At Western University the historical mean of scholarship...

At Western University the historical mean of scholarship examination scores for freshman applications is 900 . A historical population standard deviation $\sigma=180$ is assumed known. Each year, the assistant dean uses a sample of applications to determine whether the mean examination score for the new freshman applications has changed. a. State the hypotheses. b. What is the $95 \%$ confidence interval estimate of the population mean examination score if a sample of 200 applications provided a sample mean $\bar{x}=935$ ? c. Use the confidence interval to conduct a hypochesis test. Using $\alpha-.05$, what is your conclusion? d. What is the $p$-value?

At Western University the historical mean of scholarship examination scores for freshman applications is 900 . A historical population standard deviation $\sigma=180$ is assumed known. Each year, the assistant dean uses a sample of applications to determine whether the mean examination score for the new freshman applications has changed.
a. State the hypotheses.
b. What is the $95 \%$ confidence interval estimate of the population mean examination score if a sample of 200 applications provided a sample mean $\bar{x}=935$ ?
c. Use the confidence interval to conduct a hypochesis test. Using $\alpha-.05$, what is your conclusion?
d. What is the $p$-value?

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves stating a null hypothesis (typically a status quo statement) and an alternative hypothesis, followed by a test statistic analysis that concludes whether to reject the null hypothesis based on sample data.

Null and Alternative Hypotheses

In the context of testing a population mean, the null hypothesis (H?) typically asserts that the population mean is equal to a specific value, while the alternative hypothesis (H?) suggests that the mean is different from that value. This formulation is crucial for setting up the framework for a two-tailed test in which deviations in either direction are considered.

Z-Test for Population Mean

The Z-test is used when the population standard deviation is known and the sample size is sufficiently large. It involves calculating a test statistic by standardizing the difference between the sample mean and the hypothesized population mean. This test statistic is then compared to critical values from the standard normal distribution to determine whether to reject the null hypothesis.

Confidence Interval

A confidence interval provides a range of plausible values for the population parameter, based on the sample data. For a known population standard deviation and a large sample size, the interval is calculated using the Z distribution and reflects the precision of the sample estimate. It is commonly used to assess uncertainty around the estimated mean and to support or refute hypotheses regarding the population parameter.

Significance Level (?)

The significance level, denoted by alpha (?), represents the probability threshold for rejecting the null hypothesis. It quantifies the risk of committing a Type I error, which is rejecting a true null hypothesis. In hypothesis testing, if the p-value is less than ?, the null hypothesis is rejected, indicating that the result is statistically significant.

p-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one computed from the sample data, assuming that the null hypothesis is true. It serves as a measure of the strength of the evidence against the null hypothesis; a small p-value suggests that the observed data are unlikely under the null hypothesis, leading to its rejection.

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