For Each Of The 2 Majors, Conduct A Full Hypothesis Test At

For Each Of The 2 Majors Conduct A Full Hypothesis Test At The 001

Perform a full hypothesis test at the 0.01% significance level for each of the two majors regarding the claim that the average 'Cost' for a college is $160,000. Use sample data with a size of 20, and calculate standard deviations via Excel. Since the population standard deviation is unknown, utilize appropriate t-tests. Determine whether there is statistical evidence to support or reject the claim for each major.

Additionally, conduct a two-sample hypothesis test at the 10% significance level comparing Business and Engineering majors. The claim is that the average '30-Year ROI' for Business majors is less than that for Engineering majors. Assume unequal variances and apply the two-sample t-test accordingly, following all standard steps of hypothesis testing.

Furthermore, answer the following questions for your Week 8 paper:

• a. Based on the sample data from Week 2, does one of the two majors cost more on average? If yes, which one?
• b. Do the hypothesis tests support the conclusion that one major costs more than the other? If yes, which one?
• c. Why was conducting these hypothesis tests necessary if Week 2's sample data already indicated which school costs more?
• d. If a student asked, "What is it going to cost to attend a business or engineering school?" based on these results, what advice would you give?

Paper For Above instruction

Introduction

Hypothesis testing is a fundamental process in statistics that enables researchers to make informed decisions about population parameters based on sample data. In the context of educational costs and return on investment (ROI), conducting rigorous hypothesis tests allows us to ascertain whether observed differences are statistically significant or merely due to chance. This paper addresses multiple hypothesis tests concerning the costs associated with different college majors—namely, Business and Engineering—and evaluates the implications of these tests for students making educational choices.

Hypothesis Testing for College Costs

For each of the two majors, we test whether the mean cost significantly differs from the claimed average of $160,000 at a significance level of 0.01%. This involves formulating null and alternative hypotheses:

• Null hypothesis (H₀): The mean cost μ = $160,000
• Alternative hypothesis (H₁): The mean cost μ ≠ $160,000

Given a sample size of 20 and the use of Excel for calculating the standard deviations, the t-test is appropriate since the population standard deviation is unknown. The test statistic is computed as:

t = (x̄ - μ₀) / (s / √n)

where x̄ is the sample mean, s is the sample standard deviation, and n is the sample size. The critical t-value for a two-tailed test at such a stringent significance level is obtained from t-distribution tables, adjusted for degrees of freedom (n - 1).

Applying this methodology for each major yields whether the null hypothesis can be rejected or not. If rejected, it indicates that the average cost significantly differs from $160,000; if not, the data do not provide enough evidence to refute the claim.

Two-Sample Hypothesis Test Comparing ROI Between Majors

The second analysis involves comparing the 30-Year ROI between Business and Engineering majors. The claim tested is that Business majors have a lower average ROI than Engineering majors. This sets up a one-tailed test:

• Null hypothesis (H₀): μ_Business ≥ μ_Engineering
• Alternative hypothesis (H₁): μ_Business Engineering

Since variances are assumed unequal, Welch’s t-test is employed. The test statistic is calculated as:

t = (x̄₁ - x̄₂) / √(s₁² / n₁ + s₂² / n₂)

where x̄₁ and x̄₂ are the sample means, s₁² and s₂² are the sample variances, and n₁, n₂ are the sample sizes (both 20). The degrees of freedom are approximated using the Welch–Satterthwaite equation. The critical t-value for a one-tailed test at the 10% significance level guides the decision to reject or retain the null hypothesis.

Applying this approach determines if the evidence suggests that Business majors indeed have a lower 30-Year ROI than Engineering majors.

Analysis of Results and Practical Implications

Analyzing the sample means from Week 2 alongside hypothesis testing results reveals insights into the cost differences between majors. If, for example, the mean cost for Engineering exceeds that for Business and the null hypotheses are rejected accordingly, it supports the idea that Engineering tends to be more expensive on average. Conversely, if the hypothesis tests do not reject the null hypotheses despite observed differences, it suggests that the cost disparity may not be statistically meaningful.

Conducting hypothesis tests beyond descriptive statistics enhances confidence in conclusions. While Week 2's sample averages indicated potential differences, hypothesis testing accounts for variability and sample size, providing a formal statistical basis for claims.

For students deciding between majors, these results influence choices by clarifying whether observed cost differences are statistically significant and whether ROI disparities are meaningful in the long term. For instance, a student considering cost might prefer a major with lower probabilities of costing significantly more, given the evidence from hypothesis tests.

Conclusion

In summary, hypothesis testing is crucial in validating or refuting initial impressions drawn from sample data. Through individual and comparative tests concerning college costs and ROI, students and educators can better understand the significance of observed differences. These statistical tools offer a rigorous framework for making informed educational decisions, ultimately assisting students in choosing majors aligned with their financial considerations and career goals.

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