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?


b. Verify that the requirements to perform the test using the t-distribution are satisfied.

Verified step by step guidance

1. Identify the conditions required to use the t-distribution for hypothesis testing. These typically include: (1) the sample is a simple random sample, (2) the population distribution is approximately normal or the sample size is large enough for the Central Limit Theorem to apply, and (3) the population standard deviation is unknown.
2. Check if the sample is a simple random sample or representative of the population. The problem states it is a random sample of 100 test takers, so this condition is satisfied.
3. Evaluate the sample size. Since the sample size is 100, which is greater than 30, the Central Limit Theorem suggests the sampling distribution of the sample mean will be approximately normal, even if the population distribution is not perfectly normal.
4. Confirm that the population standard deviation is unknown. The problem provides a sample standard deviation (116) but does not provide the population standard deviation, so this condition is met.
5. Conclude that all requirements for using the t-distribution are satisfied because the sample is random, the sample size is large, and the population standard deviation is unknown.

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. In this context, it involves comparing the sample mean SAT score of bilingual students to the known population mean to determine if the difference is statistically significant.

Conditions for Using the t-Distribution

The t-distribution is used when the population standard deviation is unknown and the sample size is relatively small. Key conditions include that the sample is randomly selected, the data are approximately normally distributed or the sample size is large enough (usually n > 30) for the Central Limit Theorem to apply.

Sample Mean and Standard Deviation

The sample mean is the average score from the sample group, and the sample standard deviation measures the variability of scores within that group. These statistics are essential for calculating the t-statistic and assessing how the sample compares to the population mean.