How To Estimate The Difference Between Yearly Income

Question

In Order To Estimate The Difference Between The Yearly Incomes Of Mark In order to estimate the difference between the yearly incomes of marketing managers in the east and the west, the following information was gathered: East (n1= 40, x1= 72 (in $1000), s1= 6 (in $1,000)); West (n2= 45, x2= 78 (in $1000), s2= 8 (in $1000)); degrees of freedom (DF)= 80. 1. What is the point estimate for the difference between the two population means? 2. At 90% confidence, what is the margin of error? 3. What is the 90% confidence interval for the difference between the two population means? 4. Using P-value approaches, test whether the yearly average income of marketing managers in the East is significantly different from that in the West. a) State the null and alternative hypotheses. b) What is the value of the test statistic? c) What is the critical value? d) What is the P-value? e) What is your conclusion regarding the null hypothesis? Additionally, an entomologist writes in a scientific journal that claims fewer than 7 in ten thousand male fireflies are unable to produce light due to a genetic mutation. Use the parameter p, the true proportion of fireflies unable to produce light, for analysis.

Dr. Jack HW Helper · Accepted Answer

Introduction Understanding the statistical differences between two populations is essential in various fields, including economics and biology. In this analysis, we examine the difference in yearly incomes between marketing managers in the East and West regions and test a scientific claim regarding firefly luminescence. Employing confidence intervals and hypothesis tests, this paper aims to interpret the data accurately and assess the significance of findings. Comparison of Yearly Incomes of Marketing Managers in East and West The data provided compares the annual incomes of marketing managers in two geographical regions—East and West. The sample size for East is 40, with a mean income of $72,000 and a standard deviation of $6,000. The West's sample comprises 45 managers, with a mean income of $78,000 and a standard deviation of $8,000. The goal is to estimate the difference between the two population means and assess if this difference is statistically significant. 1. Point Estimate of the Difference The point estimate for the difference between the population means is calculated as the difference between the sample means: $$ \hat{\mu}_1 - \hat{\mu}_2 = x_1 - x_2 = 72 - 78 = -6 \text{ (in $1000)} $$ This indicates that, on average, marketing managers in the East earn $6,000 less than their Western counterparts. 2. Margin of Error at 90% Confidence Calculating the margin of error involves identifying the critical t-value for 90% confidence with approximately 80 degrees of freedom. Using a t-distribution table, the critical value $ t_{0.05, 80} \approx 1.664 $. The standard error (SE) for the difference is: $$ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{6^2}{40} + \frac{8^2}{45}} = \sqrt{\frac{36}{40} + \frac{64}{45}} \approx \sqrt{0.9 + 1.422} \approx \sqrt{2.322} \approx 1.523 $$ The margin of error (ME): $$ ME = t_{\alpha/2, DF} \times SE = 1.664 \times 1.523 \approx 2.535 $$ Thus, the margin of error is approximately $2,535. 3. 90% Confidenc

How To Estimate The Difference Between Yearly Income ✓ Solved

In Order To Estimate The Difference Between The Yearly Incomes Of Mark

Sample Paper For Above instruction

Introduction

Comparison of Yearly Incomes of Marketing Managers in East and West

1. Point Estimate of the Difference

2. Margin of Error at 90% Confidence

3. 90% Confidence Interval for the Difference

4. P-Value Approach to Hypothesis Testing

Analysis of Fireflies' Luminescence

Conclusion

References