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The numbers of faculty at 20 randomly selected state-controlled colleges and universities with enrollment under 12,000 students are shown below. Use these data to estimate the mean number of faculty at all state-controlled colleges and universities with enrollment under 12,000 with $94\%$ confidence. Assume $\sigma=165.1$. Assume the population is normally distributed.

Round intermediate calculation steps to 2 decimal places. Round your final answers to 1 decimal place.




412		134539471211384396224337

395		121356621367408515280289






$\square < \mu < \square$

Verified Answer :

  • Calculate the sample mean: \(\bar{x} = 358.89\).
  • Determine the critical z-score: \(z^* = 1.88\).
  • Find the standard error: \(SE = 38.91\).
  • Calculate the margin of error and the final confidence interval: \(\boxed{285.7} < \mu < \boxed{432.0}\).

Explanation

  1. Calculate the sample mean

To determine the 94% confidence interval for the mean number of faculty, we first identify the sample size, calculate the sample mean, and determine the critical value. We are provided with 18 data points, so our sample size is \(n = 18\).

First, we calculate the sample mean \(\bar{x}\):
\[\bar{x} = \frac{412 + 134 + 539 + 471 + 211 + 384 + 396 + 224 + 337 + 395 + 121 + 356 + 621 + 367 + 408 + 515 + 280 + 289}{18} = \frac{6460}{18} \approx 358.89\]
Rounding to two decimal places as instructed, we have \(\bar{x} = 358.89\).

  1. Determine the critical value

Next, we identify the critical z-score \(z^*\) for a 94% confidence level. For a 94% confidence interval, the area in both tails combined is \(1 - 0.94 = 0.06\), which means there is \(0.03\) in each tail. We look for the z-score that corresponds to a cumulative area of \(1 - 0.03 = 0.97\). Using the standard normal distribution, the critical value \(z^*\) is approximately \(1.88\).

  1. Calculate the standard error

Now, we calculate the standard error of the mean (\(SE\)):
\[SE = \frac{\sigma}{\sqrt{n}} = \frac{165.1}{\sqrt{18}} \approx \frac{165.1}{4.24} \approx 38.91\]
Rounding to two decimal places, the standard error is \(SE = 38.91\).

  1. Calculate the margin of error

The margin of error (\(E\)) is calculated by multiplying the critical value by the standard error:
\[E = z^* \times SE = 1.88 \times 38.91 \approx 73.15\]
Rounding to two decimal places, the margin of error is \(E = 73.15\).

  1. Construct the confidence interval

Finally, we construct the confidence interval \(\bar{x} \pm E\):
Lower bound: \(358.89 - 73.15 = 285.74\)
Upper bound: \(358.89 + 73.15 = 432.04\)

Rounding the final bounds to one decimal place, we obtain:
\[285.7 < \mu < 432.0\]

Examples

This method of calculating confidence intervals is fundamental for researchers and policymakers. For example, if a university administrator wants to understand the staffing needs across a state, they can use this technique to estimate the average number of faculty members based on a smaller, manageable sample size. By calculating this interval with a known population standard deviation, they can make informed decisions with a high degree of confidence about the overall faculty population size.

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