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| m102790 | The variance in GMAT exam scores is 14,660 (the Graduate Management Admission Council s official website of the GMAT, June 17, 2016). A group of economics professors recently met at a conference to discuss the performance on the GMAT of undergraduate students majoring in economics. Some expected the variability in GMAT scores achieved by undergraduate students majoring in economics to be greater than the variability in GMAT scores of the general population of GMAT test takers. Others took the opposite view. The DATAfile EconGMAT contains GMAT scores for 51 randomly selected undergraduate students majoring in economics.
a. Compute the mean, variance, and standard deviation of the GMAT scores for the 51 randomly selected undergraduate students majoring in economics.
b. Develop hypotheses to test whether the sample data indicate that the variance in GMAT scores for undergraduate students majoring in economics differs from the general population of GMAT test takers. |
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| m102791 | The vice president of sales for Blasterman Cosmetics, Inc. believes that 40% of the company s orders come from customers who are less than 30 years old. A random sample of 10,000 orders will be used to estimate the proportion of customers who are less than 30 years old.
a. Assume that the vice president of sales is correct and p 5 .40. What is the sampling distribution of p for this study?
b. What is the probability that the sample proportion will be between .37 and .43?
c. What is the probability that the sample proportion will be between .39 and .41?
d. What would you conclude if the sample proportion is .36? |
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| m102792 | The visibility standard index (VSI) is a measure of Denver air pollution that is reported each day in the Denver Post. The index ranges from 0 (excellent air quality) to 200 (very bad air quality). During winter months, when air pollution is higher, the index has a mean of about 90 (rated as fair) with a standard deviation of approximately 30. Suppose that for 15 days, the following VSI measures were reported each day:
Make a control chart for the VSI, and plot the preceding data on the control chart. Identify all out-of-control signals (high or low) that you fi nd in the control chart by type (I, II, or III)? |
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| m102793 | The Wall Street Journal reported on several studies that show massage therapy has a variety of health benefits and it is not too expensive. A sample of 10 typical one-hour massage therapy sessions showed an average charge of $59. The population standard deviation for a one-hour session is s 5 $5.50.
a. What assumptions about the population should we be willing to make if a margin of error is desired?
b. Using 95% confidence, what is the margin of error?
c. Using 99% confidence, what is the margin of error? |
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| m102794 | The Wall Street Journal reported that 33% of taxpayers with adjusted gross incomes between $30,000 and $60,000 itemized deductions on their federal income tax return. The mean amount of deductions for this population of taxpayers was $16,642. Assume the standard deviation is 5 $2400.
a. What is the probability that a sample of taxpayers from this income group who have itemized deductions will show a sample mean within $200 of the population mean for each of the following sample sizes: 30, 50, 100, and 400?
b. What is the advantage of a larger sample size when attempting to estimate the population mean? |
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| m102795 | The Wall Street Journal reported that 37% of all entrepreneurs who opened new U.S. businesses in the previous year were female (The Wall Street Journal, May 13, 2015).
a. Suppose a random sample of 300 entrepreneurs who opened new U.S. businesses in the previous year will be taken to learn about which industries are most appealing to entrepreneurs. Show the sampling distribution of p, where p is the sample proportion of entrepreneurs who opened new U.S. businesses in the previous year that are female.
b. What is the probability that the sample proportion in part (a) will be within 6.05 of its population proportion?
c. Suppose a random sample of 30,000 entrepreneurs who opened new U.S. businesses in the previous year will be taken to learn about which industries are most appealing to entrepreneurs. Show the sampling distribution of p where p is the sample proportion of entrepreneurs who opened new U.S. businesses last year that are female.
d. What is the probability that the sample proportion in part (c) will be within 6.05 of its population proportion?
e. Is the probability different in parts (b) and (d)? Why? |
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| m102796 | The Wall Street Journal reported that Americans spend nearly $7 billion on Halloween costumes and decorations. Sample data showing the amount, in dollars, that 16 adults spent on a Halloween costume are as follows.
a. What is the estimate of the population mean amount adults spend on a Halloween costume?
b. What is the sample standard deviation?
c. Provide a 95% confidence interval estimate of the population standard deviation for the amount adults spend on a Halloween costume? |
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| m102797 | The Wall Street Journal reported that approximately 25% of the people who are told a product is improved will believe that it is, in fact, improved. The remaining 75% believe that this is just hype (the same old thing with no real improvement). Suppose a marketing study consists of a random sample of eight people who are given a sales talk about a new, improved product.
(a) Make a histogram showing the probability that r 5 0 to 8 people believe the product is, in fact, improved.
(b) Compute the mean and standard deviation of this probability distribution.
(c) How many people are needed in the marketing study to be 99% sure that at least one person believes the product to be improved?
