Introduction to Statistics: Chapter 9 Homework (Inferring Population Means)
A random sample of 10 colleges was taken. A 95% confidence interval for the mean admission rate was (52.8%, 75.0%). The rates of admission were Normally distributed. Which of the following statements is the correct interpretation of the confidence level, and which is the correct interpretation of the confidence interval? a. We are confident that the mean admission rate is between 52.8% and 75.0%. b. In about 95% of all samples of 10 colleges, the confidence interval will contain the population mean admission rate.
Statement (b) correctly interprets the confidence level and statement (a) correctly interprets the confidence interval.
The accompanying table shows the technology output for a two-sample t-interval for the number of TVs owned in households of random samples of students at two different community colleges. Each individual was randomly chosen independently of the others. One of the schools is in a wealthy community (MC), and the other (OC) is in a less wealthy community. Complete parts (a) through (c) below.
a. Are the conditions for using a confidence interval for the difference between two means met? Select all that apply. Ans: Yes, all conditions are met. b. State the interval in a clear and correct sentence. Complete the sentence below. Ans: I am 95% confident that the true mean difference in the number of TVs owned in households of students at the community colleges (OC minus MC) is between −0.396 and 1.129 TVs. c. Does the interval capture 0? Explain what that shows. Ans: The interval for the difference captures 0. This implies that it is plausible that the population means are the same. NOTE: Use the technology output.
9.3 The median undergraduate grade point average (GPA) for students accepted at a random sample of 8 medical schools in a country was taken. The mean GPA for these accepted students was 3.55 with a standard error of 0.06. The distribution of undergraduate GPAs is Normal. Complete parts (a) and (b).
a. Decide which of the following statements is worded correctly for the confidence interval. Fill in the blanks for the correctly worded one. Explain the error for the ones that are incorrectly worded. Select the correct choice below and fill in the answer boxes to complete your choice. Ans: We are 95% confident that the population mean is between 3.41 and 3.69. Explain the error for the choice "We are 95% confident that the sample mean is between _____ and _____." Choose the correct answer below. Ans: The wording is not correct because it references the sample mean. The value of the sample mean is known exactly and does not need a confidence interval. Explain the error for the choice "We are 95% confident that the population mean is between _____ and _____." Choose the correct answer below. Ans: There is no error. The choice is worded correctly. Explain the error for the choice "There is a 95% probability that the population mean is between _____ and _____." Choose the correct answer below. Ans: The wording is not correct because it indicates that there is a "95% probability" that the population mean is in the interval, when it should indicate that there is "95% confidence" instead. b. Based on your confidence interval, would you believe that the population mean GPA is 3.63? Why or why not? Ans: Since 3.63 is within the bounds of the confidence interval, it is plausible that the population mean GPA is 3.63.
The acceptance rate for a random sample of 20 medical schools in a country was taken. The mean acceptance rate for this sample was 5.87% with a standard error of 0.43. Assume the distribution of acceptance rates is Normal. Complete parts (a) and (b).
a. Decide which of the following statements is worded correctly for the confidence interval. Fill in the blanks for the correctly worded one. Explain the error for the ones that are incorrectly worded. Select the correct choice below and fill in the answer boxes to complete your choice. Ans: We are 95% confident that the population mean is between 4.97% and 6.77%. Explain the error for the choice "We are 95% confident that the sample mean is between _____ and _____." Choose the correct answer below. Ans: The wording is not correct because it references the sample mean. The value of the sample mean is known exactly and does not need a confidence interval. Explain the error for the choice "We are 95% confident that the population mean is between _____ and _____." Choose the correct answer below. Ans: There is no error. The choice is worded correctly. Explain the error for the choice "There is a 95% probability that the population mean is between _____ and _____." Choose the correct answer below. Ans: The wording is not correct because it indicates that there is a "95% probability" that the population mean is in the interval, when it should indicate that there is "95% confidence" instead. b. Based on your confidence interval, would you believe that the average acceptance rate for medical schools is 6.48%? Why or why not? Ans: Since 6.48% is within the bounds of the confidence interval, it is plausible that the population mean GPA is 6.48%.
