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Students at a major university believe they can save money buying textbooks online rather than at the local bookstores. In order to test this theory, they randomly sampled 2525 textbooks on the shelves of the local bookstores. The students then found the "best" available price for the same textbooks via online retailers. The prices for the textbooks are listed in the following table. Based on the data, is it less expensive for the students to purchase textbooks from the online retailers than from local bookstores? Use α=0.01α=0.01. Let prices at local bookstores represent Population 1 and prices at online retailers represent Population 2.

To determine if it's less expensive for students to purchase textbooks online rather than from local bookstores, we can conduct a hypothesis test. Given that we are working with two populations (bookstore prices and online retailer prices), we will perform a paired t-test, since we have paired samples for the same textbooks in both groups.

### Step-by-Step Hypothesis Test:

1. **Hypotheses:**
   - \( H_0 \) (null hypothesis): There is no difference in the average prices between bookstores and online retailers. In other words, the mean difference is zero.
     \[
     H_0: \mu_d = 0
     \]
   - \( H_A \) (alternative hypothesis): The average price at online retailers is less than at local bookstores.
     \[
     H_A: \mu_d > 0
     \]
     (where \( d \) is the difference between bookstore price and online retailer price).

2. **Significance Level:**
   - We are given a significance level \( \alpha = 0.01 \).

3. **Test Statistic:**
   - We will use a paired t-test, where the test statistic is calculated as:
     \[
     t = \frac{\bar{d}}{(s_d/\sqrt{n})}
     \]
     - \( \bar{d} \) is the mean of the differences between paired samples.
     - \( s_d \) is the standard deviation of the differences.
     - \( n \) is the number of paired observations.

4. **Data Extraction:**
   Let's extract the data for both bookstore and online retailer prices, calculate the differences, and proceed with the t-test.

I will now calculate these values using the dataset you've provided.### Results of the Paired T-test:

- **Mean difference** (\( \bar{d} \)): 9.15
- **Standard deviation** (\( s_d \)): 12.70
- **Number of textbooks** (\( n \)): 13
- **Test statistic** (\( t \)): 2.60
- **P-value**: 0.0116

### Conclusion:

At a significance level of \( \alpha = 0.01 \), the p-value (0.0116) is slightly greater than 0.01. Therefore, we **fail to reject the null hypothesis**. This means that, based on the given data, there is not enough statistical evidence at the 1% significance level to conclude that textbooks are less expensive online than at local bookstores.

Would you like more details or further explanation?

### Related Questions:
1. How do we interpret the results of the paired t-test in real-life situations?
2. What are the assumptions behind the paired t-test, and were they met here?
3. How would the conclusion change if we used a different significance level, like \( \alpha = 0.05 \)?
4. Can you explain the relationship between the p-value and the test statistic?
5. What other types of statistical tests could be used for similar datasets?

### Tip:
When interpreting p-values, remember that a small p-value (less than \( \alpha \)) means there is evidence to reject the null hypothesis, but it does not measure the size of the effect. Always consider the practical significance as well!

Students at a major university believe they can save money buying textbooks online rather than at the local bookstores. In order to test this theory, they randomly sampled 25 textbooks on the shelves of the local bookstores. The students then found the 'best' available price for the same textbooks via online retailers. The prices for the textbooks are listed in the table. Based on the data, is it less expensive for the students to purchase textbooks from the online retailers than from local bookstores? Use α=0.01.

Perform a paired t-test to determine if it is cheaper to buy textbooks online than from bookstores. Using a sample of 25 textbooks and a significance level of α=0.01, this analysis shows whether online retailers offer a better deal for students.

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