Answer :
Jessica should use an independent sample t-test to compare her students' post-course SAT scores to the population mean. Z-scores are used to find how many standard deviations away from the mean a score is, which can provide a way to compare scores from different tests such as the SAT and ACT. Hence the correct option would be option 3 independent sample t-test.
Jessica should use an independent sample t-test to test the effectiveness of an SAT preparation course. This is because she is comparing the SAT scores of her students after completion of the course to a known population mean and standard deviation. The independent sample t-test is appropriate when we compare the means of two separate groups and the population standard deviation is not known. In the case that the population standard deviation is known, a z-test could be used. However, in most educational and social science research, including this scenario, the t-test is preferred due to the unknown parameters of the population being approximated from the sample.
To calculate the z-score for an SAT score of 720 when the mean is 520 and the standard deviation is 115, you use the formula z = (x - \\(\\mu\\)) / o. Plugging in the numbers, z = (720 - 520) / 115, which equals approximately 1.74. This means that a score of 720 is 1.74 standard deviations above the mean.
To find what SAT score is 1.5 standard deviations above the mean, multiply the standard deviation by 1.5 and add it to the mean: 1.5 * 115 + 520 = 692.5. This score is higher than the average score and indicates a better performance.
For comparing SAT and ACT scores, you'd calculate the z-scores for each and compare them. For a score of 700 on the SAT with a mean of 514 and standard deviation of 117, and a score of 30 on the ACT with a mean of 21 and standard deviation of 5.3, the z-scores would allow you to see who scored better relative to their respective test's distribution.
