91Ó°ÊÓ

A confidence interval provides an estimated range, derived from sample data, to inform us about the population parameter with a certain level of confidence. In this exercise, we want to estimate the population mean examination score with a 95% confidence level.

The formula used is:

• \( \bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \)

Here:

• \(\bar{x}\) is the sample mean which equals 935.
• \(\sigma\) is the population standard deviation, 180 in this case.
• \(n\) is the sample size, which is 200 in this exercise.
• \(z_{\alpha/2}\) is the z-value that corresponds to the confidence level desired, here 1.96 for 95%.

By calculating the confidence interval:
\[935 \pm 1.96 \cdot \frac{180}{\sqrt{200}} \]
we find the interval to be [910.09, 959.91].

This tells us we are 95% confident that the true mean examination score lies between 910.09 and 959.91.

The null hypothesis \( H_0 \) represents the default scenario or status quo that there is no effect or difference. It is what we assume to be true until we have enough evidence to prove otherwise.

In the context of this exercise, our null hypothesis stating that the population mean exam score for freshmen is 900:

• \( H_0: \mu = 900 \)

The alternative hypothesis \( H_a \) suggests that there has been a change in the population mean:

• \( H_a: \mu eq 900 \)

When conducting hypothesis testing, if our observed data (the sample mean) falls in the extreme end of our confidence interval, we have grounds to reject \( H_0 \).

In this case, since the computed confidence interval [910.09, 959.91] for the sample mean does not include 900, we reject the null hypothesis, suggesting a significant change in mean examination scores.

The p-value is a critical concept in hypothesis testing that helps measure the strength of the evidence against the null hypothesis. It indicates the probability of observing results as extreme as (or more extreme than) the ones attained, assuming the null hypothesis is true.

In this exercise, the p-value plays a key role in decision-making. If the p-value is less than the significance level \( \alpha = 0.05 \), we reject the null hypothesis.

Given that the sample mean of 935 is outside the calculated confidence interval when considering the null hypothesis mean of 900, this suggests the p-value is indeed less than 0.05.

• This low p-value indicates strong evidence against the null hypothesis.
• Consequently, we conclude that the mean examination score for new freshmen applicants has changed from the historical mean of 900.

Understanding p-values helps us make informed decisions about the reliability and significance of our test results.