introduction to Statistics 1114-02
Determine whether the underlined number is a statistic or a parameter. In a study of all 4283 students at a college, it is found that 45% own a television.
1.2.11 Parameter because the value is a numerical measurement describing a characteristic of a population.
Determine whether the underlined number is a statistic or a parameter. A sample of professors is selected and it is found that 40% own a vehicle.
1.2.11 Statistic because the value is a numerical measurement describing a characteristic of a sample.
State whether the data described below are discrete or continuous, and explain why. The number of majors offered by colleges
1.2.14 The data are discrete because the data can only take on specific values.
A particular country has 40 total states. If the areas of 35 states are added and the sum is divided by 35, the result is 192,825 square kilometers. Determine whether this result is a statistic or a parameter.
1.2.9
Identify the type of sampling used (random, systematic, convenience, stratified, or cluster sampling) in the situation described below. In a poll conducted by a certain research center, 1089 adults were called after their telephone numbers were randomly generated by a computer, and 24% were able to correctly identify the attorney general.
1.3.11 random sampling
Identify which type of sampling is used: random, systematic, convenience, stratified, or cluster. To determine customer opinion of their check-in service, American Airlines randomly selects 60 flights during a certain week and surveys all passengers on the flights.
1.3.13 cluster
Twenty different statistics students are randomly selected. For each of them, their body temperature (°C) is measured and their head circumference (cm) is measured. a. For this sample of paired data, what does r represent, and what does ρ represent? b. Without doing any research or calculations, estimate the value of r. c. Does r change if body temperatures are converted to Fahrenheit degrees?
10.1.1 r is a statistic that represents the value of the linear correlation coefficient computed from the paired sample data, and ρ is a parameter that represents the value of the linear correlation coefficient that would be computed by using all of the paired data in the population of all statistics students. the value of r is estimated to be 0, because it is likely that there is no correlation between body temperature and head circumference. The value of r does not change, because r is not affected by converting all values of a variable to a different scale.
Use the given data set to complete parts (a) through (c) below. (Use α=0.05.)
10.1.10
Listed below are numbers of Internet users per 100 people and numbers of scientific award winners per 10 million people for different countries. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value of r. Determine whether there is sufficient evidence to support a claim of linear correlation between the two variables. Use a significance level of α=0.05.
10.1.13-t
Police sometimes measure shoe prints at crime scenes so that they can learn something about criminals. Listed below are shoe print lengths, foot lengths, and heights of males. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value of r. Determine whether there is sufficient evidence to support a claim of linear correlation between the two variables. Based on these results, does it appear that police can use a shoe print length to estimate the height of a male? Use a significance level of α=0.05.
10.1.17-t
Listed below are annual data for various years. The data are weights (metric tons) of imported lemons and car crash fatality rates per 100,000 population. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using α=0.05. Is there sufficient evidence to conclude that there is a linear correlation between lemon imports and crash fatality rates? Do the results suggest that imported lemons cause car fatalities?
10.1.19-t
Listed below are amounts of bills for dinner and the amounts of the tips that were left. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value of r. Determine whether there is sufficient evidence to support a claim of linear correlation between the two variables. Use a significance level of α=0.05. If everyone were to tip with the same percentage, what should be the value of r?
10.1.25-t
The accompanying table lists the numbers of Internet users per 100 people and numbers of scientific award winners per 10 million people for different countries. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value of r. Determine whether there is sufficient evidence to support a claim of linear correlation between the two variables. Use a significance level of α=0.05.
10.1.29-t
If we find that there is a linear correlation between the concentration of carbon dioxide in our atmosphere and the global temperature, does that indicate that changes in the concentration of carbon dioxide cause changes in the global temperature?
10.1.3 No. The presence of a linear correlation between two variables does not imply that one of the variables is the cause of the other variable.
Match these values of r with the accompanying scatterplots: −0.711, −1, 0.363, −0.363, and −0.998.
10.1.4
Fifty-four wild bears wereanesthetized, and then their weights and chest sizes were measured and listed in a data set. Results are shown in the accompanying display. Is there sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chestsizes? When measuring an anesthetizedbear, is it easier to measure chest size thanweight? Ifso, does it appear that a measured chest size can be used to predict theweight? Use a significance level of α=0.05.
10.1.5
A data set includes weights of garbage discarded in one week from 62 different households. The paired weights of paper and glass were used to obtain the results shown to the right. Is there sufficient evidence to support the claim that there is a linear correlation between weights of discarded paper and glass? Use a significance level of α=0.05.
10.1.7
Use the given data set to complete parts (a) through (c) below. (Use α=0.05.)
10.1.9
The display provided from technology available below results from using data for a smartphone carrier's data speeds at airports to test the claim that they are from a population having a mean less than 4.00 Mbps. Conduct the hypothesis test using these results. Use a 0.05 significance level. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim.
2.3.9
When testing for current in a cable with eight color-coded wires, the author used a meter to test four wires at a time. How many different tests are required for every possible pairing of four wires?
