This problem involves testing whether graduate students access library resources more frequently than undergraduate students. It is a hypothesis test for the difference between two proportions.

Let's go step by step:

(a) Null and Alternative Hypotheses:

• Null Hypothesis ($H_0$): $p_1 \leq p_2$ (The proportion of graduate students who access library resources is less than or equal to the proportion of undergraduate students who do).

• Alternative Hypothesis ($H_A$): $p_1 > p_2$ (The proportion of graduate students who access library resources is greater than the proportion of undergraduate students who do).

Where:

• $p_1$ is the proportion of graduate students who access library resources weekly.
• $p_2$ is the proportion of undergraduate students who access library resources weekly.

(b) Test for Significance (at $\alpha = 0.05$):

We will use a z-test for the difference between two proportions to determine if there is enough evidence to reject the null hypothesis.

Step 1: Sample Proportions

• Sample proportion of graduate students: $\hat{p_1} = \frac{135}{250} = 0.54$
• Sample proportion of undergraduate students: $\hat{p_2} = \frac{120}{250} = 0.48$

Step 2: Combined Proportion (Pooled Proportion)

The combined proportion $\hat{p}$ is used under the null hypothesis: $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{135 + 120}{250 + 250} = \frac{255}{500} = 0.51$

Step 3: Standard Error (SE) of the Difference

$SE = \sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} = \sqrt{0.51(1 - 0.51)\left(\frac{1}{250} + \frac{1}{250}\right)}$

Step 4: Calculate the z-Statistic

The z-statistic is given by: $z = \frac{\hat{p_1} - \hat{p_2}}{SE}$

Let me calculate the standard error and the z-value.The calculated z-statistic is approximately $z = 1.34$, and the standard error (SE) is approximately $0.0447$.

Step 5: Determine the Critical Value and Make a Decision

For a one-tailed test at the $\alpha = 0.05$ level of significance, the critical value of $z$ from the standard normal distribution is approximately $1.645$.

• Since $1.34 < 1.645$, we fail to reject the null hypothesis.

Conclusion:

At the 0.05 level of significance, there is not enough evidence to conclude that graduate students access library resources more frequently than undergraduate students.

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Would you like further details on any part of this solution? Here are five related questions to expand on:

1. How does the choice of significance level ($\alpha$) affect hypothesis testing?
2. What would the result be if a two-tailed test were conducted?
3. What are the assumptions behind using a z-test for two proportions?
4. How does sample size impact the z-statistic in hypothesis testing?
5. Can this result change with a different level of significance?

Tip: In hypothesis testing, failing to reject the null hypothesis doesn't mean it's true; it simply means there's insufficient evidence against it at the given significance level.