In Country A, the population mean height for 3-year-old boys is 37 inches. Suppose a random sample of 15 3-year-old boys from Country B showed a sample mean of 36.1 inches with a standard deviation of 4 inches. The boys were independently sampled. Assume that heights are Normally distributed in the population. Complete parts a through c below.
a. Determine whether the population mean for Country B boys is significantly different from the Country A mean. Use a significance level of 0.05. Ans: H0: μ = 37 and Ha: μ ≠ 37; t= -0.87; p=0.398; Do not reject H0. There is no reason to believe that 37 in. is not the population mean at a significance level of 0.05. b. Now suppose the sample consists of 30 boys instead of 15 and repeat the test. Ans: t=1.23; p=0.228; Do not reject H0. There is no reason to believe that 37 in. is not the population mean at a significance level of 0.05. c. Explain why the t-values and p-values for parts a and b are different. Choose the correct answer below. Ans: A larger n causes a smaller standard error (narrower sampling distribution) with less area in the tails, as shown by the smaller p-value.
In Country A, the population mean height for 10-year-old girls is 54.9 inches with a standard deviation of 2.1 inches. Suppose a random sample of 15 10-year-old girls from Country B is taken and that these girls had a sample mean height of 54.7 inches with a standard deviation of 2.6 inches. Assume that heights are Normally distributed. Complete parts (a) through (c) below.
a. Determine whether the population mean for height for 10-year-old girls from Country B is significantly different from the Country A population mean. Use a significance level of 0.05. Ans: H0: μ = 54.9 and Ha: μ ≠ 54.9; The sample is random and the observations are independent. The distribution of the heights is Normally distributed.; t=-0.30; p = 0.770; Since the p-value is greater than the significance level, do not reject H0. There is insufficient evidence to conclude that the height for 10-year-old girls from Country B is significantly different from the Country A population mean at a significance level of 0.05. b. Now suppose the sample consists of 45 girls instead of 15. Repeat the test. Ans: H0: μ = 54.9 and Ha: μ ≠ 54.9; t=-0.52; p=0.608; Since the p-value is greater than the significance level, do not reject H0. There is insufficient evidence to conclude that the height for 10-year-old girls from Country B is significantly different from the Country A population mean at a significance level of 0.05. c. Explain why the t-values and p-values for parts a and b are different. Choose the correct answer below. Ans: A larger n causes a smaller standard error (narrower sampling distribution) with less area in the tails, as shown by the smaller p-value.
According to an online source, the average shower in a country lasts 8.6 minutes. Assume that this is correct and assume the standard deviation of 3 minutes. Complete parts (a) through (c) below.
a. Do you expect the shape of the distribution of shower lengths to be Normal, right-skewed, or left-skewed? Explain your reasoning. Ans: The distribution will be right-skewed because the values will be mostly small but there will be a few very large values. b. Suppose we survey a random sample of 100 people to find the length of their last shower. We calculate the mean length from this sample and record the value. We repeat this 500 times. What will be the shape of the distribution of these sample means? Ans: The distribution will be approximately Normal because the values will be distributed symmetrically. c. Refer to part (b). What will be the mean and standard deviation of the distribution of these sample means? Ans: The mean will be 8.6 minutes and the standard deviation will be 3.3 minutes.
A random sample of 88 men's resting pulse rates showed a mean of 73.4 beats per minute and standard deviation of 15.4. Assume that pulse rates are Normally distributed. Use the table to complete parts (a) through (c) below.
a. Find a 95% confidence interval for the population mean pulse rate of men, and report it in a sentence. Choose the correct answer below and, if necessary, fill in the answer boxes to complete your choice. Ans: We are 95% confident that the population mean resting pulse rate is between 60.5 and 86.3. b. Find a 90% confidence interval and report it in a sentence. Choose the correct answer below and, if necessary, fill in the answer boxes to complete your choice. Ans: We are 90% confident that the population mean resting pulse rate is between 63.1 and 83.7. c. Which interval is wider and why? Ans: The 95% interval is wider because it has a greater confidence level, and so we use a bigger value of t*, which creates a wider interval. NOTE: Use the T-distribution table.