4.4.14
When testing for current in a cable with twelve color-coded wires, the author used a meter to test five wires at a time. How many different tests are required for every possible pairing of five wires?
4.4.14 on calculator do 12 nCr 5 = 792
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.
5.1.12 Yes, the table shows a probability distribution.
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.
5.1.19
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.
5.1.24 Since the probability of getting 4 or more sleepwalkers is the probability of the given or more extreme result, the result from part (b) is the relevant probability. Yes, since the appropriate probability is less than 0.05, it is a significantly high number.
For 100 births, P(exactly 56 girls)=0.0390 and P(56 or more girls)=0.136. Is 56 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.
5.1.4 p(56 or more girls) is not greater than
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. Eight 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.
5.2.10 No, because the trials of the experiment are not independent and the probability of success differs from trial to trial.
A data set includes 109 body temperatures of healthy adult humans having a mean of 98.2°F and a standard deviation of 0.62°F. Construct a 99% confidence interval estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of 98.6°F as the mean body temperature?
5.2.11
Assume that a procedure yields a binomial distribution with n=3 trials and a probability of success of p=0.30. Use a binomial probability table to find the probability that the number of successes x is exactly 1.
5.2.15
Assume that when adults with smartphones are randomly selected, 52% use them in meetings or classes. If 9 adult smartphone users are randomly selected, find the probability that at least 3 of them use their smartphones in meetings or classes.
5.2.23-t
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 23 couples. Complete parts (a) through (c) below.
5.2.29
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.
6.1.11-t
Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.
6.1.14-t
Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.
6.1.16-t
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 −1.53.
6.1.23-t
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 2.25 minutes.
6.1.5
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.
6.2.5
A ski gondola carries skiers to the top of a mountain. Assume that weights of skiers are normally distributed with a mean of 185 lb and a standard deviation of 35 lb. The gondola has a stated capacity of 25 passengers, and the gondola is rated for a load limit of 3500 lb. Complete parts (a) through (d) below. a. Given that the gondola is rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the gondola is filled to the stated capacity of 25 passengers? The maximum mean weight is 140140 lb. (Type an integer or a decimal. Do not round.) b. If the gondola is filled with 25 randomly selected skiers, what is the probability that their mean weight exceeds the value from part (a)? The probability is 11. (Round to four decimal places as needed.) c. If the weight assumptions were revised so that the new capacity became 20 passengers and the gondola is filled with 20 randomly selected skiers, what is the probability that their mean weight exceeds 175 lb, which is the maximum mean weight that does not cause the total load to exceed 3500 lb? The probability is . 8997.8997. (Round to four decimal places as needed.) d. Is the new capacity of 20 passengers safe? Since the probability of overloading is over 50 % commaover 50%, the new capacity does not appear does not appear to be safe enough.
6.4.13T
Suppose that an airline uses a seat width of 16.3 in. Assume men have hip breadths that are normally distributed with a mean of 14.5 in. and a standard deviation of 1 in. Complete parts (a) through (c) below.
6.4.15-E answer- part c Part a should because the seats are occupied by individuals rather than means.
An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 130 lb and 181 lb. The new population of pilots has normally distributed weights with a mean of 137 lb and a standard deviation of 30.2 lb.
6.4.17-T Part (a) because the seat performance for a single pilot is more important.
An airliner carries 100 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8 in. Complete parts (a) through (d).
6.4.19-t The probability from part (a) is more relevant because it shows the proportion of male passengers that will not need to bend. 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 given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. The probability of no more than 35 defective CDs
6.6-10 The area to the left of 35.5
The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. The probability of exactly 44 green marbles
6.6-8 The area between 43.5 and 44.5
The value given below is discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. Probability of more than 3 passengers who do not show up for a flight
6.6.1 The area to the right of 3.5
Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 149 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 39 voted
6.6.11-t
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, 831 births consisted of 427 baby girls and 404 baby boys. In analyzing these results, assume that boys and girls are equally likely. a. Find the probability of getting exactly 427 girls in 831 births. b. Find the probability of getting 427 or more girls in 831 births. If boys and girls are equally likely, is 427 girls in 831 births unusually high? c. Which probability is relevant for trying to determine whether the technique is effective: the result from part (a) or the result from part (b)? d. Based on the results, does it appear that the gender-selection technique is effective?
6.6.13t No, because 427 girls in 831 births is not far from what is expected, given the probability of having a girl or a boy. 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. no, because the probability of having 427or more girls in 831 births is not unlikely, andthus, is attributable to random chance.
Based on a smartphone survey, assume that 54% of adults with smartphones use them in theaters. In a separate survey of 228 adults with smartphones, it is found that 120 use them in theaters. a. If the 54% rate is correct, find the probability of getting 120 or fewer smartphone owners who use them in theaters. b. Is the result of 120 significantly low?
