Performing a Sign Test In Exercises 7–22, (b) find the critical value,
[APPLET] SAT Scores A guidance counselor claims that students who take the SAT twice will improve their scores the second time that they take the SAT. The table shows both math SAT scores for 12 students who took the SAT twice. At α=0.01, can you support the guidance counselor’s claim?

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Step 1: Understand the problem. The goal is to test the guidance counselor's claim that students improve their SAT scores the second time they take the test. This is a paired data problem, and we will use the Sign Test to evaluate the claim at a significance level of α = 0.01.

Step 2: Calculate the differences between the second SAT score and the first SAT score for each student. For each pair of scores, subtract the first SAT score from the second SAT score. Record whether the difference is positive, negative, or zero.

Step 3: Count the number of positive differences (denoted as S+), the number of negative differences (denoted as S−), and the number of zero differences. Zero differences are excluded from the test.

Step 4: Determine the critical value for the Sign Test at α = 0.01. Use the binomial distribution table or a Sign Test critical value table for the given sample size (n = number of non-zero differences) and α = 0.01. The critical value is the maximum number of negative signs (or positive signs) allowed to reject the null hypothesis.

Step 5: Compare the smaller of S+ and S− to the critical value. If the smaller value is less than or equal to the critical value, reject the null hypothesis (no improvement). Otherwise, fail to reject the null hypothesis. Interpret the result in the context of the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sign Test

The Sign Test is a non-parametric statistical method used to determine if there is a significant difference between paired observations. It focuses on the direction of changes (increases or decreases) rather than the magnitude, making it suitable for ordinal data or when the assumptions of normality are not met. In this context, it will help assess whether students' SAT scores improved after retaking the test.

Critical Value

The critical value is a threshold that determines the boundary for rejecting the null hypothesis in hypothesis testing. It is derived from the significance level (α), which indicates the probability of making a Type I error. For a significance level of α=0.01, the critical value will help identify whether the observed test statistic falls into the rejection region, thus supporting or refuting the guidance counselor's claim about score improvement.

Hypothesis Testing

Hypothesis testing is a statistical procedure that evaluates two competing hypotheses: the null hypothesis (H0), which states there is no effect or difference, and the alternative hypothesis (H1), which suggests there is an effect or difference. In this scenario, the null hypothesis would assert that there is no improvement in SAT scores after retaking the test, while the alternative would claim that scores do improve, guiding the analysis of the data collected.

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