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Chapter 10: Problem 20

Annual income of CEOs A study analyzes the total annual pay of CEOs (in pounds) for a sample of UK companies over the period \(2003-2006\) categorized according to the number of compensation consultants employed. The sample included 311 firms having one consultant and 203 firms having two consultants. Software output shows the following results: Two sample T hypothesis test: \(\mu_{1}:\) Mean of CEO total pay in firms with one compensation consultant \(\mu_{2}:\) Mean of CEO total pay in firms with two compensation consultants a. Does it seem plausible that income has a normal distribution for each firms' category? Explain. b. What effect does the answer to part a have on inference comparing population means? What assumptions are made for the inferences in this table? c. A \(95 \%\) confidence interval for the difference in the population means for CEOs (men and women) is \((拢-370.19, 拢 128.19) .\) Interpret, indicating the relevance of \(拢 0\) falling in the interval.

Short Answer

Expert verified
Normality is plausible for large samples; the confidence interval suggests no significant difference in pay.

Step by step solution

01

Analyze Normal Distribution Plausibility

To determine if the income distribution is normal for each category (one consultant and two consultants), we observe the sample sizes: 311 firms with one consultant and 203 firms with two consultants. Generally, larger sample sizes help ensure that the sampling distribution is approximately normal due to the Central Limit Theorem. Thus, it seems plausible that income has a normal distribution in each category.
02

Effect on Inference and Assumptions

The assumption of normality affects the validity of the two-sample t-test used for inference. The test assumes that the populations are normally distributed or that the sample sizes are large enough for the Central Limit Theorem to apply, which allows normality of the sampling distribution of the means. Large sample sizes like these provide robustness against normality assumptions.
03

Interpret the Confidence Interval

The 95% confidence interval for the difference in population means is given as \((-拢370.19, 拢128.19)\). This interval means we are 95% confident that the difference in the mean annual CEO pay between firms with one and two compensation consultants is between 拢-370.19 and 拢128.19. Since 拢0 falls within this interval, it suggests there may be no significant difference in mean pay between the two groups.

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Key Concepts

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

Normal Distribution
Understanding the concept of a normal distribution is crucial in statistics education. A normal distribution is often depicted as a bell-shaped curve that is symmetrical around its mean. It describes how data values are dispersed around the mean. In this exercise, we focus on CEO salaries being normally distributed. Why is normality important? If the CEO incomes follow a normal distribution, our statistical analyses, like hypothesis testing and confidence intervals, are more accurate.

For larger sample sizes, the Central Limit Theorem assists us by indicating that the sampling distribution of the mean approaches normality, even if the data itself isn鈥檛 perfectly normal. With 311 and 203 firms respectively, these sample sizes are large enough to apply this theorem, making it plausible and credible to assume a normal distribution for the inference process.
Two-Sample T-Test
A two-sample t-test is a statistical method used to compare the means of two independent groups to determine if they are significantly different from each other. In our exercise, we're comparing the mean income of CEOs from companies with one consultant to those with two. A few essential considerations:
  • Independence: Observations in one sample do not affect observations in the other.
  • Normality: Each population should ideally be normally distributed, but large samples can cushion deviations with the help of the Central Limit Theorem.
  • Equality of variances: Known as homogeneity of variance, it's sometimes assumed when conducting the test.
Interpreting the results involves examining means and p-values to determine if any observed differences are statistically significant.
Confidence Interval
A confidence interval provides a range of values that is likely to contain the population parameter with a specific probability. In our case, a 95% confidence interval for the difference in CEO pay is given as developed with the formula: \[ CI = \bar{x}_1 - \bar{x}_2 \pm t^* \times SE(\bar{x}_1 - \bar{x}_2) \] where \( \bar{x} \) is the sample mean, \( t^* \) is the t-score for the desired confidence level, and \( SE \) is the standard error.The interval developed as \((-拢370.19, 拢128.19)\) indicates that we can be 95% certain the true difference in means lies within these bounds. Importantly, 拢0 falling within this interval suggests no definitive difference between the groups on a population level. This prompts us not to reject the null hypothesis of equal means.
Central Limit Theorem
The Central Limit Theorem (CLT) is a cornerstone in statistics, particularly useful when dealing with large samples. It states that the sampling distribution of the sample mean will approximate a normal distribution as the sample size becomes larger. Importantly, this holds true regardless of the original population's distribution. This theorem underpins the reliability of the inference methods used in our exercise. The large sample sizes (311 and 203 firms) allow us to comfortably use the normal distribution framework to test our hypotheses and construct confidence intervals, even if the income data isn't perfectly normal. By leveraging the CLT, statisticians ensure that their analyses remain robust and applicable under less-than-ideal conditions. It allows flexibility and confidence in conclusions drawn from sample data, making it a critical tool in statistics education and practice.

