the null hypothesis is assumed to be true.
Explain the difference between statistical significance and practical significance.
Statistical significance means that the sample statistic is not likely to come from the population whose parameter is stated in the null hypothesis. Practical significance refers to whether the difference between the sample statistic and the parameter stated in the null hypothesis is large enough to be considered important in an application.
(b) We are 82% to 98% confident 64% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.
The interpretation is flawed. The interpretation indicates that the level of confidence is varying.
In a survey of 2045 adults in a certain country conducted during a period of economic uncertainty, 64% thought that wages paid to workers in industry were too low. The margin of error was 8 percentage points with 90% confidence. For parts (a) through (d) below, which represent a reasonable interpretation of the survey results? For those that are not reasonable, explain the flaw. (a) We are 90% confident 64% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.
The interpretation is flawed. The interpretation provides no interval about the population proportion.
(d) In 90% of samples of adults in the country during the period of economic uncertainty, the proportion who believed wages paid to workers in industry were too low is between 0.56 and 0.72.
The interpretation is flawed. The interpretation suggests that this interval sets the standard for all the other intervals, which is not true.
A group conducted a poll of 2014 likely voters just prior to an election. The results of the survey indicated that candidate A would receive 47% of the popular vote and candidate B would receive 46% of the popular vote. The margin of error was reported to be 3%. The group reported that the race was too close to call. Use the concept of a confidence interval to explain what this means.
The margin of error suggests candidate A may receive between 44% and 50% of the popular vote and candidate B may receive between 43% and 49% of the popular vote. Because the poll estimates overlap when accounting for margin of error, the poll cannot predict the winner.
The points at x=_______ and x=_______ are the inflection points on the normal curve.
The points are x=μ−σ and x=μ+σ.
What happens to the probability of making a Type II error, β, as the level of significance, α, decreases? Why?
The probability increases. Type I and Type II errors are inversely related.
According to a study, the proportion of people who are satisfied with the way things are going in their lives is 0.80. Suppose that a random sample of 100 people is obtained. Complete parts (a) through (e) below. a) Suppose the random sample of 100 people is asked, "Are you satisfied with the way things are going in your life?" Is the response to this question qualitative or quantitative? Explain.
The response is qualitative because the responses can be classified based on the characteristic of being satisfied or not.
(b) Explain why the sample proportion, p, is a random variable. What is the source of the variability?
The sample proportion p hat is a random variable because the value of p hat varies from sample to sample. The variability is due to the fact that different people feel differently regarding their satisfaction.
In randomized, double-blind clinical trials of a new vaccine, infants were randomly divided into two groups. Subjects in group 1 received the new vaccine while subjects in group 2 received a control vaccine. After the second dose, 118 of 655 subjects in the experimental group (group 1) experienced fever as a side effect. After the second dose, 75 of 536 of the subjects in the control group (group 2) experienced fever as a side effect. Does the evidence suggest that a higher proportion of subjects in group 1 experienced fever as a side effect than subjects in group 2 at the α=0.10 level of significance?
The sample size is less than 5% of the population size for each sample. The samples are independent. n1p11−p1≥10 and n2p21−p2≥10
To test the belief that sons are taller than their fathers, a student randomly selects 13 fathers who have adult male children. She records the height of both the father and son in inches and obtains the following data. Are sons taller than their fathers?
The sampling method results in a dependent sample. The sample size is no more than 5% of the population size. The differences are normally distributed or the sample size is large. do not reject; is not; is not; are taller than
The normal curve is symmetric about its mean, μ.
The statement is true. The normal curve is a symmetric distribution with one peak, which means the mean, median, and mode are all equal. Therefore, the normal curve is symmetric about the mean, μ.
The mean of the pressure required to open a certain valve is known to be μ=8.4 psi. Due to changes in the manufacturing process, the quality-control manager feels that the average pressure has increased. The null hypothesis was not rejected.
There is not sufficient evidence that the mean of the pressure required to open a certain valve has increased.
Six years ago, 11.1% of registered births were to teenage mothers. A sociologist believes that the percentage has increased since then.
There is not sufficient evidence to conclude that the percentage of teenage mothers has increased.