6.6.15-e b. Is the result of 120 significantly low? No, because the probability of this event is greater than the probability cutoff that corresponds to a significant event, which is 0.05.
When a scientist conducted a genetics experiments with peas, one sample of offspring consisted of 929 peas, with 727 of them having red flowers. If we assume, as the scientist did, that under these circumstances, there is a 3/4 probability that a pea will have a red flower, we would expect that 696.75 (or about 697) of the peas would have red flowers, so the result of 727 peas with red flowers is more than expected. a. If the scientist's assumed probability is correct, find the probability of getting 727 or more peas with red flowers. b. Is 727 peas with red flowers significantly high? c. What do these results suggest about the scientist's assumption that 3/4 of peas will have red flowers?
6.6.17-t Yes, because the probability of this event is less than the probability cutoff that corresponds to a significantevent, which is 0.05. Since the result of 727 peas with red flowers is significantly high, it is strong evidence against the scientist's assumption that 3/4 of peas will have red flowers.
In a survey of 1315 people, 901 people said they voted in a recent presidential election. Voting records show that 66% of eligible voters actually did vote. Given that 66% of eligible voters actually did vote, (a) find the probability that among 315 randomly selected voters, at least 901 actually did vote. (b) What do the results from part (a) suggest?
6.6.19 Some people are being less than honest because P(x≥901) is less than 5%.
If np≥5 and nq≥5, estimate P(fewer than 5) with n=14 and p=0.6 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.
6.6.5 A
If np≥5 and nq≥5, estimate P(more than 5) with n=13 and p=0.3 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.
6.6.7-T B
The continuity correction is used to compensate for the fact that a ________ distribution is used to approximate a ________ distribution.
6.6.9 continuous; discrete
Use a normal approximation to find the probability of the indicated number of voters. In this case, assume that 151 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 37 voted
6.6.9-t
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?
7.1.1 The confidence level
Express the confidence interval (0.020,0.094) in the form of p−E<p<p+E.
7.1.11
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=904 and x=523 who said "yes." Use a 95% confidence level.
7.1.13 One has 95% 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). In a study of cell phone use and brain hemispheric dominance, an Internet survey was e-mailed to 2437 subjects randomly selected from an online group involved with ears. 1152 surveys were returned. Construct a 95% confidence interval for the proportion of returned surveys.
7.1.15 One has 95% 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 600 babies were born, and 330 of them were girls. Use the sample data to construct a 99% confidence interval estimate of the percentage of girls born. Based on the result, does the method appear to be effective?
7.1.17
A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 460 babies were born, and 253 of them were girls. Use the sample data to construct a 99% confidence interval estimate of the percentage of girls born. Based on the result, does the method appear to be effective?
7.1.17 depends on the question. either A or B
In a study of the accuracy of fast food drive-through orders, Restaurant A had 210 accurate orders and 66 that were not accurate. a. Construct a 95% confidence interval estimate of the percentage of orders that are not accurate. b. Compare the results from part (a) to this 95% confidence interval for the percentage of orders that are not accurate at Restaurant B: 0.213<p<0.306. What do you conclude?
7.1.19-t Since the two confidence intervals overlap, neither restaurant appears to have a significantly different percentage of orders that are not accurate.
A magazine provided results from a poll of 1000 adults who were asked to identify their favorite pie. Among the 1000 respondents, 13% chose chocolate pie, and the margin of error was given as ±3 percentage points. Describe what is meant by the statement that "the margin of error was given as ±3 percentage points."
7.1.2 The statement indicates that the interval 13%±3% is likely to contain the true population percentage of people that prefer chocolate pie.
In a science fair project, Emily conducted an experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily's hand without seeing it and without touching it. Among 266 trials, the touch therapists were correct 119 times. Complete parts (a) through (d).
7.1.21-t a) .5 b) find p (x/n) c) D) Since the confidence interval is not entirely above 0.5, there does not appear to be sufficient evidence that touch therapists can select the correct hand by sensing energy fields.
A study of 420,059 cell phone users found that 139 of them developed cancer of the brain or nervous system. Prior to this study of cell phone use, the rate of such cancer was found to be 0.0348% for those not using cell phones. Complete parts (a) and (b).
7.1.23 B0 depends on the question
A magazine provided results from a poll of 1500 adults who were asked to identify their favorite pie. Among the 1500 respondents, 11% chose chocolate pie, and the margin of error was given as ±4 percentage points. What values do p, q, n, E, and p represent? If the confidence level is 99%, what is the value of α? The value of p is the sample proportion.the sample proportion. The value of q is found from evaluating 1 minus ModifyingAbove p with caret .found from evaluating 1−p. The value of n is the sample size.the sample size. The value of E is the margin of error.the margin of error. The value of p is the population proportion.the population proportion. If the confidence level is 99%, what is the value of α? α=. 01.01 (Type an integer or a decimal. Do not round.)
