Q6. An international company is aware that the mean Employee Satisfaction Index (ESI) within its Denmark-based branches is excellent, being 90, on a scale from 0 to 100. A union representative wishes to ascertain if the same mean of 90 is achieved or not within the company’s U.S.-based branches. Here are some summary statistics and a histogram of the ESIs obtained from a random sample of 25 U.S. employees.

Background

Topic: Hypothesis Testing for Means (One-Sample t-Test)

This question tests your understanding of hypothesis testing for a population mean, including setting up hypotheses, checking assumptions, calculating test statistics and p-values, and interpreting confidence intervals. The histogram helps assess normality, which is important for the t-test.

Key Terms and Formulas

• Null Hypothesis (): The mean ESI in U.S. branches is 90 ().

• Alternative Hypothesis (): The mean ESI in U.S. branches is not 90 ().

• Test Statistic:

• Confidence Interval:

• Assumptions: Random sample, approximately normal distribution (check histogram), independence.

Step-by-Step Guidance

1. Write the null and alternative hypotheses:

2. Check assumptions: Is the sample random? Is the sample size large enough or is the data approximately normal? Use the histogram to assess normality.

3. Calculate the test statistic using the formula: Where , , , .

4. Find the p-value associated with your calculated t-statistic using the t-distribution with degrees of freedom.

5. Construct a 95% confidence interval for the mean ESI using: Where is the critical value for 95% confidence and 24 degrees of freedom.

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