math 119
Assume that when adults with smartphones are randomly selected, 58% use them in meetings or classes. If 5 adult smartphone users are randomly selected, find the probability that exactly 2 of them use their smartphones in meetings or classes.
. The probability is 0.2492
Express the confidence interval (0.014,0.092) in the form of p−E<p<p+E.
0.053 - 0.039 < p <0.053 + 0.039
A statistics professor plans classes so carefully that the lengths of her classes are uniformly distributed between 45.0 and 55.0 minutes. Find the probability that a given class period runs between 50.75 and 51.5 minutes.
0.075
The waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 9 minutes. Find the probability that a randomly selected passenger has a waiting time greater than 4.25 minutes.
0.523
A simple random sample from a population with a normal distribution of 98body temperatures has x=98.20°F and s=0.67°F. Construct a 99% confidence interval estimate of the standard deviation of body temperature of all healthy humans.
0.56°F<σ<0.80°F
If you select a simple random sample of M&M plain candies and construct a normal quantile plot of their weights, what pattern would you expect in the graphs?
Approximately a straight line.
Which statement below indicates the area to the left of 19.5 before a continuity correction is used?
At most 19
Which of the following is not true?
A z-score is an area under the normal curve.
Indicate a similarity between a Poisson distribution and a binomial distribution.
Both distributions have an independence requirement.
Multiple-choice questions each have five possible answers (a, b, c, d, e), one of which is correct. Assume that you guess the answers to three such questions. a. Use the multiplication rule to find P(WCC),where C denotes a correct answer and W denotes a wrong answer. b. Beginning with WCC, make a complete list of the different possible arrangements of two correct answers and one wrong answer,then find the probability for each entry in the list. P(WCC)−see above P(CCW)= P(CWC)= c. Based on the preceding results, what is the probability of getting exactly two correct answers when three guesses are made?
C= 1/4= 0.25 W= 3/4= 0.75 a. P(WCC)= (0.75)(0.25)(0.25)= 0.0321 b. P(WCC)−see above P(CCW)= same as above P(CWC)=same as above c. P(CCW)+P(CWC)+P(WCC)= 0.032(3)= 0.096
Which of the following is a biased estimator? That is, which of the following does not target the population parameter?
Median
For 100 births, P(exactly 55 girls)=0.0485 and P(55 or more girls)=0.184. Is 55 girls in 100 births a significantly high number of girls? Which probability is relevant to answering that question? Consider a number of girls to be significantly high if the appropriate probability is 0.05 or less. The relevant probability is _____, so 55 girls in 100 births ____ a significantly high number of girls because the relevant probability is _____ 0.05.
P(55 or more girls) is not greater than
If np≥5 and nq≥5, estimate P(fewer than 4) with n=14 and p=0.5 by using the normal distribution as an approximation to the binomial distribution; if np<5 or nq<5, then state that the normal approximation is not suitable.
P(fewer than 4)=0.0307
If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. This represents the _______.
Rare Event Rule
The _______ states that if, under a given assumption, the probability of a particular observed event is exceptionally small (such as less than 0.05), we conclude that the assumption is probably not correct.
Rare Event Rule for Inferential Statistics
Which of the following is NOT needed to determine the minimum sample size required to estimate a population proportion?
Standard Deviation
A newspaper provided a "snapshot" illustrating poll results from 1910 professionals who interview job applicants. The illustration showed that 26% of them said the biggest interview turnoff is that the applicant did not make an effort to learn about the job or the company. The margin of error was given as ±3 percentage points. What important feature of the poll was omitted?
The confidence level
What requirements are necessary for a normal probability distribution to be a standard normal probability distribution?
The mean and standard deviation have the values of μ=0 and σ=1
Which of the following is NOT a property of the chi-square distribution?
The mean of the chi-square distribution is 0.
If np≥5 and nq≥5, estimate P(more than 7) with n=11 and p=0.7 by using the normal distribution as an approximation to the binomial distribution; if np<5 or nq<5, then state that the normal approximation is not suitable.
The normal distribution cannot be used.
Why must a continuity correction be used when using the normal approximation for the binomial distribution?
The normal distribution is a continuous probability distribution being used as an approximation to the binomial distribution which is a discrete probability distribution.
What is different about the normality requirement for a confidence interval estimate of σ and the normality requirement for a confidence interval estimate of μ?
The normality requirement for a confidence interval estimate of σ is stricter than the normality requirement for a confidence interval estimate of μ. Departures from normality have a greater effect on confidence interval estimates of σthan on confidence interval estimates of μ. That is, a confidence interval estimate of σis less robust against a departure from normality than a confidence interval estimate of μ.
Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 175 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted. Probability that exactly 44 voted
The probability that exactly 44 of 175 eligible voters voted is 0.0435
Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 158 eligible voters aged 18-24 are randomly selected. Suppose a previous study showed that among eligible voters aged 18-24, 22% of them voted. Probability that fewer than 41 voted
The probability that fewer than 41 of 158 eligible voters voted is 0.8643
The table to the right lists probabilities for the corresponding numbers of girls in three births. What is the random variable, what are its possiblevalues, and are its values numerical? x p(x) 0 0.125 1 0.375 2 0.375 3 0.125
The random variable is x, which is the number of girls in three births. The possible values of x are 0, 1, 2, and 3. The values of the random value x are numerical.
