SAT Verbal ScoresDo students who learned English and another language simultaneously score worse on the SAT Critical Reading exam than the general population of test takers? The mean score among all test takers on the SAT Critical Reading exam is 501. A random sample of 100 test takers who learned English and another language simultaneously had a mean SAT Critical Reading score of 485 with a standard deviation of 116. Do these results suggest that students who learn English as well as another language simultaneously score worse on the SAT Critical Reading exam?


a. State the appropriate null and alternative hypotheses.

Verified step by step guidance

1. Identify the parameter of interest: the population mean SAT Critical Reading score for students who learned English and another language simultaneously.
2. State the null hypothesis (\(H_0\)) as the claim that the mean score for these students is equal to the general population mean, i.e., \(H_0: \mu = 501\).
3. State the alternative hypothesis (\(H_a\)) as the claim that the mean score for these students is less than the general population mean, i.e., \(H_a: \mu < 501\).
4. Explain that this is a one-tailed test because we are specifically testing if the mean score is lower, not just different.
5. Clarify that these hypotheses set the framework for conducting a hypothesis test to determine if the observed sample mean provides enough evidence to conclude that students who learned English and another language simultaneously score worse.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a presumed statement (null hypothesis) about a population parameter. It involves formulating a null hypothesis (no effect or difference) and an alternative hypothesis (presence of an effect or difference), then using sample data to assess their plausibility.

Null and Alternative Hypotheses

The null hypothesis (H0) represents the default assumption, often stating no difference or no effect, while the alternative hypothesis (Ha) represents the claim we want to test. In this context, H0 would state that the mean SAT score of bilingual students equals the general population mean, and Ha would state it is lower.

Sampling Distribution and Standard Error

The sampling distribution describes how sample means vary around the population mean. The standard error measures the typical distance between a sample mean and the population mean, calculated as the sample standard deviation divided by the square root of the sample size. It is crucial for determining how unusual the observed sample mean is under the null hypothesis.