The mean age of all 2571 students at a small college is 22.7 years with a standard deviation of 3.5 years, and the distribution is right-skewed. A random sample of 4 students' ages is obtained, and the mean is 23.3 with a standard deviation of 2.9 years. Complete parts (a) through (c) below.
a. Find μ, σ, s, or x-bar. μ=22.7; σ=3.5; s=23.3; x-bar=2.9 b. Is μ a parameter or a statistic? Ans: The value of μ is a parameter because it is found from the population. c. Are the conditions for using the CLT (Central Limit Theorem) fulfilled? Select all that apply. Ans: No, because the large sample condition is not satisfied. What would be the shape of the approximate sampling distribution of many means, each from a sample of 4 students? Ans: Right-skewed
The mean age of all 607 used cars for sale in a newspaper one Saturday last month was 7.3 years, with a standard deviation of 6.8 years. The distribution of ages is right-skewed. For a study to determine the reliability of classified ads, a reporter randomly selects 40 of these used cars and plans to visit each owner to inspect the cars. He finds that the mean age of the 40 cars he samples is 7.8 years and the standard deviation of those 40 cars is 4.9 years. Complete parts a through c.
a. From the problem statement, which of the values 7.3, 6.8, 7.8, and 4.9 are parameters and which are statistics? The value 7.3 is a parameter. The value 6.8 is a parameter. The value 77.8 is a statistic. The value 4.9 is a statistic. b. Find μ, σ, s, or x-bar. μ=7.3; σ=6.8; s=77.8; x-bar=4.9 c. Are the conditions for using the CLT (Central Limit Theorem) fulfilled? Ans: Yes, all the conditions for using the CLT are fulfilled. What would be the shape of the approximate sampling distribution of a large number of means, each from a sample of 40 cars? Ans: Normal
A random sample of 14 college women and a random sample of 19 college men were separately asked to estimate how much they spent on clothing in the last month. The accompanying table shows the data. Suppose the null hypothesis that the mean amount spent by men and the mean amount spent by women for clothing are the same could not be rejected using two-tailed test at a significance level of 0.05. Complete parts a through c below.
a. If a 95% confidence interval for the difference between means is found, would it capture 0? Explain. Ans: The 95% interval would capture 0, because the hypothesis that the mean amounts spent on clothing are the same could not be rejected by the hypothesis test. b. If a 99% confidence interval for the difference between means is found, would it capture 0? Explain. Ans: Yes, because a 99% interval is wider than a 95% interval and centered at the same value, and based on the results of the hypothesis test, a 95% interval would capture 0. c. Find a 95% confidence interval for the difference between means, and explain what it shows. Ans: The 95% confidence interval for women−men is (−48.9, 54.6).; Because the interval does capture 0, the hypothesis that the mean difference in spending on clothing is 0 cannot be rejected, which shows that the hypothesis that the means are the same cannot be rejected, and there is not a significant difference. Note: Use the data table.
A random sample of male college baseball players and a random sample of male college soccer players were obtained independently and weighed. The accompanying table shows the unstacked weights (in pounds). The distributions of both data sets suggest that the population distributions are roughly Normal. Using a two-tailed test with a significance level of 0.05, the null hypothesis that the mean weights of soccer and baseball players are equal is rejected. Complete parts (a) through (c) below.
a. If a 95% confidence interval for the difference between means was found, would it capture 0? Explain. Ans: The 95% interval would not capture 0, because the hypothesis that the mean weights are the same is rejected by the hypothesis test. b. If a 90% confidence interval for the difference between means was found, would it capture 0? Explain. Ans: No, because a 90% interval is narrower than a 95% interval and centered at the same value, and based on the results of the hypothesistest, a 90% interval would not capture 0. c. Find a 95% confidence interval for the difference between means, and explain what it shows. Ans: The 95% confidence interval for the difference between means (Baseball minus Soccer) is (11.03, 26.01).; Because the interval does not capture 0, it shows that the mean weights for soccer and baseball players are significantly different. NOTE: Use the data table.