A researcher collects a simple random sample of grade-point averages of statistics students, and she calculates the mean of this sample. Under what conditions can that sample mean be treated as a value from a population having a normal distribution?
The sample has more than 30 grade-point averages. If the population of grade-point averages has a normal distribution.
Which of the following is NOT a requirement for using the normal distribution as an approximation to the binomial distribution?
The sample is the result of conducting several dependent trials of an experiment in which the probability of success is p.
___________ is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population.
The sampling distribution of a statistic is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population.
Which of the following is NOT required to determine minimum sample size to estimate a population mean?
The size of the population, N
Which of the following is NOT a requirement for constructing a confidence interval for estimating the population proportion?
The trials are done without replacement.
Which of the following is not a requirement of the binomial probability distribution?
The trials must be dependent.
Which of the following is NOT one of the three methods for finding binomial probabilities that is found in the chapter on discrete probability distributions?
Use a simulation
Which concept below is NOT a main idea of estimating a population proportion?
Using a sample statistic to estimate the population proportion is utilizing descriptive statistics.
Which of the following would be a correct interpretation of a 99% confidence interval such as 4.1<μ<5.6?
We are 99% confident that the interval from 4.1 to 5.6 actually does contain the true value of μ.
Refer to the accompanying data display that results from a sample of airport data speeds in Mbps. The results in the screen display are based on a 95% confidence level. Write a statement that correctly interprets the confidence interval. T Interval (13.046,22.15) x= 17.598 Sx= 16.01712719 n= 50
We have 95% confidence that the limits of 13.05 Mbps and 22.15 Mbps contain the true value of the mean of the population of all data speeds at the airports.
Assume that we want to construct a confidence interval. Do one of the following, as appropriate: (a) find the critical value tα/2, (b) find the critical value zα/2, or (c) state that neither the normal distribution nor the t distribution applies. The confidence level is 90%, σ=3603 thousand dollars, and the histogram of 56 player salaries (in thousands of dollars) of football players on a team is as shown.
Za/2= 1.64 https://www.calculators.org/math/z-critical-value.php
If you are asked to find the 85th percentile, you are being asked to find _____.
a data value associated with an area of 0.85 to its left
Assume that all grade-point averages are to be standardized on a scale between 0 and 4. How many grade-point averages must be obtained so that the sample mean is within 0.013 of the population mean? a. Assume that a 98% confidence level is desired. If using the range rule of thumb, σ can be estimated as range4=4−04=1. b. Does the sample size seem practical?
a. 32,024 ± 10 32,124 ± 10 32,014 ± 10 32027 ± 10 b. No, because the required sample size is a fairly large number.
Use the sample data and confidence level given below to complete parts (a) through (d). A drug is used to help prevent blood clots in certain patients. In clinical trials, among 4722 patients treated with the drug, 158 developed the adverse reaction of nausea. Construct a 90% confidence interval for the proportion of adverse reactions. a. Find the best point estimate of the population proportion p. b. Identify the value of the margin of error E. c. Construct the confidence interval. d. Write a statement that correctly interprets the confidence interval. Choose the correct answer below.
a. 0.033 b 0.004 c. 0.029 < p < 0.038 d. One has 90% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.
Use the sample data and confidence level given below to complete parts (a) through (d). A research institute poll asked respondents if they felt vulnerable to identity theft. In the poll, n=1042 and x=500 who said "yes." Use a 90% confidence level. a. Find the best point estimate of the population proportion p. b. Identify the value of the margin of error E. c. Construct the confidence interval. d. Write a statement that correctly interprets the confidence interval. Choose the correct answer below.
a. 0.480 b. E= 0.025 c. 0.455 < p < 0.505 d. One has 90% confidence that the interval from the lower bound to the upper bound actually does contain the true value of the population proportion.
A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 415 babies were born, and 332 of them were girls. a. Use the sample data to construct a 99% confidence interval estimate of the percentage of girls born. b. Based on the result, does the method appear to be effective?
a. 0.749 < p < 0.851 b. Yes, the proportion of girls is significantly different from 0.5.
In a survey of 3312 adults aged 57 through 85 years, it was found that 84.7% of them used at least one prescription medication. a. How many of the 3312 subjects used at least one prescription medication? b. Construct a 90% confidence interval estimate of the percentage of adults aged 57 through 85 years who use at least one prescription medication. c. What do the results tell us about the proportion of college students who use at least one prescription medication?
a. 2805 b. https://sample-size.net/confidence-interval-proportion/ 83.6% < p < 85.7 c. The results tell us nothing about the proportion of college students who use at least one prescription medication.