7.1.3
In a study of government financial aid for college students, it becomes necessary to estimate the percentage of full-time college students who earn a bachelor's degree in four years or less. Find the sample size needed to estimate that percentage. Use a 0.05 margin of error and use a confidence level of 95%. Complete parts (a) through (c) below.
7.1.33-t No, using the additional survey information from part (b) only slightly reduces the sample size
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 95% confident that the sample percentage is within 4.5 percentage points of the true population percentage. Complete parts (a) and (b) below.
7.1.35-t
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 90% confident that his estimate is within eight percentage points of the true population percentage? Complete parts (a) through (c) below.
7.1.37-t 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.
Use technology to find the P-value for the hypothesis test described below. The claim is that for a smartphone carrier's data speeds at airports, the mean is μ=18.00 Mbps. The sample size is n=26 and the test statistic is t=−2.454.
8.3.5-t tcdf(2.079,9999,12) answer * 2 tcdf (-9999, -2.454, 25) answer *2
A magazine provided results from a poll of 1000 adults who were asked to identify their favorite pie. Among the 1000 respondents, 14% chose chocolate pie, and the margin of error was given as ±5 percentage points. Given specific sample data, which confidence interval is wider: the 95% confidence interval or the 80% confidence interval? Why is it wider?
7.1.4 A 95% confidence interval must be wider than an 80% confidence interval in order to be more confident that it captures the true value of the population proportion.
Find the critical value zα/2 that corresponds to the given confidence level. 82%
7.1.5 1.34 is the answer
Find the critical value zα/2 that corresponds to the confidence level 89%.
7.1.7-t answer is 1.6
Express the confidence interval 0.666<p<0.888 in the form p±E.
7.1.9
Refer to the accompanying data display that results from a sample of airport data speeds in Mbps. Complete parts (a) through (c) below.
7.2.1 Because the sample size of 50 is greater than 30, the distribution of sample means can be treated as a normal distribution.
A data set includes 110 body temperatures of healthy adult humans having a mean of 98.0°F and a standard deviation of 0.74°F. Construct a 99% confidence interval estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of 98.6°F as the mean body temperature?
7.2.11
A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 16 subjects had a mean wake time of 100.0 min. After treatment, the 16 subjects had a mean wake time of 91.5 min and a standard deviation of 22.7 min. Assume that the 16 sample values appear to be from a normally distributed population and construct a 99% confidence interval estimate of the mean wake time for a population with drug treatments. What does the result suggest about the mean wake time of 100.0 min before the treatment? Does the drug appear to be effective?
7.2.13-t does not include are different. has or.. does include are not different doesn't have
In a test of the effectiveness of garlic for lowering cholesterol, 49 subjects were treated with garlic in a processed tablet form. Cholesterol levels were measured before and after the treatment. The changes (before−after) in their levels of LDL cholesterol (in mg/dL) have a mean of 4.1 and a standard deviation of 17.6. Construct a 95% confidence interval estimate of the mean net change in LDL cholesterol after the garlic treatment. What does the confidence interval suggest about the effectiveness of garlic in reducing LDL cholesterol?
7.2.14 The confidence interval limits contain 0, suggesting that the garlic treatment did not affect the LDL cholesterol levels. depends on the question.
In a study of speed dating, male subjects were asked to rate the attractiveness of their female dates, and a sample of the results is listed below (1=not attractive; 10=extremely attractive). Construct a confidence interval using a 90% confidence level. What do the results tell about the mean attractiveness ratings of the population of all adult females?
7.2.17-tx The results tell nothing about the population of all adult females, because participants in speed dating are not a representative sample of the population of all adult females.
A food safety guideline is that the mercury in fish should be below 1 part per million (ppm). Listed below are the amounts of mercury (ppm) found in tuna sushi sampled at different stores in a major city. Construct a 95% confidence interval estimate of the mean amount of mercury in the population. Does it appear that there is too much mercury in tuna sushi?
7.2.19-t yes, because it is possible that the mean is greater than 1 ppm. Also, at least one of the sample values exceeds 1 ppm, so at least some of the fish have too much mercury
Homework:Section 7.2 (part 2) Homework Listed below are student evaluation ratings of courses, where a rating of 5 is for "excellent." The ratings were obtained at one university in a state. Construct a confidence interval using a 90% confidence level. What does the confidence interval tell about the population of all college students in the state?
7.2.23-t The results tell nothing about the population of all college students in the state, since the sample is from only one university.
An IQ test is designed so that the mean is 100 and the standard deviation is 15 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95% confidence that the sample mean is within 3 IQ points of the true mean. Assume that σ=15 and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation.
7.2.29-t Yes. This number of IQ test scores is a fairly small number.
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.
7.2.3 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 all grade-point averages are to be standardized on a scale between 0 and 6. How many grade-point averages must be obtained so that the sample mean is within 0.012 of the population mean? Assume that a 99% confidence level is desired. If using the range rule of thumb, σ can be estimated as range4=6−04=1.5. Does the sample size seem practical?
