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?


d. Write a conclusion based on your results to part (c).

Verified step by step guidance

1. Identify the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). Here, \(H_0\): The mean SAT score of bilingual students is equal to the general population mean, i.e., \(\mu = 501\). \(H_a\): The mean SAT score of bilingual students is less than the general population mean, i.e., \(\mu < 501\) (since we want to test if they score worse).
2. Determine the appropriate test statistic to use. Since the population standard deviation is unknown and the sample size is large (n=100), use the one-sample t-test. The test statistic formula is: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] where \(\bar{x}\) is the sample mean, \(\mu_0\) is the population mean under \(H_0\), \(s\) is the sample standard deviation, and \(n\) is the sample size.
3. Calculate the test statistic by substituting the values: sample mean \(\bar{x} = 485\), population mean \(\mu_0 = 501\), sample standard deviation \(s = 116\), and sample size \(n = 100\).
4. Find the critical value or p-value corresponding to the test statistic from the t-distribution with \(n-1 = 99\) degrees of freedom, for a one-tailed test at the chosen significance level (commonly \(\alpha = 0.05\)).
5. Compare the test statistic to the critical value or compare the p-value to \(\alpha\). If the test statistic falls in the rejection region or the p-value is less than \(\alpha\), reject the null hypothesis and conclude that students who learned English and another language simultaneously score worse on the SAT Critical Reading exam. Otherwise, do not reject the null hypothesis.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis about a population parameter. It involves formulating a null hypothesis (e.g., no difference in scores) and an alternative hypothesis (e.g., scores are lower), then using sample data to assess the likelihood of observing the results if the null is true.

Sampling Distribution and Standard Error

The sampling distribution describes the distribution of sample means from repeated samples of the same size. The standard error measures the variability of the sample mean around the population mean, calculated as the sample standard deviation divided by the square root of the sample size. It helps determine how unusual the sample mean is.

Significance Level and p-value

The significance level (commonly 0.05) is the threshold for deciding whether to reject the null hypothesis. The p-value quantifies the probability of obtaining a sample result at least as extreme as the observed one, assuming the null hypothesis is true. A p-value less than the significance level indicates strong evidence against the null.