In a survey of 1472 people, 1025 people said they voted in a recent presidential election. Voting records show that 67% of eligible voters actually did vote. Given that 67% of eligible voters actually did vote a. find the probability that among 1472 randomly selected voters, at least 1025 actually did vote b. What do the results from part (a) suggest?
a. P(X≥1025)=0.017 b. Some people are being less than honest because P(x≥1025)is less than 5%.
The probability of flu symptoms for a person not receiving any treatment is 0.031. In a clinical trial of a common drug used to lower cholesterol, 36 of 1068people treated experienced flu symptoms. a. Assuming the drug has no effect on the likelihood of flu symptoms, estimate the probability that at least 36 people experience flu symptoms. b. What do these results suggest about flu symptoms as an adverse reaction to the drug?
a. P(X≥36)=0.3372 b. The drug has no effect on flu symptoms because x≥36 is not highly unlikely
You are the operations manager for an airline and you are considering a higher fare level for passengers in aisle seats. How many randomly selected air passengers must you survey? Assume that you want to be 90% confident that the sample percentage is within 4.5 percentage points of the true population percentage. a. Assume that nothing is known about the percentage of passengers who prefer aisle seats. b. Assume that a prior survey suggests that about 37% of air passengers prefer an aisle seat.
a. n= 335 b. n= 311
The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be 95% confident that his estimate is within five percentage points of the true population percentage? a. Assume that nothing is known about the percentage of adults who have heard of the brand. b. Assume that a recent survey suggests that about 83% of adults have heard of the brand. c. Given that the required sample size is relatively small, could he simply survey the adults at the nearest college?
a. n= 385 b. 217 c. No, a sample of students at the nearest college is a convenience sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults.
The pulse rates of 169 randomly selected adult males vary from a low of 41 bpm to a high of 113 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult males. Assume that we want 90% confidence that the sample mean is within 3 bpm of the population mean a. Find the sample size using the range rule of thumb to estimate σ. b. Assume that σ=11.7bpm, based on the value s=11.7bpm from the sample of 169 male pulse rates. c. Compare the results from parts (a) and (b). Which result is likely to be better?
a. n= 98 b. n= 42 c. The result from part (a) is larger than the result from part (b). The result from part (b)is likely to be better because it uses a better estimate of sigma .
A magazine provided results from a poll of 2000 adults who were asked to identify their favorite pie. Among the 2000 respondents, 14% chose chocolate pie, and the margin of error was given as ±5 percentage points. What does ____ represent a. p hat b. q hat c. n d. E e. p f. If the confidence level is 99%, what is the value of α?
a. the sample proportion b. found from evaluating 1- p hat c. the sample size d. the margin of error e. the population of proportion f. α=0.01
A gender-selection technique is designed to increase the likelihood that a baby will be a girl. In the results of the gender-selection technique, 852 births consisted of 434 baby girls and 418 baby boys. In analyzing these results, assume that boys and girls are equally likely. a. Find the probability of getting exactly 434 girls in 852 births. b. Find the probability of getting 434 or more girls in 852 births. c. If boys and girls are equally likely, is 434 girls in 852 births unusually high? d. Which probability is relevant for trying to determine whether the technique is effective: the result from part (a) or the result from part (b)? e. Based on the results, does it appear that the gender-selection technique is effective?
https://stattrek.com/online-calculator/binomial.aspx a. The probability of getting exactly 434 girls in 852 births is 0.0235 b. The probability of getting 434 or more girls in 852 births is 0.3037 c. No, because 434 girls in 852 births is not far from what is expected, given the probability of having a girl or a boy. d. The result from part (b) is more relevant, because one wants the probability of a result that is at least as extreme as the one obtained. e. No, because the probability of having 434 or more girls in 852 births is not unlikely, and thus, is attributable to random chance.
The data given to the right includes data from 38 candies, and 8 of them are red. The company that makes the candy claims that 33% of its candies are red. a. Construct a 90% confidence interval estimate of the percentage of red candies. b. Is the result consistent with the 33% rate that is reported by the candy maker?
https://www.calculators.org/math/z-critical-value.php a. 10.2% < p < 31.9% b. No, because the confidence interval does not include 33%.
A _____________ is a graph of points (x,y) where each x-value is from the original set of sample data, and each y-value is the corresponding z-score that is a quantile value expected from the standard normal distribution.
normal quantile plot
A _______ is a single value used to approximate a population parameter.
point estimate
Express the confidence interval 0.222<p<0.888 in the form p±E.
p±E= 0.555 ±0.333
The _______ is the best point estimate of the population mean.
sample mean
Which is NOT a criterion for distinguishing between results that could easily occur by chance and those results that are highly unusual?
the sample size is less than 5% of the size of the population
A continuity correction is made to a discrete whole number x in the binomial distribution by representing the discrete whole number x by which of the following intervals?
x−0.5 to x+0.5
A critical value, zα, denotes the _______.
z-score with an area of alpha to its right.
Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. Draw a graph and find the probability of a bone density test score greater than 0.81.
z= 0.81 (Right) The probability is 0.209
Find the critical value zα/2 that corresponds to the given confidence level. 80%
zα/2=1.28
Determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). If the procedure is not binomial, identify at least one requirement that is not satisfied. Nine different senators from the current U.S. Congress are randomly selected without replacement and whether or not they've served over 2 terms is recorded. Does the probability experiment represent a binomial experiment?
No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.
Which of the following groups has terms that can be used interchangeably with the others?
Percentage, Probability, and Proportion
Where would a value separating the top 15% from the other values on the graph of a normal distribution be found?
the right side of the horizontal scale of the graph
A continuous random variable has a _______ distribution if its values are spread evenly over the range of possibilities.
uniform
Determine whether or not the procedure described below results in a binomial distribution. If it is not binomial, identify at least one requirement that is not satisfied. Seven hundred different voters in a region with two major political parties, A and B, are randomly selected from the population of 5.6 million registered voters. Each is asked if he or she is a member of political party A, recording Yes or No.
Yes, the result is a binomial probability distribution.
Pulse rates of women are normally distributed with a mean of 77.5 beats per minute and a standard deviation of 11.6 beats per minute. Answer the following questions. a. What are the values of the mean and standard deviation after converting all pulse rates of women to z scores using z=(x−μ)/σ? b. The original pulse rates are measure with units of "beats per minute". What are the units of the corresponding z scores? Choose the correct choice below.
a. μ=0 σ=1 b. The z scores are numbers without units of measurement.
Assume that the Poisson distribution applies and that the mean number of hurricanes in a certain area is 7.8 per year. a. Find the probability that, in a year, there will be 6 hurricanes. b. In a 45-year period, how many years are expected to have 6 hurricanes? c. How does the result from part (b) compare to a recent period of 45 years in which 5 years had 6 hurricanes? Does the Poisson distribution work well here?
a. The probability is 0.128 b. The expected number of years with 6 hurricanes is 5.8 c. The result from part (b) is close to the number of hurricanes in the recent period of 45 years, so the Poisson distribution does appear to work well in the given situation.
A survey found that women's heights are normally distributed with mean 63.7 in. and standard deviation 3.5 in. The survey also found that men's heights are normally distributed with mean 68.1 in. and standard deviation 3.8 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 56 in. and a maximum of 63 in. Complete parts (a) and (b) below. a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement park?
***
When conducting research on color blindness in males, a researcher forms random groups with five males in each group. The random variable x is the number of males in the group who have a form of color blindness. Determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is notgiven, identify the requirements that are not satisfied. x p(x) 0 0.655 1 0.291 2 0.048 3 0.005 4 0.001 5 0.000 1. Does the table show a probability distribution? Select all that apply. 2. Find the mean of the random variable x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. 3. Find the standard deviation of the random variable x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
1. --> Do they add up to 1 a. Yes, the table shows a probability distribution. 2. --> Multiply: (X)(p(x)) then add sum. (Ans: 0.376) a. 0.4 males 3. --> Multiply (x^2)(p(x)) then add sum. (Ans: 0.514) Then square root (part b)-(part a)^2 square root (0.514)-(0.376)^2= 0.6 a. 0.6 males
Groups of adults are randomly selected and arranged in groups of three. The random variable x is the number in the group who say that they would feel comfortable in a self-driving vehicle. Determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied. x p(x) 0 0.358 1 0.448 2 0.168 3. 0.026 1. Does the table show a probability distribution? Select all that apply. 2. Find the mean of the random variable x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. 3. Find the standard deviation of the random variable x. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
1. --> Do they add up to 1 a. Yes, the table shows a probability distribution. 2. --> Multiply: (X)(p(x)) then add sum. (Ans: 0.862) a. 0.9 adult(s) 3. --> Multiply (x^2)(p(x)) then add sum. (Ans: 1.354) Then square root (part b)-(part a)^2 square root (1.354)-(0.862)^2= 0.7816 a. 0.8 adult(s)
If calculations are time-consuming and if a sample size is no more than 5% of the size of the population, the _______ states to treat the selections as being independent (even if the selections are technically dependent).
5% Guideline for Cumbersome Calculations
About ______% of the area is between z=−3.5and z=3.5 (or within 3.5 standard deviations of the mean).
99.98%
Which of the following would be information in a question asking you to find the area of a region under the standard normal curve as a solution?
A distance on the horizontal axis is given
Which of the following is NOT a procedure for determining whether it is reasonable to assume that sample data are from a normally distributed population?
Checking that the probability of an event is 0.05 or less
Which of the following is NOT a requirement for a density curve?
The graph is centered around 0.
The accompanying table describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children. Complete parts (a) through (d) below. a. Find the probability of getting exactly 1 girl in 8 births b. Find the probability of getting 1 or fewer girls in 8 births c. Which probability is relevant for determining whether 1 is a significantly low number of girls in 8 births: the result from part (a) or part (b)?