9.5 State whether each situation has independent or paired (dependent) samples. a. A researcher wants to compare the hand-eye coordination of men and women. She finds a random sample of 50 men and 50 women, and measures their hand-eye coordination. b. A researcher wants to know whether professors with tenure have fewer office hours than professors without tenure. She observes the number of office hours for professors with and without tenure.
a. Independent samples b. Independent samples
The accompanying data table shows reaction distances in centimeters for the dominant hand for a random sample of 40 independently chosen college students. Smaller distances indicate quicker reactions. Complete parts (a) through (c) below.
a. Make a graph of the distribution of the sample, and describe its shape. Make a graph of the distribution. For values on the boundary of two bins, place that value into the bin on the left. Choose the correct graph below. Ans: Slightly right-skewed graph; The distribution is slightly skewed to the right. b. Find, report, and interpret a 95% confidence interval for the population mean. Ans: The 95% confidence interval is 7.91 cm to 10.86 cm.; If the sample were repeated many times, 95% of all the confidence intervals would contain the true population mean reaction distance. c. Suppose a professor said that the population mean should be 10 centimeters. Test the hypothesis that the population mean is not 10 cm, with a significance level of 0.05. Determine the null and alternative hypotheses. Choose the correct answer below. Ans: H0: μ = 10 and Ha: μ ≠ 10; The test statistic is −0.84.; The p-value is 0.406.; Since the p-value is greater than the significance level, do not reject H0. There is insufficient evidence to conclude that the population mean is different from 10 cm at a significance level of 0.05. NOTE: Use the data table.
The weights of four randomly and independently selected bags of potatoes labeled 2020 pounds were found to be 21.6, 22.1, 20.4, and 22. Assume Normality. a. Using a two-sided alternative hypothesis, should you be able to reject the hypothesis that the population mean is 20 pounds using a significance level of 0.05? Why or why not? The confidence interval is reported here: I am 95% confident the population mean is between 20.3 and 22.8 pounds. b. Test the hypothesis that the population mean is not 20. Use a significance level of 0.05. c. Choose one of the following conclusions: i. We cannot reject a population mean of 20 pounds. ii. We can reject a population mean of 20 pounds. iii. The population mean is 21.525 pounds.
a. Reject the hypothesis, because the interval does not contain the proposed mean. b. H0: μ = 20 and Ha: μ ≠ 20; The test statistic is 3.91.; The p-value is 0.030. c. We can reject a population mean of 20 pounds.
A hamburger chain sells large hamburgers. When we take a sample of 30 hamburgers and weigh them, we find that the mean is 0.55 pounds and the standard deviation is 0.3 pound. a. A technology input menu for calculating a confidence interval requires a sample size, a sample mean, and a sample standard deviation. State how you would fill in these numbers. b. Using the accompanying technology output, report the confidence interval in a carefully worded sentence.
a. Sample Size: 30; Sample mean: .55; Standard Deviation: .3000 b. Choose the correct interpretation of the confidence interval below and fill in the answer boxes to complete your choice. Ans: We are 95% confident that the population mean is between .4380 and .6620. NOTE: Use the technology output.
9.1 A study of all the students at a small college showed a mean age of 20.7 and a standard deviation of 1 years. a. Are these numbers statistics or parameters? Explain. b. Label both numbers with their appropriate symbol (such as x-bar, μ, s, or σ).
a. The numbers are parameters because they are for all the students, not a sample. b. μ=20.7; σ=1
A survey of 100 random full-time students at a large university showed the mean number of semester units that students were enrolled in was 10.4 with a standard deviation of 2 units. a. Are these numbers statistics or parameters? Explain. b. Label both numbers with their appropriate symbol (such as x-bar, μ, s, or σ).
a. The numbers are statistics because they are for a sample of students, not all students. b. x-bar=10.4; s = 2
According to a 2018 magazine, the average income in a state is $50,682. Suppose the standard deviation is $4,000 and the distribution of income is right-skewed. Repeated random samples of 400 are taken, and the sample mean income is calculated for each sample. Complete parts (a) through (c).
a. The population distribution is right-skewed. Will the distribution of sample means be Normal? Why or why not? Ans: Since the conditions to the Central Limit Theorem are satisfied, the distribution is Normal because the sample size is large. b. Find and interpret a z-score that corresponds with a sample mean of $50,082. Ans: Since the standard error of the sampling distribution of the means is SE=$200, the z-score is −3. So, $50,082 is 3 standard error(s) below the average income in the state. c. Would it be unusual to find a sample mean of $51,870? Why or why not? Ans: It would be unusual, because $51,870 is between 5 and 6 standard error(s) away from the mean. A random sample with a z-score that far from zero is highly unlikely.
