College Athletics: Graduation Rate Women athletes at the...

College Athletics: Graduation Rate Women athletes at the University of Colorado, Boulder, have a long-term graduation rate of $67 \%$ (Source: Chronicle of Higher Education). Over the past several years, a random sample of 38 women athletes at the school showed that 21 eventually graduated. Does this indicate that the population proportion of women athletes who graduate from the University of Colorado, Boulder, is now less than $67 \% ?$ Use a $5 \%$ level of significance.

Key Concepts

p-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It provides a measure of the strength of evidence against the null hypothesis, and if it is smaller than the significance level, the null hypothesis is rejected.

Test Statistic (Z-test)

The test statistic in a Z-test for proportions quantifies how far the observed sample proportion is from the hypothesized population proportion, in units of its standard error. This standardized value is compared against a critical value from the normal distribution to decide whether to reject the null hypothesis.

Significance Level

The significance level, often denoted by alpha (?), is the probability threshold for rejecting the null hypothesis when it is actually true. It quantifies the risk of a Type I error. Commonly set at 5%, it defines the critical region in the test statistic’s distribution.

Sampling Distribution

The sampling distribution refers to the probability distribution of a statistic (like the sample proportion) calculated from a random sample. Under the Central Limit Theorem, when the sample size is sufficiently large, the distribution of the sample proportion approximates a normal distribution, which is essential for conducting a Z-test.

Null and Alternative Hypotheses

In any hypothesis test, the null hypothesis (H0) is a statement of no change or status quo, while the alternative hypothesis (H1) is what the researcher is seeking evidence for. In tests involving proportions, H0 might state that the true proportion equals a given value, and H1 states that it is either less than, greater than, or not equal to that value.

Population Proportion

The population proportion is a parameter representing the fraction of individuals in a population that exhibit a certain characteristic. In statistical tests concerning proportions, researchers assess whether the observed sample proportion deviates significantly from a hypothesized population proportion.

Hypothesis Testing

Hypothesis testing is a statistical framework used to make inferences about a population parameter based on sample data. It involves formulating a null hypothesis that represents a baseline assumption and an alternative hypothesis that reflects the claim or effect being tested, then using a test statistic to determine whether to reject the null hypothesis.

One-tailed Test

A one-tailed test is used when the research hypothesis predicts a specific direction of the effect. For instance, if the claim is that the true proportion is less than a specified value, a one-tailed test is appropriate, focusing all the rejection region on one side of the distribution.

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