Girls (x) P(x) 0 0.003 1 0.026 2 0.104 3 0.191 4 0.352 5 0.191 6 0.104 7 0.026 8 0.003 a. P(1)= 0.026 The probability of getting 1 or fewer girls in 8 births is 0.026 b. P(0)+P(1) = 0.003+0.026 =0.029 The probability of getting 1 or fewer girls in 8 births is 0.029 c. Since getting 0 girls is an even lower number of girls than getting 1 girl, the result from part (b) is the relevant probability. d. Yes, since the appropriate probability is less than 0.05, it is a significantly low number.
Which of the following is not a commonly used practice?
If the distribution of the sample means is normally distributed, and n>30, then the population distribution is normally distributed.
Which of the following is NOT true in regards to using a normal quantile plot to determine whether or not a distribution is normal?
If the plot is bell-shaped, the population distribution is normal.
Which of the following does NOT describe the standard normal distribution?
The graph is uniform.
** Assume that random guesses are made for eight multiple choice questions on an SAT test, so that there are n=8 trials, each with probability of success (correct) given by p=0.4. Find the indicated probability for the number of correct answers. Find the probability that the number x of correct answers is fewer than 4. P(X<4)=0.5941
P(X<4)= P(3 or 2 or 1 or 0)= P(3)+P(2)+P(1)+P(0) P(X<4)=0.5941
Assume that a randomly selected subject is given a bone density test. Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1. Draw a graph and find P9, the 9th percentile. This is the bone density score separating the bottom 9% from the top
P9 (Left) (Less than half) The bone density score corresponding to P9 is -1.34
The _______ distribution is a discrete probability distribution that applies to the number of occurrences of some event over a specified interval.
Poisson
What's wrong with the following statement? "Because the digits 0, 1, 2, . . . , 9 are the normal results from lottery drawings, such randomly selected numbers have a normal distribution."
Since the probability of each digit being selected is equal, lottery digits have a uniform distribution, not a normal distribution.
Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1. z= -0.81 and z= 1.24
The area of the shaded region is 0.6835
Find the area of the shaded region. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. IQ= 85-120
The area of the shaded region is 0.7495
Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1. z= -1.08 (Right)
The area of the shaded region is 0.8599
Find the area of the shaded region. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. IQ= 125
The area of the shaded region is 0.9525
Which of the following is NOT a property of the sampling distribution of the variance?
The distribution of sample variances tends to be a normal distribution.
Which of the following is NOT a conclusion of the Central Limit Theorem?
The distribution of the sample data will approach a normal distribution as the sample size increases.
Which of the following is NOT a property of the sampling distribution of the sample mean?
The distribution of the sample mean tends to be skewed to the right or left.
Assume that adults have IQ scores that are normally distributed with a mean of 104.4 and a standard deviation 16.6. Find the first quartile Q1,which is the IQ score separating the bottom 25% from the top 75%.
The first quartile is 93.3
Which of the following is NOT a descriptor of a normal distribution of a random variable?
The graph is centered around 0.
Engineers want to design seats in commercial aircraft so that they are wide enough to fit 95% of all males. (Accommodating 100% of males would require very wide seats that would be much too expensive.) Men have hip breadths that are normally distributed with a mean of 14.9 in. and a standard deviation of 1.1 in. Find P95. That is, find the hip breadth for men that separates the smallest 95% from the largest 5%.
The hip breadth for men that separates the smallest 95% from the largest 5% is P95=16.71 in.
Find the indicated IQ score. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15. x= 0.65
The indicated IQ score, x, is 105.9
Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. area= 0.2514 (Left)
The indicated z score is -0.67
Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. area: 0.8212 (Right)
The indicated z score is -0.92
Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. area= 0.9864
The indicated z score is 2.21
Which of the following is NOT a requirement of the Poisson Distribution?
The occurrences must be dependent.
In a recent year, an author wrote 172 checks. Use the Poisson distribution to find the probability that, on a randomly selected day, he wrote at least one check.
The probability is 0.376
Assume that when adults with smartphones are randomly selected, 62% use them in meetings or classes. If 25 adult smartphone users are randomly selected, find the probability that exactly 15 of them use their smartphones in meetings or classes.
The probability is 0.1578
Assume that adults have IQ scores that are normally distributed with a mean of 95.3 and a standard deviation of 21.4. Find the probability that a randomly selected adult has an IQ greater than 121.6
The probability that a randomly selected adult from this group has an IQ greater than 121.6 is 0.1093
or the purposes of constructing modified boxplots, outliers are defined as data values that are above Q3 by an amount greater than 1.5×IQR or below Q1 by an amount greater than 1.5×IQR, where IQR is the interquartile range. Using this definition of outliers, find the probability that when a value is randomly selected from a normal distribution, it is an outlier.
The probability that a randomly selected value taken from a normal distribution is considered an outlier is 0.0071
A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 50 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 3000 aspirin tablets actually has a 4% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?
The probability that this whole shipment will be accepted is 0.4005 The company will accept 40% of the shipments and will reject 60% of the shipments, so many of the shipments will be rejected
___________ is the distribution of sample proportions, with all samples having the same sample size n taken from the same population.
The sampling distribution of the proportion is the distribution of sampleproportions, with all samples having the same sample size n taken from the same population.