9.2 Some sources report that the weights of full-term newborn babies in a certain town have a mean of 7 pounds and a standard deviation of 1.2 pounds and are normally distributed. a. What is the probability that one newborn baby will have a weight within 1.2 pounds of the mean—that is, between 5.8 and 8.2 pounds, or within one standard deviation of the mean? b. What is the probability that the average of four babies' weights will be within 1.2 pounds of the mean; will be between 5.8 and 8.2 pounds? c. Explain the difference between (a) and (b).
a. The probability is 0.6827 b. The probability is 0.9545. c. The distribution of means is taller and narrower than the original distribution. Therefore, the distribution of means will have more observations located closer to the center of the distribution.
A random sample of 40 12th-grade students was asked how long it took to get to school. The sample mean was 15.5 minutes and the sample standard deviation was 11.1 minutes. a. Find a 95% confidence interval for the population mean time it takes 12th-grade students to get to school. b. Would a 90% confidence interval based on this sample data be wider or narrower than the 95% confidence interval? Explain. Check your answer by constructing a 90% confidence interval and comparing the width of the interval with the width of the 95% confidence interval you found in part (a).
a. The 95% confidence interval is from 11.95 minutes to 19.05 minutes. b. The 95% interval is wider because it has a greater confidence level, and therefore uses a larger critical value, which creates a wider interval.; The 90% confidence interval is from 12.54 minutes to 18.46 minutes.; This interval is narrower than the interval in part a.
a. A researcher collects a sample of 18 measurements from a population and wishes to find a 99% confidence interval for the population mean. What value should he use for t*? b. If he instead decides to use a 95% confidence interval, will the interval be wider or be narrower or stay the same? Why?
a. To find a 99% confidence interval for the population mean, he should use 2.898 for t*. b. The interval will be narrower because t* is smaller, which makes the margin of error smaller. NOTE: Use the T-distribution table.
A statistics student collected data on the prices of the same items at a military commissary and a nearby corporate store. The items were matched for content, manufacturer, and size and were priced separately. a. Report and compare the sample means. b. Assume that they are a random sample of items, and use a significance level of 0.05 to test the hypothesis that the military commissary has a lower mean price. Assume that the population distribution of differences is approximately Normal.
a. x-bar corporate=3.64; x-bar navy=3.06; The sample mean for the corporate store is greater. b. Determine the hypotheses for this test. Let μdifference be the population mean difference between the corporate price and the commissary price (μdifference=μCorporate−μNavy). Choose the correct answer below.; H0: μdifference=0;μdifference>0; t=3.48; p-value=0.001; Reject H0. There is sufficient evidence to conclude that the military commissary has a lower mean price.
A 95% confidence interval for the ages of six consecutive presidents at their inaugurations is about (51.7,59.3). Either interpret the interval or explain why it should not be interpreted.
It should not be interpreted. The data are not a random sample and so inference based on a confidence interval is not possible.
Thirty GPAs from a randomly selected sample of statistics students at a college are given in the accompanying table. Assume that the population distribution is approximately Normal. The technician in charge of records claimed that the population mean GPA for the whole college is 2.81. A one-sided test with a significance level of 0.05 had previously been performed on the data and the null hypothesis was rejected. Use the data to find a 95% confidence interval for the mean GPA. If a two-sided alternative had been used with a significance level of 0.05, would the hypothesized mean of 2.81 have been rejected?
First, find the 95% confidence interval for the mean GPA. Ans: The confidence interval is 2.97 to 3.23. Next, test the two-sided alternative with a significance level of 0.05. Determine the null and alternative hypotheses. Choose the correct answer below. Ans: H0: μ = 28.1 and Ha: μ ≠ 28.1 The test statistic is 4.46.; The p-value is 0.00.; Since the p-value is less than or equal to the significance level, reject H0. There is sufficient evidence to conclude that the mean GPA for statistics students is different from 2.81 at a significance level of 0.05. NOTE: Use the data table.