7.2.31-t No, because the required sample size is a fairly large number.
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 is not known, and the normal quantile plot of the 17 salaries (in thousands of dollars) of basketball players on a team is as shown.
7.2.5-t
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=4088 thousand dollars, and the histogram of 65 player salaries (in thousands of dollars) of football players on a team is as shown.
7.2.7-t
Here are summary statistics for randomly selected weights of newborn girls: n=223, x=29.6 hg, s=7.9 hg. Construct a confidence interval estimate of the mean. Use a 95% confidence level. Are these results very different from the confidence interval 27.9 hg<μ<31.9 hg with only 15 sample values, x=29.9 hg, and s=3.6 hg?
7.2.9-t No, because the confidence interval limits are similar.
A bottle contains a label stating that it contains pills with 500 mg of vitamin C, and another bottle contains a label stating that it contains pills with 325 mg of aspirin. When testing claims about the mean contents of the pills, which would have more serious implications: rejection of the vitamin C claim or rejection of the aspirin claim? Considering only a type I error and using the same sample size, is it wise to use the same significance level for hypothesis tests about the mean amount of vitamin C and the mean amount of aspirin?
8.1.1 aspirin aspirin vitamin C smaller
Claim: Most adults would erase all of their personal information online if they could. A software firm survey of 499 randomly selected adults showed that 60% of them would erase all of their personal information online if they could. Find the value of the test statistic.
8.1.13
Claim: The mean pulse rate (in beats per minute) of adult males is equal to 69 bpm. For a random sample of 144 adult males, the mean pulse rate is 68.1 bpm and the standard deviation is 11.1 bpm. Find the value of the test statistic.
8.1.15
The test statistic of z=1.95 is obtained when testing the claim that p>0.5. a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed. b. Find the P-value. c. Using a significance level of α=0.01, should we reject H0 or should we fail to reject H0?
8.1.17 right tailed Fail to reject H0. There is not sufficient evidence to support the claim that p>0.2.
The test statistic of z=2.94 is obtained when testing the claim that p≠0.168. a. Identify the hypothesis test as being two-tailed, left-tailed, or right-tailed. b. Find the P-value. c. Using a significance level of α=0.01, should we reject H0 or should we fail to reject H0?
8.1.19 Reject H0. There is sufficient evidence to support the claim that p≠0.168.
The test statistic of z=−2.12 is obtained when testing the claim that p<0.72. a. Using a significance level of α=0.01, find the critical value(s). b. Should we reject H0 or should we fail to reject H0?
8.1.22
The test statistic of z=−3.19 is obtained when testing the claim that p<0.37. a. Using a significance level of α=0.05, find the critical value(s). b. Should we reject H0 or should we fail to reject H0?
8.1.22
The test statistic of z=−1.83 is obtained when testing the claim that p=1/2. a. Using a significance level of α=0.05, find the critical value(s). b. Should we reject H0 or should we fail to reject H0?
8.1.24
Assume a significance level of α=0.05 and use the given information to complete parts (a) and (b) below. Original claim: More than 43% of adults would erase all of their personal information online if they could. The hypothesis test results in a P-value of 0.1923.
8.1.25 Fail to reject H0 because theP-value is greater than α. There is not sufficient evidence to support the claim that the percentage of adults that would erase all of their personal information online if they could is more than 43%.
Assume a significance level of α=0.1 and use the given information to complete parts (a) and (b) below. Original claim: The mean pulse rate (in beats per minute) of a certain group of adult males is 74 bpm. The hypothesis test results in a P-value of 0.0019.
8.1.27 Reject H0 because theP-value is less than or equal to α. There is sufficient evidence to warrant rejection of the claim that the mean pulse rate (in beats per minute) of the group of adult males is 74 bpm
Assume a significance level of α=0.1 and use the given information to complete parts (a) and (b) below. Original claim: The standard deviation of pulse rates of a certain group of adult males is more than 14 bpm. The hypothesis test results in a P-value of 0.3164.
8.1.28
Claim: A minority of adults would erase all of their personal information online if they could. A software firm survey of 540 randomly selected adults showed that 44% of them would erase all of their personal information online if they could. Complete parts (a) and (b) below.
8.1.5 A) P< .5 B)H0:p =.5 H1: p<. 5
Claim: The mean pulse rate (in beats per minute) of adult males is equal to 68.9 bpm. For a random sample of 167 adult males, the mean pulse rate is 67.8 bpm and the standard deviation is 11.2 bpm. Complete parts (a) and (b) below.
8.1.7 A) μ=68.9 bpm B)H0: p= 68.9 H1: p≠ 68.9
Assume that adults were randomly selected for a poll. They were asked if they "favor or oppose using federal tax dollars to fund medical research using stem cells obtained from human embryos." Of those polled, 482 were in favor, 396 were opposed, and 115 were unsure. A politician claims that people don't really understand the stem cell issue and their responses to such questions are random responses equivalent to a coin toss. Exclude the 115 subjects who said that they were unsure, and use a 0.10 significance level to test the claim that the proportion of subjects who respond in favor is equal to 0.5. What does the result suggest about the politician's claim?
