Master-Level Statistics Solutions from Our Experts

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Problem 1: Hypothesis Testing & Decision Making

A company claims that the average time spent by users on its website is more than 15 minutes per session. A sample of 40 users was observed, and their average session time was found to be 16.2 minutes with a standard deviation of 3.5 minutes. Test the company’s claim at a 5% significance level using the appropriate hypothesis test.

Expert Solution:

To test this claim, we formulate our hypotheses as follows:

  • Null Hypothesis (H₀): (The true mean session time is 15 minutes or less)

  • Alternative Hypothesis (H₁): (The true mean session time is greater than 15 minutes)

Since the sample size is large (n = 40), we use a one-sample t-test:

 

Substituting the values:

 

The critical t-value for a one-tailed test at a 5% significance level with 39 degrees of freedom is 1.685 (from t-tables). Since 2.17 > 1.685, we reject the null hypothesis.

Conclusion: There is sufficient evidence to support the claim that the average session time exceeds 15 minutes.


Problem 2: Regression Analysis & Predictive Modeling

A researcher is analyzing the relationship between the advertising budget (X) and product sales (Y) in thousands of dollars. Using a sample of 50 data points, the following simple linear regression equation was obtained:

 

If the advertising budget is increased by $10,000, predict the change in product sales.

Expert Solution:

The regression equation provided is:

 

Here, the coefficient of X (2.8) represents the change in product sales for every unit increase in the advertising budget. Since X is measured in thousands of dollars, increasing the budget by $10,000 (or 10 units) results in:

 

Conclusion: An increase of $10,000 in the advertising budget is expected to result in an increase of $28,000 in product sales.


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