Question

The ACT test has a mean score of 23 and a standard deviation of 6, while the SAT test has a mean score of 1,065 and a standard deviation of 215.

a) Melissa took both tests. She scored 25 on the ACT and 1,335 on the SAT. Did Melissa do better on the ACT or the SAT? Use z-scores to find out.

b) Eddy took the SAT and scored 2.2 standard deviations above the mean. What was Eddy’s score?

c) Alisha scored in the 80th percentile on the SAT, which means her score was higher than 80% of the scores. What was Alisha’s SAT score?

d) What proportion of the population of students who take the SAT score between 700 and 800?

e) What proportion of the population of students who take the SAT score between 1,000 and 1,200?
f) What is the probability that a random sample of 9 students would have a mean SAT score below 900?

g) Theoretically, if you were to repeatedly select samples of 25 students taking the SAT, what would the average of all the sample means be?

h) How much error would you expect, on average, between the sample mean and the population mean for a sample of 25 students on the SAT?

Answer 1

a) Melissa did better on the SAT. b) Eddy's score is approximately 771.4. c) Alisha's SAT score is approximately 617.88.

Step-by-step explanation:

a) Melissa: To determine if Melissa did better on the ACT or SAT, we will calculate the z-scores for each test score. For the ACT, the z-score is calculated as (ACT score - mean) / standard deviation = (25 - 23) / 6 = 0.33. For the SAT, the z-score is calculated as (SAT score - mean) / standard deviation = (1335 - 1065) / 215 = 1.26. Since the z-score for the SAT is higher than the z-score for the ACT, Melissa did better on the SAT.

b) Eddy: To find Eddy’s score, we can use the z-score formula. Given that the z-score is 2.2 standard deviations above the mean, we can compute Eddy’s score as mean + (z-score x standard deviation) = 514 + (2.2 x 117) = 514 + 257.4 ≈ 771.4. Therefore, Eddy’s score is approximately 771.4.

c) Alisha: To find Alisha’s SAT score, we need to find the score that falls in the 80th percentile. The 80th percentile means that 80% of students scored below Alisha’s score. Using a z-table, we can find the z-score that corresponds to the 80th percentile, which is approximately 0.84. Using the formula z = (x - mean) / standard deviation, we can solve for x to find the SAT score: x = mean + (z x standard deviation) = 514 + (0.84 x 117) ≈ 617.88. Therefore, Alisha’s SAT score is approximately 617.88.