8.2.11-t
Use technology to find the P-value for the hypothesis test described below. The claim is that for 12 AM body temperatures, the mean is μ>98.6°F. The sample size is n=8 and the test statistic is t=2.504.
8.3.6-t tcdf( 2.504, 9999, 7)
In a study of cell phone usage and brain hemispheric dominance, an Internet survey was e-mailed to 6978 subjects randomly selected from an online group involved with ears. There were 1328 surveys returned. Use a 0.01 significance level to test the claim that the return rate is less than 20%. Use the P-value method and use the normal distribution as an approximation to the binomial distribution.
8.2.15-t
Consider a drug testing company that provides a test for marijuana usage. Among 322 tested subjects, results from 28 subjects were wrong (either a false positive or a false negative). Use a 0.10 significance level to test the claim that less than 10 percent of the test results are wrong.
8.2.16-t
Trials in an experiment with a polygraph include 97 results that include 23 cases of wrong results and 74 cases of correct results. Use a 0.01 significance level to test the claim that such polygraph results are correct less than 80% of the time. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P-value method. Use the normal distribution as an approximation of the binomial distribution.
8.2.19-t
In a study of 420,089 cell phone users, 133 subjects developed cancer of the brain or nervous system. Test the claim of a somewhat common belief that such cancers are affected by cell phone use. That is, test the claim that cell phone users develop cancer of the brain or nervous system at a rate that is different from the rate of 0.0340% for people who do not use cell phones. Because this issue has such great importance, use a 0.001 significance level. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P-value method and the normal distribution as an approximation to the binomial distribution.
8.2.23-t
An online poll asked: "Do you believe the Loch Ness monster exists?" Among 20,688 responses, 63% were "yes." Use a 0.10 significance level to test the claim that most people believe that the Loch Ness monster exists. How is the conclusion affected by the fact that Internet users who saw the question could decide whether to respond?
8.2.29-t
In a recent court case it was found that during a period of 11 years 888 people were selected for grand jury duty and 39% of them were from the same ethnicity. Among the people eligible for grand jury duty, 80.2% were of this ethnicity. Use a 0.05 significance level to test the claim that the selection process is biased against allowing this ethnicity to sit on the grand jury. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim. Use the P-value method and the normal distribution as an approximation to the binomial distribution.
8.2.31-t
A certain drug is used to treat asthma. In a clinical trial of the drug, 15 of 281 treated subjects experienced headaches (based on data from the manufacturer). The accompanying calculator display shows results from a test of the claim that less than 10% of treated subjects experienced headaches. Use the normal distribution as an approximation to the binomial distribution and assume a 0.05 significance level to complete parts (a) through (e) below.
8.2.5
A poll of 2,087 randomly selected adults showed that 95% of them own cell phones. The technology display below results from a test of the claim that 91% of adults own cell phones. Use the normal distribution as an approximation to the binomial distribution, and assume a 0.01 significance level to complete parts (a) through (e).
8.2.7
In a study of the accuracy of fast food drive-through orders, one restaurant had 33 orders that were not accurate among 383 orders observed. Use a 0.10 significance level to test the claim that the rate of inaccurate orders is equal to 10%. Does the accuracy rate appear to be acceptable?
8.2.9-t calculator STAT TESTS 5
Twelve different video games showing drug use were observed. The duration times of drug use were recorded, with the times (seconds) listed below. What requirements must be satisfied to test the claim that the sample is from a population with a mean greater than 80 sec? Are the requirements all satisfied?
8.3.1
Park officials make predictions of times to the next eruption of a particular geyser, and collect data for the errors (minutes) in those predictions. The display from technology available below results from using the prediction errors to test the claim that the mean prediction error is equal to zero. Comment on the accuracy of the predictions. Use a 0.05 significance level. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim.
8.3.11
A data set about speed dating includes "like" ratings of male dates made by the female dates. The summary statistics are n=190, x=6.82, s=2.01. Use a 0.05 significance level to test the claim that the population mean of such ratings is less than 7.00. Assume that a simple random sample has been selected. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim.
8.3.14-t
In a test of the effectiveness of garlic for lowering cholesterol, 64 subjects were treated with raw garlic. Cholesterol levels were measured before and after the treatment. The changes (before minus after) in their levels of LDL cholesterol (in mg/dL) have a mean of 0.4 and a standard deviation of 1.69. Use a 0.01 significance level to test the claim that with garlic treatment, the mean change in LDL cholesterol is greater than 0. What do the results suggest about the effectiveness of the garlic treatment? Assume that a simple random sample has been selected. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim.
8.3.15-t
different video games showing violence were observed. The duration times of violence (in seconds) were recorded. When using this sample for a t test of the claim that the population mean is greater than 88 sec, what does df denote, and what is its value?
