To test whether the proportion of the letter "a" has changed in modern times compared to the 1800s, we can set up a hypothesis test using the 0.05 level of significance. Here's how you can approach this:
1. **Formulate the Hypotheses:**
- Null Hypothesis (H0): The proportion of a's in modern times is the same as in the 1800s (p = 0.08).
- Alternative Hypothesis (H1): The proportion of a's in modern times is different from the 1800s (p ≠ 0.08).
2. **Calculate the Sample Proportion:**
- Sample proportion (p-hat) = Number of a's in the sample / Total sample size
- p-hat = 25/500 = 0.05
3. **Check Conditions for a Hypothesis Test:**
- Random Sample: The newspaper sample is considered random.
- Sample Size: The sample size is sufficiently large (500 > 30) for normal approximation to apply.
- Independence: Each letter in the sample should be independent.
4. **Perform the Hypothesis Test:**
- Calculate the test statistic using the formula z = (p-hat - p) / sqrt[(p * (1 - p)) / n], where p is the hypothesized proportion (0.08) and n is the sample size (500).
- Calculate the critical z-value at the 0.05 level of significance for a two-tailed test.
- Compare the calculated z-value with the critical z-value to determine statistical significance.
5. **Make a Conclusion:**
- If the calculated z-value falls in the rejection region (outside the critical values), we reject the null hypothesis.
- If the calculated z-value does not fall in the rejection region, we fail to reject the null hypothesis.
By following these steps, you can evaluate whether there is sufficient evidence to conclude that the proportion of a's has changed in modern times compared to the 1800s.
