Questions: What is the probability that the sample mean score is less than 500 round the answer to at least four decimal places. The probability that the sample mean score is less than 500 is assume that a given test mathematics SAT score has a standard deviation was 116. A sample of 65 scores is chosen.

What is the probability that the sample mean score is less than 500 round the answer to at least four decimal places. The probability that the sample mean score is less than 500 is assume that a given test mathematics SAT score has a standard deviation was 116. A sample of 65 scores is chosen.
Transcript text: What is the probability that the sample mean score is less than 500 round the answer to at least four decimal places. The probability that the sample mean score is less than 500 is assume that a given test mathematics SAT score has a standard deviation was 116. A sample of 65 scores is chosen.
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Solution

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Solution Steps

Step 1: Define the Problem

We need to find the probability that the sample mean score is less than 500 for a normally distributed population with a mean \( \mu = 500 \) and a standard deviation \( \sigma = 116 \). The sample size is \( n = 65 \).

Step 2: Calculate the Standard Error

The standard error (SE) of the sample mean is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{116}{\sqrt{65}} \approx 14.3538 \]

Step 3: Determine Z-scores

To find the Z-scores for the bounds, we use the formula: \[ Z = \frac{X - \mu}{SE} \] For the upper bound \( X = 500 \): \[ Z_{end} = \frac{500 - 500}{14.3538} = 0.0 \] For the lower bound, which approaches negative infinity: \[ Z_{start} = -\infty \]

Step 4: Calculate the Probability

Using the cumulative distribution function \( \Phi \), we find: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.0) - \Phi(-\infty) = 0.5 - 0 = 0.5 \]

Final Answer

The probability that the sample mean score is less than 500 is \[ \boxed{P = 0.5} \]

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