8.3.2
In the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. The treatment group consisted of 201,229 children given the Salk vaccine for polio, and the other 200,745 children were given a placebo. Among those in the treatment group, 33 developed polio, and among those in the placebo group, 115 developed polio. If we want to use the methods for testing a claim about two population proportions to test the claim that the rate of polio is less for children given the Salk vaccine, are the requirements for a hypothesis test satisfied? Explain.
9.1.1 The requirements are satisfied; the samples are simple random samples that are independent, and for each of the two groups, the number of successes is at least 5 and the number of failures is at least 5.
A study was conducted to determine the proportion of people who dream in black and white instead of color. Among 324 people over the age of 55, 64 dream in black and white, and among 312 people under the age of 25, 18 dream in black and white. Use a 0.01 significance level to test the claim that the proportion of people over 55 who dream in black and white is greater than the proportion for those under 25. Complete parts (a) through (c) below.
9.1.11-t
Rhino viruses typically cause common colds. In a test of the effectiveness of echinacea, 42 of the 49 subjects treated with echinacea developed rhinovirus infections. In a placebo group, 83 of the 99 subjects developed rhinovirus infections. Use a 0.05 significance level to test the claim that echinacea has an effect on rhinovirus infections. Complete parts (a) through (c) below.
9.1.15-t
In a study of treatments for very painful "cluster" headaches, 151 patients were treated with oxygen and 156 other patients were given a placebo consisting of ordinary air. Among the 151 patients in the oxygen treatment group, 121 were free from headaches 15 minutes after treatment. Among the 156 patients given the placebo, 24 were free from headaches 15 minutes after treatment. Use a 0.05 significance level to test the claim that the oxygen treatment is effective. Complete parts (a) through (c) below.
9.1.19-t
In a large clinical trial, 398,388 children were randomly assigned to two groups. The treatment group consisted of 200,437 children given a vaccine for a certain disease, and 32 of those children developed the disease. The other 197,951 children were given a placebo, and 123 of those children developed the disease. Consider the vaccine treatment group to be the first sample. Identify the values of n1, p1, q1, n2, p2, q2, p, and q.
9.1.2
In a random sample of males, it was found that 28 write with their left hands and 207 do not. In a random sample of females, it was found that 72 write with their left hands and 433 do not. Use a 0.01 significance level to test the claim that the rate of left-handedness among males is less than that among females. Complete parts (a) through (c) below.
9.1.21-t
In a large clinical trial, 395,202 children were randomly assigned to two groups. The treatment group consisted of 197,580 children given a vaccine for a certain disease, and 36 of those children developed the disease. The other 197,622 children were given a placebo, and 117 of those children developed the disease. Consider the vaccine treatment group to be the first sample. Complete parts (a) through (d) below.
9.1.4 A hypothesis test is better. -p-value and critical value -The assumption that the two population proportions are equal -estimated the values of the population proportions c. 0.01 = 98% 0.05 = 90% 0.10= 80% d. -does not contain -0 -appears to be -only of values less than -0 - less than - sufficient
A newspaper published an article about a study in which researchers subjected laboratory gloves to stress. Among 255 vinyl gloves, 60% leaked viruses. Among 255 latex gloves, 13% leaked viruses. Using the accompanying display of the technology results, and using a 0.01 significance level, test the claim that vinyl gloves have a greater virus leak rate than latex gloves. Let vinyl gloves be population 1.
9.1.5
Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, and then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Among 2177 passenger cars in a particular region, 249 had only rear license plates. Among 375 commercial trucks, 53 had only rear license plates. A reasonable hypothesis is that commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars. Use a 0.01 significance level to test that hypothesis. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval.
9.1.7-t
Determine whether the samples are independent or dependent. A data set includes the morning and evening temperature for the last 180 days.
9.2.1 the samples are dependent because there is a natural pairing between the 2 samples.
Given in the table are the BMI statistics for random samples of men and women. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. Use a 0.01 significance level for both parts.
9.2.11-t
Listed in the data table are IQ scores for a random sample of subjects with medium lead levels in their blood. Also listed are statistics from a study done of IQ scores for a random sample of subjects with high lead levels. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. LOADING... Click the icon to view the data table of IQ scores.
9.2.12-t
Listed below are time intervals (min) between eruptions of a geyser. Assume that the "recent" times are within the past few years, the "past" times are from around 20 years ago, and that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Does it appear that the mean time interval has changed? Is the conclusion affected by whether the significance level is 0.10 or 0.01?
9.2.19-e
The following data lists the ages of a random selection of actresses when they won an award in the category of Best Actress, along with the ages of actors when they won in the category of Best Actor. The ages are matched according to the year that the awards were presented. Complete parts (a) and (b) below.
9.3.5-t
Large samples of women and men are obtained, and the hemoglobin level is measured in each subject. Here is the 95% confidence interval for the difference between the two population means, where the measures from women correspond to population 1 and the measures from men correspond to population 2: −1.76 g/dL<μ1−μ2<−1.62 g/dL. Complete parts (a) through (c) below.