Note that P(r ( 1) = 0.99 is equivalent to 1 - P(0) - 0.99, or P(0) - 0.01. |
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| m102798 | The webpage "Bottlenose Dolphin" produced by the National Geographic Society provides information about the bottlenose dolphin. A random sample of 50 adult bottlenose dolphins have a mean length of 12.04 ft with a standard deviation of 1.03 ft. Find and interpret a 90% confidence interval for the mean length of all adult bottlenose dolphins. |
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| m102799 | The Wind Mountain archaeological site is in southwest New Mexico. Prehistoric Native Americans called Anasazi once lived and hunted small game in this region. A stemmed projectile point is an arrowhead that has a notch on each side of the base. Both stemmed and stemless projectile points were found at the Wind Mountain site. A random sample of n1 = 55 stemmed projectile points showed the mean length to be x̅1 = 3.0 cm, with sample standard deviation s1 = 0.8cm. Another random sample of n2 = 51 stemless projectile points showed the mean length to be x̅2 = 2.7cm, with s2 = 0.9cm Do these data indicate a difference (either way) in the population mean length of the two types of projectile points? Use a 5% level of signifi cance.
(a) What is the level of signifi cance? State the null and alternate hypotheses.
(b) What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding distribution value.
(c) Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically signifi cant at level a?
(e) Interpret your conclusion in the context of the application.
For degrees of freedom d.f. not in the Student s t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and the |
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| m102800 | The winner of the 2012-2013 National Basketball Association (NBA) championship was the Miami Heat. One possible starting lineup for that team is as follows.
a. Find the population mean height of the five players.
b. For samples of size 2, construct a table similar to Table 7.2 on page 310. Use the letter in parentheses after each player s name to represent each player.
c. Draw a dotplot for the sampling distribution of the sample mean for samples of size 2.
d. For a random sample of size 2, what is the chance that the sample mean will equal the population mean?
e. For a random sample of size 2, obtain the probability that the sampling error made in estimating the population mean by the sample mean will be 1 inch or less; that is, determine the probability that x̅ will be within 1 inch of μ. Interpret your result in terms of percentages.
Exercises 7.11-7.23 are intended solely to provide concrete illustrations of the sampling distribution of the sample mean. For that reason, the populations considered are unrealistically small. In each exercise, assume that sampling is without replacement. |
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| m102801 | The winner of the 2012-2013 National Basketball Association (NBA) championship was the Miami Heat. One possible starting lineup for that team is as follows.
a. Determine the population mean height, μ, of the five players.
b. Consider samples of size 2 without replacement. Use your answer to Exercise 7.11(b) on page 312 and Definition 3.11 on page 140 to find the mean, μ x̅ , of the variable x̅.
c. Find μx̅, using only the result of part (a).
Exercises 7.41-7.45 require that you have done Exercises 7.11-7.15, respectively. |
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| m102802 | The World Series in baseball is won by the first team to win four games (ignoring the 1903 and 1919-1921 World Series, when it was a best of nine). Thus it takes at least four games and no more than seven games to establish a winner. From the document World Series History on the Baseball Almanac website, as of November 2013, the lengths of the World Series are as given in the following table.
a. If X denotes the number of games that it takes to complete a World Series, identify the possible values of the random variable X.
b. Do the first and third columns of the table provide a probability distribution for X? Explain your answer.
c. Historically, what is the most likely number of games it takes to complete a series?
d. Historically, for a randomly chosen series, what is the probability that it ends in five games?
e. Historically, for a randomly chosen series, what is the probability that it ends in five or more games?
f. The data in the table exhibit a statistical oddity. If the two teams in a series are evenly matched and one team is ahead three games to two, either team has the same chance of winning game number six. Thus there should be about an equal number of six- and seven-game series. If the teams are not evenly matched, the series should tend to be shorter, ending in six or fewer games, not seven games. Can you explain why the series tend to last longer than expected? |
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| m102803 | The World Series in baseball is won by the first team to win four games (ignoring the 1903 and 1919-1921 World Series, when it was a best of nine). From the document World Series History on the Baseball Almanac website, as of November 2013, the lengths of the World Series are as given in the following table.
Let X denote the number of games that it takes to complete a World Series, and let Y denote the number of games that it took to complete a randomly selected World Series from among those considered in the table.
a. Determine the mean and standard deviation of the random variable Y. Interpret your results.
b. Provide an estimate for the mean and standard deviation of the random variable X. Explain your reasoning. |
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| m102804 | There are fi ve multiple-choice questions on an exam, each with four possible answers. Determine the number of possible answer sequences for the fi ve questions. Only one of the sets can contain all fi ve correct answers. If you are guessing, so that you are as likely to choose one sequence of answers as another, what is the probability of getting all fi ve answers correct? |
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| m102805 | There are several extensions of linear regression that apply to exponential growth and power law models. Problems 22-25 will outline some of these extensions. First of all, recall that a variable grows linearly over time if it adds a fi xed increment during each equal time period. Exponential growth occurs when a variable is multiplied by a fi xed number during each time period. This means that exponential growth increases by a fixed multiple or percentage of the previous amount. College algebra can be used to show that if a variable grows exponentially, then its logarithm grows linearly. The exponential growth model is y = αβx, where a and b are fixed constants to be estimated from data.
