To answer this question, we need to use the confidence interval for the difference in two independent proportions, because we are comparing the proportions of two populations (freshmen psychology majors and freshmen business majors) who changed their major before they graduated.

Here’s how we can approach this:

Step 1: Define the sample proportions and sample sizes.

• Population 1 (psychology majors):

  • $n_1 = 50$ (total number of psychology majors)
  • $x_1 = 23$ (number of psychology majors who changed their major)
  • Sample proportion for psychology majors: $\hat{p}_1 = \frac{x_1}{n_1} = \frac{23}{50} = 0.460$

• Population 2 (business majors):

  • $n_2 = 62$ (total number of business majors)
  • $x_2 = 29$ (number of business majors who changed their major)
  • Sample proportion for business majors: $\hat{p}_2 = \frac{x_2}{n_2} = \frac{29}{62} = 0.468$

Step 2: Calculate the standard error for the difference in proportions.

The standard error for the difference between two independent proportions is given by:

$SE = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$

Substitute the values:

$SE = \sqrt{\frac{0.460(1 - 0.460)}{50} + \frac{0.468(1 - 0.468)}{62}}$

$SE = \sqrt{\frac{0.460 \times 0.540}{50} + \frac{0.468 \times 0.532}{62}}$

$SE = \sqrt{\frac{0.2484}{50} + \frac{0.2489}{62}} = \sqrt{0.004968 + 0.004015} = \sqrt{0.008983} = 0.0948$

Step 3: Calculate the critical value (z-value) for a 95% confidence interval.

For a 95% confidence level, the critical value $z$ is approximately 1.96 (from standard normal distribution tables).

Step 4: Compute the margin of error.

The margin of error (ME) is calculated as:

$ME = z \times SE = 1.96 \times 0.0948 = 0.1858$

Step 5: Calculate the confidence interval.

The difference in sample proportions is:

$\hat{p}_1 - \hat{p}_2 = 0.460 - 0.468 = -0.008$

Now, calculate the 95% confidence interval:

$\text{Lower bound} = (\hat{p}_1 - \hat{p}_2) - ME = -0.008 - 0.1858 = -0.1938$ $\text{Upper bound} = (\hat{p}_1 - \hat{p}_2) + ME = -0.008 + 0.1858 = 0.1778$

Thus, the 95% confidence interval for the difference in proportions is:

$-0.194 < \hat{p}_1 - \hat{p}_2 < 0.178$

Step 6: Conclusion based on the confidence interval.

Since the confidence interval contains zero, it suggests that there is no statistically significant difference between the proportions of psychology majors and business majors who change their major before they graduate, at a 95% confidence level. Therefore, we conclude:

Freshmen psychology majors are just as likely to change their major before they graduate compared to freshmen business majors, with 95% confidence.

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Do you have any questions or need more details?

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Five related questions:

1. How do we interpret the margin of error in a confidence interval?
2. What happens if the confidence interval does not contain zero?
3. Why do we use a z-value of 1.96 for a 95% confidence interval?
4. How can increasing the sample size affect the width of the confidence interval?
5. How would the results differ if the confidence level was changed to 99%?

Tip:

When the confidence interval includes zero, it means there is no clear evidence to suggest a difference between the two proportions being compared.