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Most popular questions from this chapter

An AP story (April 9,2005\()\) with headline Study: Attractive People Make More stated that "A study concerning weight showed that women who were obese earned 17 percent lower wages than women of average weight." a. Identify the two variables stated to have an association. b. Identify a control variable that might explain part or all of this association. If you had the original data including data on that control variable, how could you check whether the control variable does explain the association?

The table shows results from the 2014 General Social Survey on gender and whether one believes in an afterlife. $$ \begin{array}{lccc} \hline & \ {\text { Belief in Afterlife }} & \\ { 2 - 3 } \text { Gender } & \text { Yes } & \text { No } & \text { Total } \\\ \hline \text { Female } & 1026 & 207 & 1233 \\ \text { Male } & 757 & 252 & 1009 \\ \hline\end{array}$$ a. Denote the population proportion who believe in an afterlife by \(p_{1}\) for females and by \(p_{2}\) for males. Estimate \(p_{1}, p_{2},\) and \(\left(p_{1}-p_{2}\right)\) b. Find the standard error for the estimate of \(\left(p_{1}-p_{2}\right)\). Interpret. c. Construct a \(95 \%\) confidence interval for \(\left(p_{1}-p_{2}\right)\). Can you conclude which of \(p_{1}\) and \(p_{2}\) is larger? Explain. d. Suppose that, unknown to us, \(p_{1}=0.81\) and \(p_{2}=0.72 .\) Does the confidence interval in part c contain the parameter it is designed to estimate? Explain.

A study \(^{14}\) compared personality characteristics between 49 children of alcoholics and a control group of 49 children of nonalcoholics who were matched on age and gender. On a measure of well-being, the 49 children of alcoholics had a mean of \(26.1(s=7.2)\) and the 49 subjects in the control group had a mean of \(28.8(s=6.4) .\) The difference scores between the matched subjects from the two groups had a mean of \(2.7(s=9.7)\). a. Are the groups to be compared independent samples or dependent samples? Why? b. Show all steps of a test of equality of the two population means for a two- sided alternative hypothesis. Report the P-value and interpret. c. What assumptions must you make for the inference in part b to be valid?

Steve Solomon, the owner of Leonardo's Italian restaurant, wonders whether a redesigned menu will increase, on the average, the amount that customers spend in the restaurant. For the following scenarios, pick a statistical method from this chapter that would be appropriate for analyzing the data, indicating whether the samples are independent or dependent, which parameter is relevant, and what inference method you would use: a. Solomon records the mean sales the week before the change and the week after the change and then wonders whether the difference is "statistically significant." b. Solomon randomly samples 100 people and shows them each both menus, asking them to give a rating between 0 and 10 for each menu. c. Solomon randomly samples 100 people and shows them each both menus, asking them to give an overall rating of positive or negative to each menu. d. Solomon randomly samples 100 people and randomly separates them into two groups of 50 each. He asks those in Group 1 to give a rating to the old menu and those in Group 2 to give a rating to the new menu, using a 0 to 10 rating scale.

Two new programs were recently proposed at the University of Florida for treating students who suffer from math anxiety. Program A provides counseling sessions, one session a week for six weeks. Program \(\mathrm{B}\) supplements the counseling sessions with short quizzes that are designed to improve student confidence. For ten students suffering from math anxiety, five were randomly assigned to each program. Before and after the program, math anxiety was measured by a questionnaire with 20 questions relating to different aspects of taking a math course that could cause anxiety. The study measured, for each student, the drop in the number of items that caused anxiety. The sample values were Program A: 0,4,4,6,6 Program B: 6,12,12,14,16 Using software, analyze these data. Write a report, summarizing the analyses and interpretations.
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