How do we know when we are dealing with exponential growth, and how can we estimate α and β? Please read on. Populations of living things such as bacteria, locusts, sh, panda bears, and so on, tend to grow (or decline) exponentially. However, these populations can be restricted by outside limitations such as food, space, pollution, disease, hunting, and so on. Suppose we have data pairs (x, y) for which there is reason to believe the scatter plot is not linear, but rather exponential, as described above. This means the increase in y values begins rather slowly but then seems to explode. Note: For exponential growth models, we assume all y > 0.
Consider the following data, where x = time in hours and y = number of bacteria in a laboratory culture at the end of x hours.
(a) Look at the |
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| m102806 | Thirty years ago, the Bureau of Justice Statistics reported in Profile of Jail Inmates that the median educational attainment of jail inmates was 10.2 years. Ten current inmates are randomly selected and found to have the following educational attainments, in years.
Assume that educational attainments of current jail inmates have a symmetric, non-normal distribution. At the 10% significance level, do the data provide sufficient evidence to conclude that this year s median educational attainment has changed from what it was 30 years ago?
a. Use the t-test.
b. Use the Wilcoxon signed-rank test.
c. If this year s median educational attainment has in fact changed from what it was 30 years ago, how do you explain the discrepancy between the two tests? |
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| m102807 | Thirty-six percent of all Americans drink bottled water more than once a week (Natural Resources Defense Council, December 4, 2015). Suppose you have been hired by the Natural Resources Defense Council to investigate bottled water consumption in St. Paul. You plan to select a sample of St. Paulites to estimate the proportion who drink bottled water more than once a week. Assume the population proportion of St. Paulites who drink bottled water more than once a week is .36, the same as the overall proportion of Americans who drink bottled water more than once a week.
a. Suppose you select a sample of 540 St. Paulites. Show the sampling distribution of p.
b. Based upon a sample of 540 St. Paulites, what is the probability that the sample proportion will be within .04 of the population proportion?
c. Suppose you select a sample of 200 St. Paulites. Show the sampling distribution of p.
d. Based upon the smaller sample of only 200 St. Paulites, what is the probability that the sample proportion will be within .04 of the population proportion?
e. As measured by the increase in probability, how much do you gain in precision by taking the larger sample in parts (a) and (b) rather than the smaller sample in parts (c) and (d)? |
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| m102808 | This cumulative review problem uses material from Chapters 3, 5, and 10. Recall that the Poisson distribution deals with rare events. Death from the kick of a horse is a rare event, even in the Prussian army. The following data are a classic example of a Poisson application to rare events.
A reproduction of the original data can be found in C. P. Winsor, Human Biology, Vol. 19, pp. 154-161. The data represent the number of deaths from the kick of a horse per army corps per year for 10 Prussian army corps for 20 years (1875-1894). Let x represent the number of deaths and f the frequency of x deaths.
(a) First, we t the data to a Poisson distribution (see Section 5.4).
Poission distribution:
Where λ ≈ x̅ (sample mean of x values) From our study of weighted averages (see section 3.1),
Verify that x̅ ≈ 0.61 For the category 3 or more, use 3.
(b) Now we have
1, 2, 3. p
Find P(0), P(1), P(2), and P(3 ≤ x). Round to three places after the decimal.
(c) The total number of observations is ∑f = 200.
For a given x, the expected frequency of x deaths is 200P(x). The following table gives the observed frequencies O and the expected frequencies E = 200P(x).
(d) State the null and alternate hypotheses for a chi-square goodness-of- t test. Set the level of signifi cance to be a = 0.01. Find the P-value for a goodness-of- t test. Interpret your conclusion in the context of this application. Is there reason to believe that the Poisson distribution fits the raw data pro |
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| m102809 | This exercise requires that you have first done Exercise 3.199. In Example 3.12 on page 109, we found that, considering the five starting players on Team II a sample of all male starting college basketball players, the mean and standard deviation of the heights are 75 inches and 6.2 inches, respectively. Explain why, numerically, the sample mean of 75 inches is the same as the population mean found in Exercise 3.199(a) but that the sample standard deviation of 6.2 inches differs from the population standard deviation found in Exercise 3.199(b). |
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