The standard deviation of the distribution of sample means is _______.
The standard deviation of the distribution of sample means is σ/√n
Determine whether the given procedure results in a binomial distribution. If it is not binomial, identify the requirements that are not satisfied. Surveying 200 teenagers and recording if they have ever committed a crime
Yes, because all 4 requirements are satisfied.
a. Use a calculator or computer software to generate a normal quantile plot for the data in the accompanying table b. Determine whether the data come from a normally distributed population.
a. starts at -2 & curves slightly over straight line b. The distribution is not normal. The points show a systematic pattern that is not a straight-line pattern
The ages (years) of three government officials when they died in office were 55, 46, and 60. a. Assuming that 2 of the ages are randomly selected with replacement, list the different possible samples. b. Find the range of each of the samples, then summarize the sampling distribution of the ranges in the format of a table representing the probability distribution. c. Compare the population range to the mean of the sample ranges. Choose the correct answer below. d. Do the sample ranges target the value of the population range? In general, do sample ranges make good estimators of population ranges? Why or why not?
a. (55,55), (55,46),(55,60),(46,55),(46,46),(46,60),(60,55),(60,46),(60,60) b. sample probability 0 3/9 5 2/9 9 2/9 14 2/9 c. The population range is not equal to the mean of the sample ranges (it is also not equal to the age of the oldest official or age of the youngest official at the time of death). d. The sample ranges do not target the population range, therefore, sample ranges do not make good estimators of population ranges
A common design requirement is that an environment must fit the range of people who fall between the 5th percentile for women and the 95th percentile for men. In designing an assembly work table, the sitting knee height must be considered, which is the distance from the bottom of the feet to the top of the knee. Males have sitting knee heights that are normally distributed with a mean of 21.2 in. and a standard deviation of 1.1 in. Females have sitting knee heights that are normally distributed with a mean of 19.7in. and a standard deviation of 1.0 in. a. What is the minimum table clearance required to satisfy the requirement of fitting 95% of men? b. Determine if the following statement is true or false. If there is clearance for 95% of males, there will certainly be clearance for all women in the bottom 5%. c. The author is writing this exercise at a table with a clearance of 23.2 in. above the floor. What percentage of men fit this table? d. What percentage of women fit this table? e. Does the table appear to be made to fit almost everyone? Choose the correct answer below.
a. 23.0in b. The statement is true because the 95th percentile for men is greater than the 5th percentile for women. c. 96.56 d. 99.99% e. The table will fit almost everyone except about 3% of men with the largest sitting knee heights.
Examine the normal quantile plot and determine whether it depicts sample data from a population with a normal distribution. --> starts at -2 & curves slightly over straight line a. Does the normal quantile plot depict sample data from a population with a normal distribution?
a. No. The points exhibit some systematic pattern that is not a straight-line pattern
A rare form of malignant tumor occurs in 11 children in a million, so its probability is 0.000011. Four cases of this tumor occurred in a certain town, which had 11,639 children. a. Assuming that this tumor occurs as usual, find the mean number of cases in groups of 11,639 children. b. Using the unrounded mean from part (a), find the probability that the number of tumor cases in a group of 11,639 children is 0 or 1. c. What is the probability of more than one case? d. Does the cluster of four cases appear to be attributable to random chance? Why or why not?
a. The mean number of cases is 0.128 b. The probability that the number of cases is exactly 0 or 1 is 0.992 c. The probability of more than one case is 0.008 d. No, because the probability of more than one case is very small.
Last year, a person wrote 139 checks. Let the random variable x represent the number of checks he wrote in one day, and assume that it has a Poisson distribution a. What is the mean number of checks written per day? b. What is the standard deviation? c. What is the variance?
a. The mean number of checks written per day is 0.381. b. The standard deviation is 0.617 c. The variance is 0.381
When studying radioactive material, a nuclear engineer found that over 365 days, 1,000,000 radioactive atoms decayed to 976,956 radioactive atoms, so 23,044 atoms decayed during 365 days. a. Find the mean number of radioactive atoms that decayed in a day. b. Find the probability that on a given day, 51 radioactive atoms decayed.
a. The mean number of radioactive atoms that decay per day is 63.134 b. The probability that on a given day, 51 radioactive atoms decayed, is 0.015997
he weights of a certain brand of candies are normally distributed with a mean weight of 0.8596g and a standard deviation of 0.0511g. A sample of these candies came from a package containing 447 candies, and the package label stated that the net weight is 381.5g. (If every package has 447 candies, the mean weight of the candies must exceed 381.5/447=0.8535g for the net contents to weigh at least 381.5g.) a. If 1 candy is randomly selected, find the probability that it weighs more than 0.8535 g. b. If 447 candies are randomly selected, find the probability that their mean weight is at least 0.8535g. c. Given these results, does it seem that the candy company is providing consumers with the amount claimed on the label?
a. The probability is 0.5478 b. The probability that a sample of 447 candies will have a mean of 0.8535g or greater is 0.9941 c. Yes, because the probability of getting a sample mean of 0.8535g or greater when 447 candies are selected is not exceptionally small.
