SAT Verbal ScoresDo students who learned English and another l...

Question

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.

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

Accepted Answer

Welcome back everyone. In this problem, researchers select a random sample of 60 high school seniors who attended an SAT prep course. The average SAT math score is 570 with a sample standard deviation of 90. The national mean SAT math score is 550. Can the T distribution be used to test if the prep course students score higher than the national average? Verify all necessary conditions. A says yes, All requirements for the tea distribution are satisfied. B says no. The sample size is too small for the t distribution. C. No, the sample standard deviation is not given, and D no, the sample is not random. Now, let's look at our problem statement and try to figure out if the information that we're given satisfies the conditions for us to use a T distribution. What do we know? Well, for starters, we're told that researchers select a random sample. So the sample is random. What else do we know? Well, next it says that random sample has 60 high school seniors, so N is equal to 60 or sample size, and that is greater than 30. Therefore, that means the central limit theorem applies here, which means that we can treat this as a normal distribution. Now what else do we know? We're told that the average score is 570 with a sample standard deviation of 90. So in this case, since we know our sample standard deviation S, which is 90, but we don't know what our population standard deviation is, OK. We don't know what our population deviation is. In other words, we don't know what sigma would be, and this is one of the conditions for us to use the T distribution. If the population standard deviation is unknown, while the sample standard deviation is given. Now another thing that helps. we're told here that well if if our sample size is 60 we know for sure that that is much less than 10% of all high school seniors, OK? So in other words, N is going to be less than 10% of the population. So that means independence is reasonable. And with our sample size N equals 60, the T test is robust to moderate skewness and odd layers. Therefore, all requirements for using the T distribution are satisfied. A is the correct answer. If we look back at our answer choices, we can read through the rest to make sure that A is correct. In B, it says no because the sample size is too small for the T distribution, but that is sufficient. That sample size is sufficient for a T distribution. In C, it says the sample standard deviation is not given, but that's not true because we know it is 90 and D says the sample is not random, but the problem statement told us the sample is random. So that's how we're sure A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Hypothesis Testing

Conditions for Using the t-Distribution

Sample Mean and Standard Deviation

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