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A confidence interval provides a range of values within which we can be fairly certain the true proportion exists. In hypothesis testing, when comparing a sample statistic to a claimed population parameter, confidence intervals help validate our findings.To calculate it, we use the formula: \[ \hat{p} \pm z_{\alpha/2} \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]where:

• \( \hat{p} \) is the observed sample proportion. In this case, \( \hat{p} = \frac{117}{484} \approx 0.2415 \).
• \( z_{\alpha/2} \) is the z-value for the desired confidence level (e.g., 1.96 for a 95% confidence level).
• \( n \) is the sample size, here 484.

The Confidence Interval provides insights about the population parameter. Here, it ranges from 0.2025 to 0.2805, indicating where the true proportion of engineers planning further study might lie. Since 0.5 (Fortune's claim) is not within this range, it suggests that their claim is inconsistent with the data.

The p-value is a critical concept in hypothesis testing. It quantifies how likely the observed data would occur if the null hypothesis is true. A small p-value indicates that the observed data is unlikely under the null hypothesis.To find the p-value, we use the calculated test statistic, which, for proportions, is represented by the z-value:\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]With our example, the z-value is approximately \(-11.39\).Looking up this z-value in the standard normal distribution, we find that a value this extreme would occur less than 0.05% of the time if the null hypothesis were true. Thus, the p-value is nearly 0, much smaller than the predetermined \( \alpha = 0.05 \).This small p-value leads us to conclude that there is strong evidence against the null hypothesis, indicating that Fortune's claim is likely inaccurate for the sample studied.

A proportion test is used when we aim to compare a sample proportion against a claimed population proportion. It helps determine if there is a significant difference between the two proportions.In this exercise, our goal is to see if the sample proportion (engineers planning further studies) contradicts the proportion claimed by Fortune (0.5).The steps include:

• State the null hypothesis \( H_0: p = 0.5 \) and the alternative hypothesis \( H_a: p eq 0.5 \).
• Calculate the sample proportion \( \hat{p} = \frac{117}{484} \).
• Use the z-test formula to obtain the test statistic.
• Decide based on the p-value or confidence interval if there’s enough evidence to reject the null hypothesis.

The outcome in our example shows that the data significantly deviates from the claimed proportion, as evidenced by both a p-value near 0 and a confidence interval excluding 0.5.

The null hypothesis is a statement of no effect or no difference, serving as a starting point in hypothesis testing.In many testing scenarios, including the current exercise, it assumes a specific value for a population parameter. Here it suggests that \( H_0: p = 0.5 \), meaning the true proportion of engineers planning further studies is 50% or equivalent to the Fortune claim.The role of the null hypothesis:

• Provides a basis for comparison. We assess if the observed data significantly deviates from the presumption made by the null hypothesis.
• Enables us to form an alternative hypothesis, here \( H_a: p eq 0.5 \), to check for possible deviations.
• Directs the decision-making in hypothesis testing. If the observed results produce a small p-value, we reject the null hypothesis.

In our exercise, rejecting the null hypothesis indicates that the observed fraction of graduates planning further study deviates significantly from Fortune's claim, suggesting their statement isn't supported by our data.