The expected frequencies in a chi-square test for independence are found using the formula below. Expected frequency=(row total)(column total)table total
True. It is a simplification of multiplying the proportion of a row variable by the proportion of the column variable to find the proportion for a cell, then multiplying by the table total
The following data represent the muzzle velocity (in feet per second) of rounds fired from a 155-mm gun. For each round, two measurements of the velocity were recorded using two different measuring devices, resulting in the following data. Why are these matched-pairs data?
Two measurements (A and B) are taken on the same round. u=0 0=/0
Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. P(X>38)
Z = (X - μ) / σ Z = (38 - 50) / 7 Z= -1.71429 -1.71429=0.0436 1-.0436=0.9564
For students who first enrolled in two year public institutions in a recent semester, the proportion who earned a bachelor's degree within six years was 0.393. The president of a certain college believes that the proportion of students who enroll in her institution have a lower completion rate. (a) Determine the null and alternative hypotheses. (b) Explain what it would mean to make a Type I error. (c) Explain what it would mean to make a Type II error.
a) State the hypotheses. H0:p=0.393 H1:p<0.393 b) The president rejects the hypothesis that the proportion of students who earn a bachelor's degree within six years is 0.393, when, in fact, the proportion is 0.393. c) The president fails to reject the hypothesis that the proportion of students who earn a bachelor's degree within six years is 0.393, when, in fact, the proportion is less than 0.393.
The time required for an automotive center to complete an oil change service on an automobile approximately follows a normal distribution, with a mean of 19 minutes and a standard deviation of 3 minutes. a. The automotive center guarantees customers that the service will take no longer than 20 minutes. If it does take longer, the customer will receive the service for half-price. What percent of customers receive the service for half-price? (b) If the automotive center does not want to give the discount to more than 3% of its customers, how long should it make the guaranteed time limit?
a. (x-mean)/st.dev find z score on standard normal table subtract from 1 b. 100-percentage find .0000 number on st normal table make 0.00 number =(x-19)/3
The number of chocolate chips in an 18-ounce bag of chocolate chip cookies is approximately normally distributed with mean 1252 and standard deviation 129 chips. a. What is the probability that a randomly selected bag contains between 1000 and 1400 chocolate chips?
a. find z scores of both values (1000 and 1400), and then find their values on the normal table. then subtract larger initial value (?) from smaller value.
The graph to the right is the uniform probability density function for a friend who is x minutes late. Find the probability that the friend is between 5 and 20 minutes late. It is 10 A.M. There is a 50% probability the friend will arrive within how many minutes?
a. length*width 15*(1/30)=0.5 b. percentage/width from above 0.5/(1/30)=15
State the requirements to perform a goodness-of-fit test.
all=> 1 80%=>5
The _______ _______ is a statement we are trying to find evidence to support.; no change, no effect, no difference
alternative hypothesus; null hypothesis
A professor wanted to determine whether an online homework system improved scores on a final exam. In the fall semester, he taught a class using the online homework system (which meant students did their homework online and received instant feedback about their answers along with helpful guidance). In the spring semester, he taught a class without the homework system (which meant students were responsible for doing their homework the old-fashioned way - paper and pencil). The professor made sure to teach the two courses identically (same text, syllabus, tests, meeting time, meeting location, and so on). The table summarizes the results of the two classes on their final exam.
completely randomized design response variable: final exam scores treatments: hw system controlled: text, test, location, teacher, syllabus students "randomly" enrolled
expected count for each outcome
compute>n*pi
(b) We are 95% confident that the mean number of hours worked by adults in this country in the previous week was between 41.9 hours and 45.4 hours
correct
A sampling method is when the individuals selected for one sample are used to determine the individuals in the second sample.
dependent
The normal probability plot ▼ does not suggest suggests the data could come from a normal population because 0.939▼ greater than>less than<nothing and the boxplot ▼ shows does not show outliers, so a t-interval ▼ could not could be constructed
does not suggest; <; 0.959; shows; could not
Suppose there are n independent trials of an experiment with k>3 mutually exclusive outcomes, where pi represents the probability of observing the ith outcome. What would be the formula of an expected count in this situation?
e=np
Sample evidence can prove that a null hypothesis is true.
false
Suppose that X is a binomial random variable. To approximate P(3≤X<7) using the normal probability distribution, we compute P(3.5≤X<7.5).