9.2.2 does not include 0 is There is 95% confidence that the interval from −1.76 g/dL to −1.62 g/dL actually contains the value of the difference between the two population means μ1−μ2. all positive numbers
Listed below are the numbers of years that archbishops and monarchs in a certain country lived after their election or coronation. Assume that the two samples are independent simple random samples selected from normally distributed populations. Do not assume that the population standard deviations are equal. Use a 0.10 significance level to test the claim that the mean longevity for archbishops is less than the mean for monarchs after coronation. All measurements are in years. LOADING... Click the icon to view the table of longevities of archbishops and monarchs.
9.2.21-t
For conducting a two-tailed hypothesis test with a certain data set, using the smaller of n1−1 and n2−1 for the degrees of freedom results in df=11, and the corresponding critical values are t=±2.201. Using the formula for the exact degrees of freedom results in df=19.063, and the corresponding critical values are t=±2.093. How is using the critical values of t=±2.201 more "conservative" than using the critical values of ±2.093?
9.2.4 Using the critical values of t=±2.201 is less likely to lead to rejection of the null hypothesis than using the critical values of ±2.093.
Data on the weights (lb) of the contents of cans of diet soda versus the contents of cans of the regular version of the soda is summarized to the right. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. Use a 0.01 significance level for both parts.
9.2.5-t
Data on the weights (lb) of the contents of cans of diet soda versus the contents of cans of the regular version of the soda is summarized to the right. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. Use a 0.05 significance level for both parts.
9.2.5-t
A study was done using a treatment group and a placebo group. The results are shown in the table. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below. Use a 0.01 significance level for both parts.
9.2.6-t
A study was done on proctored and nonproctored tests. The results are shown in the table. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below.
9.2.7-t
Researchers conducted a study to determine whether magnets are effective in treating back pain. The results are shown in the table for the treatment (with magnets) group and the sham (or placebo) group. The results are a measure of reduction in back pain. Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Complete parts (a) and (b) below.
9.2.9-t
Which of the following statements are true concerning the mean of the differences between two dependent samples (matched pairs)?
9.3.1 If one has fifteen matched pairs of sample data, there is a loose requirement that the fifteen differences appear to be from a normally distributed population. If one wants to use a confidence interval to test the claim that μd>0 with a 0.05 significance level, the confidence interval should have a confidence level of 90%.
Which of the following statements are true concerning the mean of the differences between two dependent samples (matched pairs)?
9.3.1 If one has fifteen matched pairs of sample data, there is a loose requirement that the fifteen differences appear to be from a normally distributed population. If one wants to use a confidence interval to test the claim that μd>0 with a 0.10 significance level, the confidence interval should have a confidence level of 80%.
Listed below are systolic blood pressure measurements (mm Hg) taken from the right and left arms of the same woman. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Use a 0.10 significance level to test for a difference between the measurements from the two arms. What can be concluded?
9.3.10-t
Researchers collected data on the numbers of hospital admissions resulting from motor vehicle crashes, and results are given below for Fridays on the 6th of a month and Fridays on the following 13th of the same month. Use a 0.05 significance level to test the claim that when the 13th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected.
9.3.12
Data on the numbers of hospital admissions resulting from motor vehicle crashes are given below for Fridays on the 6th of a month and Fridays on the following 13th of the same month. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Construct a 95% confidence interval estimate of the mean of the population of differences between hospital admissions. Use the confidence interval to test the claim that when the 13th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected.
9.3.13-t
The data below are yields for two different types of corn seed that were used on adjacent plots of land. Assume that the data are simple random samples and that the differences have a distribution that is approximately normal. Construct a 95% confidence interval estimate of the difference between type 1 and type 2 yields. What does the confidence interval suggest about farmer Joe's claim that type 1 seed is better than type 2 seed?
9.3.14-t
Listed below are body temperatures from five different subjects measured at 8 AM and again at 12 AM. Find the values of d and sd. In general, what does μd represent?
9.3.2
Listed below are the numbers of words spoken in a day by each member of eight different randomly selected couples. Complete parts (a) and (b) below. In this example, μd is the mean value of the differences d for the population of all pairs of data, where each individual difference d is defined as the words spoken by the male minus words spoken by the female. What are the null and alternative hypotheses for the hypothesis test?
9.3.8-t
Several students were tested for reaction times (in thousandths of a second) using their right and left hands. (Each value is the elapsed time between the release of a strip of paper and the instant that it is caught by the subject.) Results from five of the students are included in the graph to the right. Use a 0.10 significance level to test the claim that there is no difference between the reaction times of the right and left hands. LOADING... Click the icon to view the reaction time data table.
9.3.9
Which of the following is NOT a requirement of testing a claim about the mean of the differences from dependent samples?
9.3.RA-2 The degrees of freedom are n−2.
