5.  Do graduate students use Library resources more than undergraduate students?   In a survey of 250 graduate students and 250 undergraduate students, 135 of the graduate students said they access Library resources every week, compared to 120 of the undergraduate students.

(a)  State the null and alternative hypotheses.

(b) At a 0.05 level of significance, is there evidence that graduate students access Library resources more frequently than undergraduate students?


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.

---

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.

Do graduate students use Library resources more than undergraduate students? In a survey of 250 graduate students and 250 undergraduate students, 135 of the graduate students said they access Library resources every week, compared to 120 of the undergraduate students.

Explore a hypothesis test comparing graduate and undergraduate students' use of library resources. This problem involves hypothesis testing for the difference between two proportions, using survey data from 250 graduate and 250 undergraduate students. Learn how to perform a z-test and calculate the z-statistic at a 0.05 significance level.

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