Questions: SAT scores: Assume that in a given year the mean mathematics SAT score was 595, and the standard deviation was 134. A sample of 75 scores is chosen. Use the T1-84 Plus calculator. Part 1 of 5 (a) What is the probability that the sample mean score is less than 580? Round the answer to at least four decimal places. The probability that the sample mean score is less than 580 is . Part 2 of 5 (b) What is the probability that the sample mean score is between 550 and 600? Round the answer to at least four decimal places. The probability that the sample mean score is between 550 and 600 is .

SAT scores: Assume that in a given year the mean mathematics SAT score was 595, and the standard deviation was 134. A sample of 75 scores is chosen. Use the T1-84 Plus calculator. Part 1 of 5 (a) What is the probability that the sample mean score is less than 580? Round the answer to at least four decimal places. The probability that the sample mean score is less than 580 is . Part 2 of 5 (b) What is the probability that the sample mean score is between 550 and 600? Round the answer to at least four decimal places. The probability that the sample mean score is between 550 and 600 is .
Transcript text: SAT scores: Assume that in a given year the mean mathematics SAT score was 595, and the standard deviation was 134. A sample of 75 scores is chosen. Use the T1-84 Plus calculator. Part 1 of 5 (a) What is the probability that the sample mean score is less than 580? Round the answer to at least four decimal places. The probability that the sample mean score is less than 580 is $\square$ $\square$. Part 2 of 5 (b) What is the probability that the sample mean score is between 550 and 600? Round the answer to at least four decimal places. The probability that the sample mean score is between 550 and 600 is $\square$.
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Solution

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Solution Steps

Step 1: Probability that the Sample Mean Score is Less than 580

To find the probability that the sample mean score is less than 580, we calculated the cumulative distribution function (CDF) for a normal distribution with a mean of \( \mu = 595 \) and a standard deviation of the sample mean \( \sigma_{\bar{x}} = \frac{134}{\sqrt{75}} \). The result is:

\[ P(\bar{X} < 580) = 0.1662 \]

Step 2: Probability that the Sample Mean Score is Between 550 and 600

Next, we calculated the probability that the sample mean score is between 550 and 600. This involves finding the CDF values at both 550 and 600 and then taking the difference:

\[ P(550 < \bar{X} < 600) = P(\bar{X} < 600) - P(\bar{X} < 550) = 0.6249 \]

Final Answer

The probability that the sample mean score is less than 580 is \( \boxed{0.1662} \) and the probability that the sample mean score is between 550 and 600 is \( \boxed{0.6249} \).

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