Questions: SAT scores: Assume that in a given year the mean mathematics SAT score was 499, and the standard deviation was 112. A sample of 63 scores is chosen. Use the TI-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.

SAT scores: Assume that in a given year the mean mathematics SAT score was 499, and the standard deviation was 112. A sample of 63 scores is chosen. Use the TI-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.

Transcript text: SAT scores: Assume that in a given year the mean mathematics SAT score was 499, and the standard deviation was 112. A sample of 63 scores is chosen. Use the TI-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.

Solution

Solution Steps

Step 1: Given Information

We are given the following parameters for the SAT scores:

Mean (\( \mu \)) = 499
Standard deviation (\( \sigma \)) = 112
Sample size (\( n \)) = 63

Step 2: Calculate the Standard Error

The standard error of the mean (\( SE \)) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{112}{\sqrt{63}} \approx 14.1168 \]

Step 3: Calculate the Z-Score

To find the probability that the sample mean score is less than 481, we first calculate the Z-score for 481 using the formula: \[ Z = \frac{X - \mu}{SE} = \frac{481 - 499}{14.1168} \approx -1.2756 \]

Step 4: Calculate the Probability

Using the Z-score, we find the probability: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(-1.2756) - \Phi(-\infty) \] From the output, we have: \[ P \approx 0.101 \]