Questions: SAT Scores The national average SAT score (for verbal and math) is 1028. Suppose that nothing is known about the shape of the distribution and that the standard deviation is 100. Use The Standard Normal Distribution Table. Round the final answer to at least four decimal places and intermediate z-value calculations to two decimal places. Part 1 of 2 If a random sample of 240 scores was selected, find the probability that the sample mean is greater than 1054. Assume that the sample is taken from a large population and the correction factor can be ignored. P(X̄>1054)=0.0 Part 2 of 2 If a random sample of 240 scores were selected and the sample mean were calculated to be greater than 1054, would you be surprised? (Choose one) , since the event is (Choose one) to happen. Yes No

SAT Scores The national average SAT score (for verbal and math) is 1028. Suppose that nothing is known about the shape of the distribution and that the standard deviation is 100. Use The Standard Normal Distribution Table. Round the final answer to at least four decimal places and intermediate z-value calculations to two decimal places.

Part 1 of 2

If a random sample of 240 scores was selected, find the probability that the sample mean is greater than 1054. Assume that the sample is taken from a large population and the correction factor can be ignored.
P(X̄>1054)=0.0

Part 2 of 2

If a random sample of 240 scores were selected and the sample mean were calculated to be greater than 1054, would you be surprised?
(Choose one) , since the event is (Choose one) to happen.
Yes
No

Transcript text: SAT Scores The national average SAT score (for verbal and math) is 1028. Suppose that nothing is known about the shape of the distribution and that the standard deviation is 100. Use The Standard Normal Distribution Table. Round the final answer to at least four decimal places and intermediate $z$-value calculations to two decimal places. Part 1 of 2 If a random sample of 240 scores was selected, find the probability that the sample mean is greater than 1054. Assume that the sample is taken from a large population and the correction factor can be ignored. \[ P(\bar{X}>1054)=0.0 \] Part 2 of 2 If a random sample of 240 scores were selected and the sample mean were calculated to be greater than 1054, would you be surprised? (Choose one) $\nabla$, since the event is (Choose one) $\boldsymbol{\nabla}$ to happen. Yes No

Solution

Solution Steps

Step 1: Calculate the Z-score

To find the probability that the sample mean $ \bar{X} $ is greater than 1054, we first calculate the Z-score using the formula:

\[ Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}} \]

Substituting the values:

\[ Z = \frac{1054 - 1028}{100 / \sqrt{240}} = \frac{26}{6.4583} \approx 4.0279 \]

Step 2: Determine the Probability

Next, we find the probability that the sample mean is greater than 1054, which is given by:

\[ P(\bar{X} > 1054) = 1 - P(Z < 4.0279) \]

Using the standard normal distribution table, we find:

\[ P(Z < 4.0279) \approx 1 \]

Thus,

\[ P(\bar{X} > 1054) = 1 - 1 = 0.0 \]

Step 3: Assess the Surprising Nature of the Result

Since the probability $ P(\bar{X} > 1054) = 0.0 $, this indicates that observing a sample mean greater than 1054 is extremely unlikely. Therefore, we conclude:

Yes, since the event is unlikely to happen.