Using data from a national health survey, researchers looked at the pulse rate for nearly 800 people to see whether it is plausible that men and women have the same population mean. The data are random and independent. Technology output is shown in the accompanying table. Complete parts (a) through (c) below.
a. Are the conditions for using a confidence interval for the difference between two means met? Select all that apply. Ans: Yes, all conditions are met. b. State the interval in a clear and correct sentence. Complete the sentence below. Ans: I am 95% confident that the true mean difference in pulse rate (women minusmen) is between 2.116 and 5.822 beats per minute. c. Does the interval capture 0? Explain what that shows. Ans: The interval for the difference does not capture 0. This implies that it is not plausible that the population means are the same. NOTE: Use the technology output.
Chpater 9 Review For parts (a) through (c) below, choose a test for each situation: one-sample t-test, two-sample t-test, paired t-test, and no t-test.
a. A random sample of university students who are majoring in statistics are asked about their credit hours. Our goal is to determine whether the mean credit hours for statistics majors is significantly different from the population mean credit hours for all the students at the university. Ans: This situation should use a one-sample t-test because there is only one sample. b. Students observe the class sizes for a random sample of male professors and a random sample of untenured professors. Ans: This situation should use a two-sample t-test because there are two independent samples. c. A researcher goes to the parking lot of a large grocery chain and observes whether each person is male or female and whether they return the cart to the correct spot before leaving (yes or no). Ans: This situation should use no t-test because the data are categorical.
According to an online source, the mean time spent on smartphones daily by adults in a country is 2.65 hours. Assume that this is correct and assume the standard deviation is 1.2 hours. Complete parts (a) and (b) below.
a. Suppose 150 adults in the country are randomly surveyed and asked how long they spend on their smartphones daily. The mean of the sample is recorded. Then we repeat this process, taking 1000 surveys of 150 adults in the country. What will be the shape of the distribution of these sample means? Ans: The distribution will be approximately Normal because the values will be distributed symmetrically. b. Refer to part (a). What will be the mean and standard deviation of the distribution of these sample means? Ans: The mean will be 2.65 hours and the standard deviation will be .10 hours.
A fast-food chain sells drinks that they call HUGE. When we take a sample of 36 drinks and weigh them, we find that the mean is 36.7 ounces with a standard deviation of 1.8 ounces. a. A technology input menu for calculating a confidence interval requires the sample mean, x-bar, the sample standard deviation, Sx, and the sample size, n. State how you would fill in these numbers. b. Report the confidence interval in a carefully worded sentence.
a. x-bar: 36.7; Sx: 1.8; n:36 b. Choose the correct interpretation of the confidence interval below and fill in the answer boxes to complete your choice. Ans: We are 95% confident that the population mean is between 36.091 and 37.309.
A body mass index of more than 25 is considered unhealthful. The technology output given is from 50 randomly and independently selected people from a health agency's study. Test the hypothesis that the mean BMI is more than 25 using a significance level of 0.05. Assume that conditions are met.
H0: μ = 25 and Ha: μ > 25; The test statistic is 2.32.; The p-value is .012.; Since the p-value is less than or equal to the significance level, reject H0. There is sufficient evidence to conclude that the mean BMI is more than 25 using a significance level of 0.05.
State whether each situation has independent or paired (dependent) samples. a. A researcher wants to know whether men and women at a particular amusement park have different mean cholesterol levels. She gathers two random samples (one of cholesterol levels from 100 men and the other from 100 women.) b. A researcher wants to know whether couples have different mean cholesterol levels. He collects a sample of couples and has each person report his or her cholesterol level.
a. Independent samples b. Paired (dependent) samples
A human resources manager for a large company takes a random sample of 50 employees from the company database. She calculates the mean time that they have been employed. She records this value and then repeats the process: She takes another random sample of 50 names and calculates the mean employment time. After she has done this 1000 times, she makes a histogram of the mean employment times. Is this histogram a display of the population distribution, the distribution of a sample, or the sampling distribution of means?
