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Chapter 9: Problem 16

In the following data pairs, A represents the cost of living index for utilities and \(B\) represents the cost of living index for transportation. The data are paired by metropolitan areas in the United States. A random sample of 46 metropolitan areas gave the following information. (Reference: Statistical Abstract of the United States, 121 st edition.) \(\begin{array}{c|ccccccccc} \hline A: & 90 & 84 & 85 & 106 & 83 & 101 & 89 & 125 & 105 \\ \hline B: & 100 & 91 & 103 & 103 & 109 & 109 & 94 & 114 & 113 \\ \hline A: & 118 & 133 & 104 & 84 & 80 & 77 & 90 & 92 & 90 \\ \hline B: & 120 & 130 & 117 & 109 & 107 & 104 & 104 & 113 & 101 \\ \hline \hline A: & 106 & 95 & 110 & 112 & 105 & 93 & 119 & 99 & 109 \\ \hline B: & 96 & 109 & 103 & 107 & 103 & 102 & 101 & 86 & 94 \\ \hline A: & 109 & 113 & 90 & 121 & 120 & 85 & 91 & 91 & 97 \\ \hline B: & 88 & 100 & 104 & 119 & 116 & 104 & 121 & 108 & 86 \\ \hline A: & 95 & 115 & 99 & 86 & 88 & 106 & 80 & 108 & 90 & 87 \\ \hline B: & 100 & 83 & 88 & 103 & 94 & 125 & 115 & 100 & 96 & 127 \\ \hline \end{array}\) i. Let \(d\) be the random variable \(d=A-B\). Use a calculator to verify that \(\bar{d} \approx-5.739\) and \(s_{d} \approx 15.910 .\) ii. Do the data indicate that the U.S. population mean cost of living index for utilities is less than that for transportation in these areas? Use \(\alpha=0.05\).

Short Answer

Expert verified
The data indicate utilities have a lower mean cost index than transportation.

Step by step solution

01

Define the Hypotheses

We need to determine if the mean cost of living index for utilities is less than the mean cost for transportation. Define the null hypothesis, \(H_0: \mu_d = 0\), and the alternative hypothesis, \(H_1: \mu_d < 0\), where \(\mu_d\) is the mean of \(d = A - B\).
02

State Statistical Significance Level

The significance level \(\alpha\) is given as 0.05. This will be the threshold for determining whether we reject or fail to reject the null hypothesis.
03

Calculate the Test Statistic

The test statistic for a t-test is calculated using the formula: \( t = \frac{\bar{d}}{s_d / \sqrt{n}} \), where \( \bar{d} = -5.739 \), \( s_d = 15.910 \), and \( n = 46 \). Compute the value: \[ t = \frac{-5.739}{15.910 / \sqrt{46}} \approx -2.541 \]
04

Determine Degrees of Freedom

The degrees of freedom (df) for this t-test is \( n - 1 = 45 \). This will be used to find the critical value or p-value.
05

Find the Critical Value

For a one-tailed t-test at \(\alpha = 0.05\) and \(df = 45\), find the critical t-value from the t-distribution table, which is approximately \(-1.679\).
06

Compare Test Statistic to Critical Value

Compare the calculated t-value (-2.541) to the critical value (-1.679). Since \( -2.541 < -1.679 \), we reject the null hypothesis.
07

Conclusion

We have sufficient evidence at the \(\alpha = 0.05\) level to conclude that the mean cost of living index for utilities is less than that for transportation in these metropolitan areas.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-test
The t-test is a statistical method used to determine whether there is a significant difference between the means of two groups. In this case, we are comparing the mean cost of living index for utilities with that of transportation. By employing a t-test, we can decide if one is genuinely different from the other, based on the sample data provided.
To conduct a t-test, you need to calculate the t-statistic using the formula:
  • t = \( \frac{\bar{d}}{s_d / \sqrt{n}} \)
Here, \( \bar{d} \) is the sample mean difference between the two groups, \( s_d \) is the standard deviation of the differences, and \( n \) is the sample size.
A key aspect of a t-test is determining whether or not we reject the null hypothesis, which typically states that there is no difference between the groups being compared. In this case, we are checking if the utilities are less than transportation in cost.
cost of living index
The cost of living index is a theoretical price index that measures relative cost of living over time. This index helps to show how much it costs within a certain location to afford a typical standard of living. A higher cost of living index means it is more expensive to live in that area.
In this exercise, we are looking at two types of indices: utilities and transportation. These are indicative figures that aim to represent the average concentration of costs in each category across different metropolitan areas. For example, a higher index in transportation than in utilities indicates that transportation costs are generally higher in that area.
significance level
Significance level, often denoted as \( \alpha \), plays a crucial role in hypothesis testing. It represents the probability of rejecting the null hypothesis when it is actually true. This is also known as the Type I error rate. In statistical settings, the significance level is a threshold that determines how strong the evidence must be before we consider the results statistically significant.
  • In this exercise, the selected significance level is 0.05.
This means there is a 5% risk of concluding that a difference exists when, in fact, there is none.
Choosing a significance level involves balancing the risks of making wrong decisions. A lower value for \( \alpha \) decreases the likelihood of a Type I error, but also makes it harder to detect a real difference.
degrees of freedom
Degrees of freedom (df) are important when performing a t-test or other statistical analyses, as they define the shape and characteristics of the sampling distribution used to determine critical values. Degrees of freedom typically relate to the number of independent values or quantities which can vary in an analysis without breaking any constraints.
In a paired t-test, like the one in this exercise, degrees of freedom are calculated using the formula \( n-1 \), where \( n \) is the sample size. So for the sample of 46 metropolitan areas, we have:
  • Degrees of freedom = 46 - 1 = 45
Degrees of freedom affect the critical values obtained from the t-distribution table, which in turn affect hypothesis test results. It is important to get this right to ensure accurate testing outcomes.

