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In statistics, the sample proportion is a measure used to understand how a sample reflects certain traits or characteristics of the population. In this exercise, the sample proportion is calculated from the students surveyed who approve of the new parking arrangement. Out of the 200 students sampled, 83 gave their approval.

To find the sample proportion, you simply divide the number of favorable outcomes by the total sample size. Therefore, the sample proportion, usually denoted as \( \hat{p} \), is given by:

• \( \hat{p} = \frac{83}{200} = 0.415 \)

This statistic allows us to make inferences about the larger population. Here, it helps in understanding if there is a significant difference in approval rates before and after the modification.

The standard error (SE) quantifies the variability of the sample proportion as an estimate of the true population proportion. It's a critical part of hypothesis testing as it aids in determining how accurately your sample statistic reflects the population parameter.

In this problem, the formula for the standard error of the sample proportion \( \hat{p} \) is:

• \( SE = \sqrt{\frac{p_0(1 - p_0)}{n}} \)

This calculates to approximately \( SE = 0.034 \) using \( p_0 = 0.37 \) and \( n = 200 \).

It is essential as it provides information about the spread or dispersion of the sample proportion around the population proportion, thus forming the basis for the test statistic.

The test statistic is a function of the sample data that helps in making decisions regarding the hypotheses. In this scenario, a z-test is applied since we deal with proportions. The z-statistic, specifically, tells us how many standard errors \( \hat{p} \) is away from the null hypothesis proportion \( p_0 \).

The formula for the test statistic \( z \) in this context is:

• \( z = \frac{\hat{p} - p_0}{SE} \)

Substituting the given values, we derive \( z \approx 1.324 \).

This calculation illustrates how far the observed sample proportion is from the expected proportion, measured in terms of standard error.

The significance level, denoted by \( \alpha \), is a threshold used to judge the strength of evidence against the null hypothesis. It's a pre-determined probability that dictates how extreme the data must be to reject the null hypothesis.

Typically set at 0.05, this level implies a 5% risk of rejecting the null hypothesis when it is true. In our exercise, the significance level is \( \alpha = 0.05 \). We then compare the test statistic's z-value to its corresponding critical value. For a one-tailed test at \( \alpha = 0.05 \), the critical z-value is approximately 1.645.

In this case, since the calculated z-value of 1.324 does not exceed the critical value, we fail to reject the null hypothesis. This means there's not sufficient evidence at the 5% significance level to conclude that the parking situation improvement resulted in increased student approval.