false
To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.
false
When testing a hypothesis using the P-value Approach, if the P-value is large, reject the null hypothesis.
false
the population proportion and sample proportion always have the same value.
false
If we reject the null hypothesis when the statement in the null hypothesis is true, we have made a Type _______ error.
i
A sampling method is ______ when an individual selected for one sample does not dictate which individual is to be in the second sample.
independent
The _______ represents the expected proportion of intervals that will contain the parameter if a large number of different samples of size n is obtained. It is denoted _______.
level of confidence; (1-alpha)*100%
Suppose a simple random sample of size n is drawn from a large population with mean μ and standard deviation σ. The sampling distribution of x has mean μx=___ and standard deviation σx=___
mean, st dev/sqrt n
Since the sample size is ▼ no less no more than 5% of the population size and np(1−p)=__≥10, the distribution of p is ▼ skewed right approximately normal uniform skewed left with μp= and σp=.
no more; 16; approximately normal; 0.8; 0.04
Determine whether the following graph can represent a normal density function. [pic is an upside down parabola where the ends of it are below the x axis]
no. pnly normal density function when all points are above/below axis. also has to be symmetrical.
A _______ is a graph that plots observed data versus normal scores.
normal probability plot
Suppose the mean IQ score of people in a certain country is 101. Suppose the director of a college obtains a simple random sample of 38 students from that country and finds the mean IQ is 105.6 with a standard deviation of 13.9. Complete parts (a) through (d) below.
not reject; accept; accepts; population mean; population mean; does not reject; population mean; population mean; accepting; not rejecting
In a binomial experiment with n trials and probability of success p, if ____ __ __ the binomial random variable X is approximately normal with μX= ___ and σX= ___
np(1-p) >= 10 np sqrt(np(1-p))
A ________ ________ is the value of a statistic that estimates the value of a parameter.
point estimate
Determine the point estimate of the population mean and margin of error for the confidence interval. Lower bound is 17, upper bound is 25.
pop mean: 21 margin of error: 4
(c) We are 90% confident that the interval from 0.56 to 0.72 contains the true proportion of adults in the country during the period of economic uncertainty who believed wages paid to workers in industry were too low.
reasonable
The procedure for constructing a confidence interval about a mean is _______, which means minor departures from normality do not affect the accuracy of the interval.
robust
The _____ _____, denoted phat, is given by the formula p=_____, where x is the number of individuals with a specified characteristic in a sample of n individuals.
sample proportion; x/n
the standard deviation of the sampling distribution of x bar, denoted σx is called teh __ ___ of the ___.
standard error of the mean
(c) 95% of adults in this country worked between 41.9 hours and 45.4 hours last week.
tFlawed. This interpretation makes an implication about individuals rather than the mean
A normal score is the expected z-score of a data value, assuming the distribution of the random variable is normal. Is this statement true or false?
true
The distribution of the sample mean, x, will be normally distributed if the sample is obtained from a population that is normally distributed, regardless of the sample size.
true
The mean of the sampling distribution of p hat is p.
true
A researcher wanted to determine if carpeted or uncarpeted rooms contain more bacteria. The table shows the results for the number of bacteria per cubic foot for both types of rooms. A normal probability plot and boxplot indicate that the data are approximately normally distributed with no outliers. Do carpeted rooms have more bacteria than uncarpeted rooms at the α=0.05 level of significance?
u(carpet)=u(no carpet) u(carpet)>u(no carpet)
Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. Find the 81st percentile.
x = z*σ + µ x= 0.88*7+50 x=56.16
Assume the random variable X is normally distributed, with mean μ=46 and standard deviation σ=7. Find the 7th percentile.
x = z*σ + µ x=-1.48*7+46 x= 35.64
The mean waiting time at the drive-through of a fast-food restaurant from the time an order is placed to the time the order is received is 88.5 seconds. A manager devises a new drive-through system that he believes will decrease wait time. As a test, he initiates the new system at his restaurant and measures the wait time for 10 randomly selected orders.
yes; are; is; greater
The notation zα is the z-score that the area under the standard normal curve to the right of zα is _______.
α
The null and alternative hypotheses are given. Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. What parameter is being tested? H0:σ=6 H1:σ>6
right-tailed test; population st dev
