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Suppose 11%11\% of student veterans at a college are involved in sports. A random sample of 135135 student veterans is taken. What is the mean of the sampling distribution for the proportion of veterans in sports at this college? σ(@)\sigma^{(@)} When working with samples of size 135135, what is the standard error of the sampling distribution for the proportion of veterans in sports at this college? Round answer to 33 decimal places.

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Q. Suppose 11%11\% of student veterans at a college are involved in sports. A random sample of 135135 student veterans is taken. What is the mean of the sampling distribution for the proportion of veterans in sports at this college? σ(@)\sigma^{(@)} When working with samples of size 135135, what is the standard error of the sampling distribution for the proportion of veterans in sports at this college? Round answer to 33 decimal places.
  1. Calculate Standard Error: Now, we calculate the standard error of the sampling distribution for the proportion.\newlineStandard error (SE) = (p×(1−p))/n\sqrt{\left(p \times (1 - p)\right) / n}, where pp is the population proportion and nn is the sample size.\newlineSE = (0.11×(1−0.11))/135\sqrt{\left(0.11 \times (1 - 0.11)\right) / 135}.
  2. Calculate Standard Error: Perform the calculation for the standard error. \newlineSE=(0.11×0.89)/135SE = \sqrt{\left(0.11 \times 0.89\right) / 135}.\newlineSE=(0.0979)/135SE = \sqrt{\left(0.0979\right) / 135}.\newlineSE=0.000725185SE = \sqrt{0.000725185}.\newlineSE=0.026937SE = 0.026937, which we round to three decimal places as 0.0270.027.
  3. Calculate Sample Proportion: To determine if it is unusual that no more than 2020 veterans in the sample are involved in sports, we calculate the sample proportion.\newlineSample proportion = 20135≈0.148\frac{20}{135} \approx 0.148.
  4. Compare Sample to Population: We compare the sample proportion to the population proportion using the standard error.\newlineZ=Sample proportion−Population proportionStandard error.Z = \frac{\text{Sample proportion} - \text{Population proportion}}{\text{Standard error}}.\newlineZ=0.148−0.110.027.Z = \frac{0.148 - 0.11}{0.027}.
  5. Calculate Z-score: Calculate the Z-score.\newlineZ=0.0380.027Z = \frac{0.038}{0.027}.\newlineZ=1.407Z = 1.407, which we round to four decimal places as 1.40701.4070.
  6. Check Unusual Result: To determine if the result is unusual, we check if the ZZ-score corresponds to a probability of less than 5%5\%. A ZZ-score of 1.40701.4070 corresponds to a probability greater than 5%5\%, so it is not unusual.

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