SAT scores: Assume that in a given year the mean mathematics SAT score was 498, and the standard deviation was 112 , A sample of 63 scores is chosen. Use the $71-84$ Plus calculator. Part 1 of 5 (a) What is the probability that the sample mean score is less than 481 ? Round the answer to at least four decimal places. The probability that the sample mean score is less than 481 is 0.1141 Part 2 of 5 (b) What is the probability that the sample mean score is between 465 and 500 ? Round the answer to at least four decimal places. The probability that the sample mean score is between 465 and 500 is 0.5467 . Part 3 of 5 (c) Find the $65^{\text {th }}$ percentile of the sample mean. Round the answer to at least two decimal places. The $65^{\text {th }}$ percentile of the sample mean is 503.43 Part 4 of 5 (d) Using a cutoff of 0.05 , would it be unusual if the sample mean were greater than 510 ? Round the answer to at least four decimal places. It would not $\mathbf{V}$ be unusual if the sample mean were greater than 510 , since the probability is 0.1980 . Part: $4 / 5$ Part 5 of 5 (e) Using a cutoff of 0.05 , do you think it would be unusual for an individual to get a score greater than 510 ? Explain. Assume the variable is normally distributed, Round the answer to at least four decimal places. (Choose one) $\mathbf{V}$, because the probability that an individual gets a score greater than 510 is
Solution
Step 1 ：Given that the mean SAT score is 498, the standard deviation is 112, and the sample size is 63.
Step 2 ：For part (a), we need to find the probability that the sample mean score is less than 481. This involves calculating a z-score and then finding the corresponding probability from a standard normal distribution. The calculated probability is approximately 0.1141.
Step 3 ：For part (b), we need to find the probability that the sample mean score is between 465 and 500. This involves calculating two z-scores and finding the probability between them. The calculated probability is approximately 0.5467.
Step 4 ：For part (c), we need to find the 65th percentile of the sample mean. This involves finding the z-score corresponding to the 65th percentile and then converting it back to a score. The calculated 65th percentile of the sample mean is approximately 503.43.
Step 5 ：For part (d), we need to determine if it would be unusual for the sample mean to be greater than 510, given a significance level of 0.05. This involves calculating the z-score for 510 and finding the corresponding probability. It would not be unusual if the sample mean were greater than 510, since the probability is approximately 0.1980, which is greater than the cutoff of 0.05.
Step 6 ：For part (e), we need to determine if it would be unusual for an individual to score greater than 510, given a significance level of 0.05. This involves calculating the z-score for 510 and finding the corresponding probability. It would not be unusual for an individual to get a score greater than 510, since the probability is approximately 0.4573, which is also greater than the cutoff of 0.05.
Step 7 ：Final Answer: (a) The probability that the sample mean score is less than 481 is \(\boxed{0.1141}\). (b) The probability that the sample mean score is between 465 and 500 is \(\boxed{0.5467}\). (c) The $65^\text{th}$ percentile of the sample mean is \(\boxed{503.43}\). (d) It would not be unusual if the sample mean were greater than 510, since the probability is \(\boxed{0.1980}\). (e) It would not be unusual for an individual to get a score greater than 510, since the probability is \(\boxed{0.4573}\).