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Most popular questions from this chapter

Please read the Focus Problem at the beginning of this chapter. Recall that Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with " 1 " as the leading digit is about \(0.301\) (see the reference in this chapter's Focus Problem). Now suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of \(n=215\) numerical entries from the file and \(r=46\) of the entries had a first nonzero digit of 1 . Let \(p\) represent the population proportion of all numbers in the corporate file that have a first nonzero digit of \(1 .\) i. Test the claim that \(p\) is less than \(0.301\). Use \(\alpha=0.01\). ii. If \(p\) is in fact less than \(0.301\), would it make you suspect that there are not enough numbers in the data file with leading 1's? Could this indicate that the books have been "cooked" by "pumping up" or inflating the numbers? Comment from the viewpoint of a stockholder. Comment from the perspective of the Federal Bureau of Investigation as it looks for money laundering in the form of false profits. iii. Comment on the following statement: If we reject the null hypothesis at level of significance \(\alpha\), we have not proved \(H_{0}\) to be false. We can say that the probability is \(\alpha\) that we made a mistake in rejecting \(H_{0} .\) Based on the outcome of the test, would you recommend further investigation before accusing the company of fraud?

Prose rhythm is characterized as the occurrence of five-syllable sequences in long passages of text. This characterization may be used to assess the similarity among passages of text and sometimes the identity of authors. The following information is based on an article by D. Wishart and S. V. Leach appearing in Computer Studies of the Humanities and Verbal Behavior (Vol. 3, pp. 90-99). Syllables were categorized as long or short. On analyzing Plato's Republic, Wishart and Leach found that about \(26.1 \%\) of the five-syllable sequences are of the type in which two are short and three are long. Suppose that Greek archaeologists have found an ancient manuscript dating back to Plato's time (about \(427-347\) B.C. \() .\) A random sample of 317 five-syllable sequences from the newly discovered manuscript showed that 61 are of the type two short and three long. Do the data indicate that the population proportion of this type of fivesyllable sequence is different (either way) from the text of Plato's Republic? Use \(\alpha=0.01\)

The following is based on information from The Wolf in the Southwest: The Making of an Endangered Species, by David E. Brown (University of Arizona Press). Before 1918, the proportion of female wolves in the general population of all southwestern wolves was about \(50 \%\). However, after 1918 , southwestern cattle ranchers began a widespread effort to destroy wolves. In a recent sample of 34 wolves, there were only 10 females. One theory is that male wolves tend to return sooner than females to their old territories, where their predecessors were exterminated. Do these data indicate that the population proportion of female wolves is now less than \(50 \%\) in the region? Use \(\alpha=0.01\)

In environmental studies, sex ratios are of great importance. Wolf society, packs, and ecology have been studied extensively at different locations in the U.S. and foreign countries. Sex ratios for eight study sites in northern Europe are shown below (based on The Wolf by L. D. Mech, University of Minnesota Press). \(\begin{array}{lcc} \hline \text { Location of Wolf Pack } & \text { \% Males (Winter) } & \text { \% Males (Summer) } \\ \hline \text { Finland } & 72 & 53 \\ \text { Finland } & 47 & 51 \\ \text { Finland } & 89 & 72 \\ \text { Lapland } & 55 & 48 \\ \text { Lapland } & 64 & 55 \\ \text { Russia } & 50 & 50 \\ \text { Russia } & 41 & 50 \\ \text { Russia } & 55 & 45 \\ \hline \end{array}\) It is hypothesized that in winter, "loner" males (not present in summer packs) join the pack to increase survival rate. Use a \(5 \%\) level of significance to test the claim that the average percentage of males in a wolf pack is higher in winter.

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. What is the value of the sample test statistic? (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 significant at level \(\alpha\) ? (e) State your conclusion in the context of the application. The price to earnings ratio \((\mathrm{P} / \mathrm{E})\) is an important tool in financial work. A random sample of 14 large U.S. banks (J. P. Morgan, Bank of America, and others) gave the following \(\mathrm{P} / \mathrm{E}\) ratios (Reference: Forbes). \(\begin{array}{lllllll}24 & 16 & 22 & 14 & 12 & 13 & 17 \\\ 22 & 15 & 19 & 23 & 13 & 11 & 18\end{array}\) The sample mean is \(\bar{x} \approx 17.1\). Generally speaking, a low \(\mathrm{P} / \mathrm{E}\) ratio indicates a "value" or bargain stock. A recent copy of The Wall Street Journal indicated that the \(\mathrm{P} / \mathrm{E}\) ratio of the entire \(\mathrm{S\&P} 500\) stock index is \(\mu=19 .\) Let \(x\) be a random variable representing the \(\mathrm{P} / \mathrm{E}\) ratio of all large U.S. bank stocks. We assume that \(x\) has a normal distribution and \(\sigma=4.5\). Do these data indicate that the \(\mathrm{P} / \mathrm{E}\) ratio of all U.S. bank stocks is less than 19 ? Use \(\alpha=0.05\).
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