"[NW] Small-Sample Hypothesis TestProfessors Honey Kirk and Diane Lerma of Palo Alto College developed a “learning community curriculum that blended the developmental mathematics and the reading curriculum with a structured emphasis on study skills.” In a typical developmental mathematics course at Palo Alto College, 50% of the students complete the course with a letter grade of A, B, or C. In the experimental course, of the 16 students enrolled, 11 completed the course with a letter grade of A, B, or C. Do you believe the experimental course was effective at the α = 0.05 level of significance?


a. State the appropriate null and alternative hypotheses.


Source: Kirk, Honey and Lerma, Diane, “Reading Your Way to Success in Mathematics: A Paired Course of Developmental Mathematics and Reading.” MathAMATYC Educator, Vol. 1 No. 2, 2010."

Verified step by step guidance

1. Identify the parameter of interest: here, it is the population proportion of students who complete the course with a grade of A, B, or C. Denote this proportion as \(p\).
2. State the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)): since the typical course has a success rate of 50%, the null hypothesis assumes \(p = 0.5\). The alternative hypothesis tests whether the experimental course is more effective, so \(p > 0.5\).
3. Formally write the hypotheses as: \[H_0: p = 0.5\] \[H_a: p > 0.5\]
4. Determine the significance level, which is given as \(\alpha = 0.05\), and decide that this will be a right-tailed test because the alternative hypothesis is testing for an increase in the proportion.
5. Plan to use a small-sample hypothesis test for a proportion since the sample size is 16 (which is small). This typically involves using the binomial distribution or an exact test rather than a normal approximation.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. It involves formulating two competing statements about a population parameter and using sample data to assess their plausibility at a chosen significance level.

Null and Alternative Hypotheses

The null hypothesis (H0) represents the default or status quo assumption, often stating no effect or no difference. The alternative hypothesis (Ha) reflects the claim we want to test, such as an improvement or change. Correctly stating these hypotheses is essential for guiding the test and interpreting results.

Small-Sample Testing and Significance Level

Small-sample testing applies when the sample size is too small for normal approximation, often requiring exact or specialized tests. The significance level (α) is the threshold probability for rejecting the null hypothesis, commonly set at 0.05, indicating a 5% risk of a Type I error (false positive).