The histogram is a display of the sampling distribution of means.
One of the histograms given below is a histogram of a sample (from a population with a skewed distribution), one is the distribution of many means of repeated random samples of size 5, and one is the distribution of repeated means of random samples of size 25; all the samples are from the same population. State which is which and how you know.
The histogram that shows the simple sample is Histogram A, because the distribution has the largest standard deviation and is strongly skewed. The histogram that shows the means of samples of size 5 is Histogram C, because the distribution has neither the largest nor the smallest standard deviation and is slightly skewed. The histogram that shows the means of samples of size 25 is Histogram B, because the distribution has the smallest standard deviation and is approximately Normal. NOTE: Please view the histograms.
A researcher collects a sample of 10 measurements from a population and wishes to find a 90% confidence interval for the population mean. What value should he use for t*? (Recall that the number of degrees of freedom for a one-sample t-test is given by df=n−1, where n is the sample size.)
To find a 90% confidence interval for the population mean, he should use 1.833 for t*. NOTE: Use the T-distribution table.
Ten people went on a diet for a month. The weight losses experienced (in pounds) are given below. The negative weight loss is a weight gain. Test the hypothesis that the mean weight loss was more than 0, using a significance level of 00.05. Assume the population distribution is Normal. 2, 7, 11, 0, 3, 6, 6, 4, 2, and −3
H0: μ = 0 and Ha: μ > 0; The test statistic is 3.05.; The p-value is 0.007.; Since the p-value is less than or equal to the significance level, reject H0. There is sufficient evidence to conclude that the population mean is more than 0 pounds at a significance level of 0.05.
A health agency suggested that a healthy total cholesterol measurement should be 200 mg/dL or less. Records from 50 randomly and independently selected people from a study conducted by the agency showed the results in the technology output given below. Test the hypothesis that the mean cholesterol level is more than 200 using a significance level of 0.05. Assume that conditions are met.
H0: μ = 200 and Ha: μ > 200; The test statistic is 1.45.; The p-value is 0.077.; Since the p-value is greater than the significance level, do not reject H0. There is insufficient evidence to conclude that the mean cholesterol level is more than 200 mg/dL at a significance level of 0.05.
9.4 A random sample of 10 independent healthy people showed the body temperatures given below (in degrees Fahrenheit). Test the hypothesis that the population mean is not 98.6°F, using a significance level of 0.05. 98.5, 98.7, 99.0, 96.9, 98.9, 98.8, 97.9, 99.4, 98.8, and 97.1
H0: μ = 98.6 and Ha: μ ≠ 98.6 The sample is random and the observations are independent. The distribution of the sample is approximately Normal.; t=-.076; p-value=0.466; Since the p-value is greater than the significance level, do not reject H0. There is insufficient evidence to conclude that the population mean is not 98.6°F at a significance level of 0.05.
Several times during the year, an organization takes random samples from the population. One such survey, based on a large (several thousand) sample of randomly selected households, estimates the mean retirement income to be $21,201 per year. Suppose we were to make a histogram of all of the retirement incomes from this sample. Would the histogram be a display of the population distribution, the distribution of a sample, or the sampling distribution of means?
The histogram would be a display of the distribution of a sample.
Suppose that 200 statistics students each took a random sample (with replacement) of 100 students at their college and recorded the ages of the students in their sample. Then each student used his or her data to calculate a 95% confidence interval for the mean age of all students at the college. How many of the 200 intervals would you expect to capture the true population mean age, and how many would you expect not to capture the true population mean? Explain by showing your calculation.
The number of intervals expected to capture the true population mean is 190.; The number of intervals expected to not capture the true population mean is 10.; The expression 0.95(200) can be used to find the number of intervals expected to capture the true population mean.
For parts (a) through (c) below, choose a test for each situation: one-sample t-test, two-sample t-test, paired t-test, and no t-test.
a. A random sample of office supply stores is obtained. Then a student walks into each store wearing old clothes and finds out how long it takes (in seconds) for a salesperson to approach the student. Later, the student goes into the same store dressed very nicely and finds out how long it takes for a salesperson to approach. Ans: This situation should use a paired t-test because there are two dependent samples. b. A researcher at an elementary school selects a random sample of 4th-graders, determines whether they know all 50 states (yes or no), and records gender. Ans: This situation should use no t-test because the data is categorical. c. A researcher calls the cell phone for a random sample of staff at a city hall late at night, measures the length of the outgoing message, and records gender. Ans: This situation should use a two-sample t-test because there are two independent samples.