Assume that females have pulse rates that are normally distributed with a mean of μ=74.0 beats per minute and a standard deviation of σ=12.5 beats per minute. Complete parts (a) through (c) below. a. If 1 adult female is randomly selected, find the probability that her pulse rate is less than 77 beats per minute. b. If 25 adult females are randomly selected, find the probability that they have pulse rates with a mean less than 77 beats per minute. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?
a. The probability is 0.5948 b. The probability is 0.8849 c. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size.
An airliner carries 100 passengers and has doors with a height of 74 in. Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8 in. a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending. b. If half of the 100 passengers are men, find the probability that the mean height of the 50 men is less than 74 in. c. When considering the comfort and safety of passengers, which result is more relevant: the probability from part (a) or the probability from part (b)? Why? d. When considering the comfort and safety of passengers, why are women ignored in this case?
a. The probability is 0.9633 b. The probability is 1.0000 c. The probability from part (a) is more relevant because it shows the proportion of male passengers that will not need to bend. d. Since men are generally taller than women, a design that accommodates a suitable proportion of men will necessarily accommodate a greater proportion of women.
The lengths of pregnancies are normally distributed with a mean of 266 days and a standard deviation of 15 days. a. Find the probability of a pregnancy lasting 309 days or longer. b. If the length of pregnancy is in the lowest 2%, then the baby is premature. Find the length that separates premature babies from those who are not premature.
a. The probability that a pregnancy will last 309 days or longer is 0.0021 b. Babies who are born on or before 235 days are considered premature.
Based on a poll, 40% of adults believe in reincarnation. Assume that 6 adults are randomly selected, and find the indicated probability. Complete parts (a) through (d) below. a. What is the probability that exactly 5of the selected adults believe in reincarnation? b. What is the probability that all of the selected adults believe in reincarnation? c. What is the probability that at least 5 of the selected adults believe in reincarnation? d. If 6 adults are randomly selected, is 5 a significantly high number who believe in reincarnation?
a. The probability that exactly 5 of the 6 adults believe in reincarnation is 0.0370.037. b. The probability that all of the selected adults believe in reincarnation is 0.004 c. The probability that at least 5 of the selected adults believe in reincarnation is 0.0410.041. d. Yes, because the probability that 5 or more of the selected adults believe in reincarnation is less than 0.05.
An elevator has a placard stating that the maximum capacity is 1630 lb—10 passengers. So, 10 adult male passengers can have a mean weight of up to 1630/10=163 pounds. a. If the elevator is loaded with 10 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 163lb. b. Does this elevator appear to be safe?
a. The probability the elevator is overloaded is 0.8621 b. No, there is a good chance that 10 randomly selected people will exceed the elevator capacity.
Assume that different groups of couples use a particular method of gender selection and each couple gives birth to one baby. This method is designed to increase the likelihood that each baby will be a girl, but assume that the method has no effect, so the probability of a girl is 0.5. Assume that the groups consist of 26 couples. Complete parts (a) through (c) below. a. Find the mean and the standard deviation for the numbers of girls in groups of 26 births. b. Use the range rule of thumb to find the values separating results that are significantly low or significantly high. c. Is the result of 21 girls a result that is significantly high? What does it suggest about the effectiveness of the method?
a. The value of the mean is μ=13 The value of the standard deviation is σ=2.5 b. Values of 7.9 girls or fewer are significantly low. Values of 18.1 girls or greater are significantly high c. The result is significantly high, because 21 girls is greater than 18.1 girls. A result of 21 girls would suggest that the method is effective.
Determine whether the following value is a continuous random variable, discrete random variable, or not a random variable. a. The square footage of a pool b. The hair color of adults in the United States c. The time required to upload a file to the Internet d. The number of light bulbs that burn out in the next year in a room with 13 bulbs e. The time it takes for a light bulb to burn out f. The number of textbook authors now eating a meal
a. continuous random variable. b. not a random variable c. continuous random variable d. discrete random variable e. continuous random variable f. discrete random variable
The heights (in inches) of men listed in the accompanying table have a distribution that is approximately normal, so it appears that those heights are from a normally distributed population. a. If 9 inches is subtracted from each height, are the new heights also normally distributed? b. If each height is converted from inches to centimeters, are the heights in centimeters also normally distributed? c. Are the logarithms of normally distributed heights also normally distributed?
a. yes b. yes c. no
Assume a population of 4, 5, and 9. Assume that samples of size n=2 are randomly selected with replacement from the population. Listed below are the nine different samples. Complete parts a through d below. 4,4. 4,5. 4,9. 5,4. 5,5. 5,9. 9,4. 9,5. 9,9 a. Find the value of the population standard deviation σ. b. Find the standard deviation of each of the nine samples, then summarize the sampling distribution of the standard deviations in the format of a table representing the probability distribution of the distinct standard deviation values. Use ascending order of the sample standard deviations. c. Find the mean of the sampling distribution of the sample standard deviations. d.Do the sample standard deviations target the value of the population standard deviation? In general, do sample standard deviations make good estimators of population standard deviations? Why or why not?
a. σ= 2.160 b. s probability 0 3/9 0.707 2/9 2.828 2/9 3.536 2/9 c. The mean of the sampling distribution of the sample standard deviations is 1.571 d. The sample standard deviations do not target the population standard deviation, therefore, sample standard deviations are biased estimators.
