Chapter 9 Question 1: A Bond Analyst Is Analyzing The Intere

Chapter 9question 1 A Bond Analyst Is Analyzing The Interest Rates For

Analyze multiple hypothesis testing scenarios involving different statistical tests such as z-tests, t-tests, and F-tests examining data such as interest rates, life expectancy, and variances. Determine appropriate test values, critical values, confidence intervals, and the validity of hypotheses based on provided sample data, significance levels, and statistical formulas. Focus on interpreting test results, assessing whether evidence supports null hypotheses or suggests differences between groups, and calculating degrees of freedom and critical values for various statistical tests.

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The given set of questions revolves around hypothesis testing in diverse contexts, illustrating fundamental principles in inferential statistics such as comparing means, proportions, and variances, and understanding the application of different statistical tests, including z-tests, t-tests, and F-tests, at specified significance levels. These scenarios highlight the importance of selecting the appropriate test, correctly calculating test statistics, determining critical values, and interpreting results within the context of research hypotheses.

In the first question concerning bond interest rates, the key is to evaluate whether there is statistically significant evidence of a difference in interest rates paid by two states, based on sample means, variances, and sizes. Since population variances are known and equal, a z-test is appropriate. The test statistic is derived from the difference in sample means divided by the standard error of the difference. The computed z-statistic is approximately 1.08, which falls within the critical region for a two-tailed test at α = 0.05 (−1.96 to 1.96). Consequently, the correct conclusion is "No, because the test value 1.08 is inside the critical region," implying insufficient evidence to conclude a difference.

Similarly, the second question deals with comparing expenditures on textbooks between freshmen and seniors, with given sample means, standard deviations, and sample sizes. Here, the appropriate test is a z-test for the difference between two means, assuming known population standard deviations. The test statistic calculates to approximately -0.91, which again falls within the critical region (−1.96 to 1.96), leading to the conclusion that there's insufficient evidence to suggest a difference in textbook spending.

The third scenario involves testing if life expectancy in Africa is less than in Asia, requiring a z-test for difference between population means with known standard deviations. Calculating the test statistic yields approximately -2.33, matching the critical value at α = 0.05 for a lower-tailed test. The critical value in this case is −2.33, indicating the boundary at which the null hypothesis would be rejected, supporting the researcher’s hypothesis if the test statistic exceeds this critical value.

When constructing a 95% confidence interval for the difference in life expectancy between Africa and Asia, the parameters include sample means, standard deviations, and sample sizes for each population. Using the standard error formula for the difference in means, the margin of error is computed, and the resulting confidence interval is approximately from −13.5 to −6.3 years. This interval implies that the true difference in life expectancy is likely within this range, supporting the alternative hypothesis of a significant difference.

For the analysis comparing cholesterol-lowering drug effectiveness between genders, an appropriate z-test or t-test depends on whether population variances are known. Given the same variances and large enough sample sizes, the test statistic is computed as approximately 0.73, suggesting no significant difference in effectiveness between women and men at the 0.05 level.

In the context of spending on vacations between married and single men, the appropriate test could be an independent z-test for means with known variances or a t-test if variances are unknown. The calculated test statistic is about -1.60, which is within the noncritical region, indicating no sufficient evidence to conclude a difference in spending patterns among these groups.

The example involving wine prices compares imported and domestic wine prices using hypothesis testing for means, with known standard deviations. The critical value for a one-tailed t-test at α=0.05 with degrees of freedom approximated from the sample sizes is about −2.131. The test value, based on the sample means and standard deviations, is approximately −6.49, which falls in the rejection region, supporting the hypothesis that imported wines are less expensive on average.

Further, testing the difference between two means involving unknown variances requires calculating the t-statistic with appropriate degrees of freedom, which involves the standard deviations and sample sizes. For the wine prices, the computed t-value is approximately −2.54, indicating a significant difference at the 0.05 level.

Similarly, the questions involving variances employ the F-test, where the test value is computed as the ratio of variances, and the degrees of freedom are based on sample sizes. For example, comparing variances in car rental rates involves calculating an F-statistic around 1.41 with corresponding degrees of freedom, and comparing to critical F-values to decide whether variances are significantly different.

In conclusion, these examples underscore the importance of selecting correct statistical procedures based on the data characteristics, correctly computing test statistics, identifying critical values, and interpreting results accurately within the chosen significance levels. Proper understanding of degrees of freedom, confidence intervals, and the distinction between types of tests enables researchers to draw valid inferences from sample data, a core skill in statistical analysis and research methodology.

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