Task 4

You can find the solution to this problem in the Hypothesis Testing – task 4.xlsx file:

• Sheet: Task 4
• Sheet: Test for Variances

First, you must perform a two-sample f-test for variances to prove that assumption of unequal variances between the samples for free- and paid-plan subscribers:

The p-value indicates the probability of obtaining the observed f-value if the null hypothesis (equal variances) were true. The sample variances are not identical since the p-value in both cases is 0.

Next, we use a left-tailed t-test assuming unequal variances for paying and free-plan students. 

Paid-Plan Students

Calculate the mean, standard deviation, and sample size for paid-plan students:

Calculate the t-statistic:

Look up the critical t-value using a t-distribution table to correspond to your chosen significance level (commonly 0.05) and calculated degrees of freedom.

Compare the t-statistic to the critical t-value:

$$If-3.35\le -1.65,thenreject{H}_0$$

Conclusion: Reject  because the calculated t-statistic is lower than the critical value.

Alternatively, use the Two-Sample t-Test Assuming Unequal Variances that is part of the Data Analysis ToolPak:

$$DecisionRule:Ifp-value\le 0.05,Reject{H}_0$$

Conclusion: Reject  because the p-value is lower than the specified significance level α (0.05).

Summary: With a t-statistic of -3.05 (less than the critical value of -1.645), you would reject the null hypothesis because the negative t-statistic indicates that  (the mean minutes watched by students in Q4 2021) is significantly smaller than  (the mean minutes watched by students in Q4 2022). This is contrary to the null, so we reject it. Of course, rejecting the null hypothesis does not confirm the alternative hypothesis; it suggests that the data provide enough evidence against the null hypothesis.

Free-Plan Students

Calculate the mean, standard deviation, and sample size for free-plan students:

Calculate the t-statistic:

Look up the critical t-value using a t-distribution table to correspond to your chosen significance level (commonly 0.05) and calculated degrees of freedom.

Compare the t-statistic to the critical t-value:

$$If29.78\le -1.65,thenreject{H}_0$$

Conclusion: Fail to Reject  because the calculated t-statistic is higher than the critical value.

Alternatively, use the Two-Sample t-Test Assuming Unequal Variances that is part of the Data Analysis ToolPak to confirm the result:

For free-plan students: With a t-statistic of 29.78 (greater than the critical value of -1.645), you would fail to reject the null hypothesis. This means there’s not enough evidence to conclude that ${\mu }_1$ is smaller than ${\mu }_2$. So, the data supports the null hypothesis that ${\mu }_1$ is larger than or equal to ${\mu }_2$.

These results align with previous findings from the confidence intervals and further underscore the difference in engagement patterns between paid- and free-plan students.

Regarding the second part of the question, a Type I error (false positive) occurs when you reject the null hypothesis, which is true. In our case, this would mean concluding that engagement in 2022 is higher when it’s not. The probability of making this error is the level of significance, α. Since you (the researcher) choose the significance level of the hypothesis test, the responsibility for making this error lies solely on you.

Note that the significance level is closely related to the confidence level, representing our degree of certainty in the estimated results. It’s equal to (1 − α). For example, a significance level of 5% for a hypothesis test means a 5% probability of rejecting a true null hypothesis, corresponding to a 95% confidence level.

A Type II error (false negative) occurs when you fail to reject the null hypothesis, but it’s false. In our case, this would mean that the engagement in 2022 is not higher than it is.

The cost to the company of each type of error would depend on the implications of incorrectly concluding that engagement has increased—potentially leading to over-investment in certain features or complacency about needing to improve features—versus incorrectly concluding that engagement has not increased—potentially missing out on recognizing successful features or identifying areas that need improvement.

Task 5

You can find the solution to this problem in the Hypothesis Testing – task 5.xlsx file:

• Sheet: Task 5
• Sheet: Test for Variances

First, you must perform a two-sample f-test for variances to prove that assumption of unequal variances between the samples for free- and paid-plan subscribers:

The p-value indicates the probability of obtaining the observed f-value if the null hypothesis (equal variances) were true. The sample variances are not identical since the p-value is lower than 0. We must perform a left-tailed t-test assuming unequal variances: 

Calculate the mean, standard deviation, and sample size for both samples:

Calculate the t-statistic:

Look up the critical t-value using a t-distribution table to correspond to your chosen significance level (commonly 0.05) and calculated degrees of freedom.

Compare the t-statistic to the critical t-value:

$$If-1.21\le -1.65,thenreject{H}_0$$

Conclusion: Fail to Reject  because the calculated t-statistic is higher than the critical value.

Alternatively, use the Two-Sample t-Test Assuming Unequal Variances that is part of the Data Analysis ToolPak:

$$DecisionRule:Ifp-value\le 0.05,Reject{H}_0$$

Conclusion: Fail to Reject  because the p-value is higher than the specified significance level α (0.05).

If the hypothesis that US students watch more or an equal amount of content as Indian students is rejected, this suggests that US students watch less content on average than students in India.

This could have the following implications.

Market Differences: These details might indicate that the platform is more engaging or relevant to students in India than e US students. Understanding the reasons behind this could be valuable. Are the platform’s specific features, content, or aspects particularly appealing to Indian students? Such questions need to be addressed further but are beyond the scope of this analysis.

Growth Opportunities: If US engagement is lower, this could represent a growth opportunity. The 365 company might seek ways to increase engagement among US students, involving marketing efforts, adding more content relevant to US students, or other strategies.

Resource Allocation: This information could be helpful when deciding where to allocate resources. For example, if Indian students are more engaged, investing in more content and features targeted toward this audience might make sense.