Some sources report that the weights of full-term newborn babies have a mean of 8 pounds and a standard deviation of 1.2 pounds and are Normally distributed. In the given outputs, the shaded areas (reported as p=) represent the probability that the mean will be larger than 9.2 or smaller than 6.8. One of the outputs uses a sample size of 4, and one uses a sample size of 9. Complete parts (a) and (b) below.
a. Which is which, and how do you know? Ans: Figure A is for the sample size of 4 and Figure B is for the sample size of 9. The standard error is smaller for larger sample sizes, so when the null hypothesis is true, the test statistic is less likely to be more than 1.2 pounds away from the mean. b. These graphs are made so that they spread out to occupy the room on the face of the calculator. If they had the same horizontal axis, one would be taller and narrower than the other. Which one would that be, and why? Ans: Figure B would be taller and narrower, since the standard error is smaller. NOTE: Please view the output.
The average income in a certain region in 2013 was $72,000 per person per year. Suppose the standard deviation is $33,000 and the distribution is right-skewed. Suppose we take a random sample of 100 residents of the region. a. Is the sample size large enough to use the Central Limit Theorem for means? Explain. b. What are the mean and standard error of the sampling distribution? c. What is the probability that the sample mean will be more than $3,300 away from the population mean?
a. Yes, it is large enough because the sample size of 100 is greater than 25. b. The mean is $72000 and the standard error is $3300. c. The probability is 0.3174.
The distribution of the scores on a certain exam is N(40,10), which means that the exam scores are Normally distributed with a mean of 40 and standard deviation of 10. a. Sketch the curve and label, on the x-axis, the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations. b. Find the probability that a randomly selected score will be greater than 50. Shade the region under the Normal curve whose area corresponds to this probability.
a. graph: +10 standard deviations b. Using the Empirical Rule, the probability that a randomly selected score will be greater than 50 is about 16% (Graph: Right-shaded).
The distribution of the scores on a certain exam is N(70,5), which means that the exam scores are Normally distributed with a mean of 70 and standard deviation of 5. a. Sketch the curve and label, on the x-axis, the position of the mean, the mean plus or minus one standard deviation, the mean plus or minus two standard deviations, and the mean plus or minus three standard deviations. b. Find the probability that a randomly selected score will be between 60 and 80. Shade the region under the Normal curve whose area corresponds to this probability.
a. graph: +5 standard deviations b. The probability that a randomly selected score will be between 60 and 80 is 95% (Graph: Shaded between 60 and 80).
If you take samples of 40 lines from a random number table and find that the confidence interval for the proportion of odd-numbered digits captures 50% 37 times out of the 40 lines, is it the confidence interval or confidence level you are estimating with the 37 out of 40?
level
A random sample of male college baseball players and a random sample of male college soccer players were obtained independently and weighed. The accompanying table shows the unstacked weights (in pounds). The distributions of both data sets suggest that the population distributions are roughly Normal. Determine whether the difference in means is significant, using a significance level of 0.05.
H0: μbaseball-μsoccer=0 and Ha: μbaseball-μsoccer≠0; t=5.29;p-value=0.00; Reject H0. At the 0.05 significance level, there is sufficient evidence to conclude that the difference in mean weights is significant. NOTE: Use the data table.
A random sample of 33 professional baseball salaries from 1985 through 2015 was selected. The league of the player (American or National) was also recorded. Salary (in thousands of dollars) and league are shown in the accompanying table. Test the hypothesis that there is a difference in the mean salary of players in each league. Assume the distributions are Normal enough to use the t-test. Use a significance level of 0.05.
H0:μa=μn and Ha:μa≠μn; t=0.45; p-value=0.655; Do not reject H0. The difference in the mean salary of players in each league are not significantly different. NOTE: Use the data table.
