To answer the question, let's go through each part step-by-step.

Problem Overview

The professor wants to determine if a new software program will increase students' scores. Previous scores (without the software) averaged at 115 points out of 150. In a trial with the software, 235 students scored an average of 118 points with a standard deviation of 8.6 points.

Part a) Hypothesis Testing

We will perform a hypothesis test to see if the observed increase in scores is statistically significant.

Step 1: Define the Null and Alternative Hypotheses

Let:

• $\mu_0 = 115$: The historical mean score without the software.

The hypotheses are:

• Null Hypothesis (H₀): The software does not increase scores. $H_0: \mu = 115$
• Alternative Hypothesis (Hₐ): The software increases scores. $H_A: \mu > 115$

This is a one-tailed test since we are only interested in whether the software increases the scores, not if it decreases them.

Step 2: Test Statistic

Since we know the sample mean, population mean, sample size, and sample standard deviation, we can use a t-test to compute the test statistic.

The formula for the test statistic $t$ is: $t = \frac{\bar{X} - \mu_0}{\frac{s}{\sqrt{n}}}$ where:

• $\bar{X} = 118$: sample mean,
• $\mu_0 = 115$: population mean under $H_0$,
• $s = 8.6$: sample standard deviation,
• $n = 235$: sample size.

Let's substitute these values: $t = \frac{118 - 115}{\frac{8.6}{\sqrt{235}}}$

First, calculate the denominator: $\frac{8.6}{\sqrt{235}} \approx \frac{8.6}{15.33} \approx 0.561$

Then, calculate $t$: $t = \frac{3}{0.561} \approx 5.35$

Step 3: Compare with Critical Value

Using a significance level of $\alpha = 0.05$, we look up the critical $t$-value for a one-tailed test with $n - 1 = 234$ degrees of freedom. For large $n$, the critical value for $\alpha = 0.05$ in a one-tailed test is approximately 1.645.

Since our calculated $t$-value of 5.35 is much greater than 1.645, we reject the null hypothesis.

Conclusion

There is enough statistical evidence to suggest that the software does improve students' scores. The professor should consider spending money on the software.

Would you like more details or explanations on any step? Here are five related questions that can deepen your understanding:

1. How would the conclusion change if the sample mean was lower than 115?
2. What would the test look like if we used a two-tailed hypothesis?
3. How do changes in the sample size affect the test statistic in hypothesis testing?
4. What is the interpretation of a p-value in this context?
5. What are the implications of Type I and Type II errors in this decision?

Tip: Always check if the sample size is large enough for the Central Limit Theorem to apply when performing hypothesis testing with sample means.