The random variable x represents the number of phone calls an author receives in a day, and it has a Poisson distribution with a mean of 7.7 calls. a. What are the possible values of x? b. Is a value of x=3.5 possible?Is x a discrete random variable or a continuous random variable?
a. 0, 1, 2, 3, ... b. A value of x=3.5 is not possible because x is a discrete random variable.
In a probability histogram, there is a correspondence between _______.
area and probability
What conditions would produce a negative z-score?
a z-score corresponding to an area located entirely in the left side of the curve
A _______ random variable has infinitely many values associated with measurements.
continuous
A _______ random variable has either a finite or a countable number of values.
discrete
Finding probabilities associated with distributions that are standard normal distributions is equivalent to _______.
finding the area of the shaded region representing that probability.
In the binomial probability formula, the variable x represents the _______.
number of successes
A ______ variable is a variable that has a single numerical value, determined by chance, for each outcome of a procedure.
random
The accompanying table describes the random variable x, the numbers of adults in groups of five who reported sleepwalking. Complete parts (a) through (d) below. a. Find the probability of getting exactly 4 sleepwalkers among 5 adults. b. Find the probability of getting 4 or more sleepwalkers among 5 adults. c. Which probability is relevant for determining whether 4 is a significantly high number of sleepwalkers among 5 adults: the result from part (a) or part (b)? d. Is 4 a significantly high number of 4 sleepwalkers among 5 adults? Why or why not? Use 0.05 as the threshold for a significant event.
x P(x) 0 0.151 1 0.391 2 0.309 3 0.115 4 0.025 5 0.009 a. P(4)= 0.025 b. P(4)+P(5)= 0.025+ 0.009= 0.034 c. Since the probability of getting 5 sleepwalkers includes getting 4 sleepwalkers, the result from part (b) is the relevant probability. d. Yes, since the appropriate probability is less than 0.05, it is a significantly high number.
The accompanying table describes results from groups of 10 births from 10 different sets of parents. The random variable x represents the number of girls among 10 children. Use the range rule of thumb to determine whether 1 girl in 10 births is a significantly low number of girls. Use the range rule of thumb to identify a range of values that are not significant. x p(x) 0 0.002 1 0.011 2 0.039 3 0.116 4 0.208 5 0.235 6 0.198 7 0.116 8 0.036 9 0.015 10 0.024 1. The maximum value in this range is ___ girls. 2. The minimum value in this range is ___ girls. 3. Based on the result, is 1 girl in 10 births a significantly low number of girls? Explain.
x p(x) xp(x) (x-m)(px) 0 0.002 0 0.05202 1 0.011 0.011 0.18491 2 0.039 0.078 0.37479 3 0.116 0.348 0.51156 4 0.208 0.832 0.25168 5 0.235 1.175 0.00235 6 0.198 1.188 0.16038 7 0.116 0.812 0.41876 8 0.036 0.288 0.30276 9 0.015 0.135 0.22815 10 0.024 0.0.24 0.57624 5.107 3.0680 1. find standard deviation square root (3.0680) = 1.7517 approx. 1.8 5.1 - 2(1.8)= 1.5 girls 2. find standard deviation 5.1 + 2(1.8)= 8.7 girls 3. Yes, 1 girl is a significantly low number of girls, because 1 girl is below the range of values that are not significant.
Find the indicated critical value. z0.05
z0.05= 1.645
Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. Find the probability that a given score is less than −1.57 and draw a sketch of the region
z= -1.57 (Left) The probability is 0.0582
Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. Draw a graph and find the probability of a bone density test score between −2.01 and 2.01.
z= -2.01 - 2.01 The probability is 0.9556
Assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. Draw a graph and find the probability of a bone density test score less than 0.
z= 0 (Left) The probability is . 5000
Annual incomes are known to have a distribution that is skewed to the right instead of being normally distributed. Assume that we collect a large (n>30) random sample of annual incomes. Can the distribution of incomes in that sample be approximated by a normal distribution because the sample is large? Why or why not?
No; the sample means will be normally distributed, but the sample of incomes will be skewed to the right.
Determine whether the given procedure results in a binomial distribution (or a distribution that can be treated as binomial). If the procedure is not binomial, identify at least one requirement that is not satisfied. The YSORT method of sex selection, developed by the Genetics & IVF Institute, was designed to increase the likelihood that a baby will be a boy. When 60 couples use the YSORT method and give birth to 60 babies, the sex of the babies is recorded.
Yes, because the procedure satisfies all the criteria for a binomial distribution.
The standard deviation of the Poisson distribution is calculated using _______.
√ mean
For the binomial distribution, which formula finds the standard deviation?
√npq
