College

Use the following sample to estimate a population mean [tex]\mu[/tex].

[tex]\[

\begin{array}{|r|r|r|r|}

\hline

93.6 & 60.4 & 62.1 & 52.1 \\

\hline

79.1 & 78.8 & 67.2 & 61.9 \\

\hline

85.6 & 59.4 & 81 & 40.6 \\

\hline

64.7 & 83.6 & 100.8 & 93.1 \\

\hline

42.1 & 93.5 & 64.3 & 74.4 \\

\hline

64.2 & 38.3 & 56.7 & 79.1 \\

\hline

62.1 & 56.9 & 74.8 & 41.4 \\

\hline

71 & 63.5 & 61.2 & 78 \\

\hline

87.2 & 56.1 & 62.6 & 44.3 \\

\hline

43.7 & 87.1 & 91.5 & 73.5 \\

\hline

65.5 & 58.1 & 84.1 & 59.8 \\

\hline

56 & 80.1 & 45 & 71 \\

\hline

44.3 & 87.6 & 62.3 & 92.4 \\

\hline

49.9 & 69.5 & 48.9 & 71.1 \\

\hline

57.1 & 57.3 & 86 & 84.3 \\

\hline

71.6 & 48.5 & 81.8 & \\

\hline

\end{array}

\][/tex]

Find the [tex]99.9\%[/tex] confidence interval about the population mean. Enter your answer as a tri-linear inequality accurate to two decimal places.

[tex]\square \ \textless \ \mu \ \textless \ \square[/tex]

Answer :

To find the 99.9% confidence interval for the population mean [tex]\(\mu\)[/tex] using the given sample data, we can follow a series of steps:

1. Calculate the Sample Mean:
- The sample mean is the average of all the sample values. For this sample, the mean is approximately [tex]\(67.68\)[/tex].

2. Determine the Sample Standard Deviation:
- The sample standard deviation measures the spread of the sample data around the mean. For this sample, it is approximately [tex]\(15.93\)[/tex].

3. Find the Sample Size:
- The sample size [tex]\(n\)[/tex] is the total number of data points, which is 63 in this case.

4. Locate the Critical z-value:
- For a 99.9% confidence level and a two-tailed test, we use the z-value that corresponds to the middle 99.9% of the standard normal distribution. The critical z-value is approximately [tex]\(3.291\)[/tex].

5. Calculate the Margin of Error:
- The margin of error (ME) is calculated using the formula:
[tex]\[
\text{ME} = z \times \left(\frac{\text{sample standard deviation}}{\sqrt{n}}\right)
\][/tex]
- For this problem, the margin of error is approximately [tex]\(6.60\)[/tex].

6. Determine the Confidence Interval:
- Finally, we calculate the confidence interval using the sample mean and the margin of error:
[tex]\[
\text{Lower Limit} = \text{sample mean} - \text{margin of error} \approx 61.07
\][/tex]
[tex]\[
\text{Upper Limit} = \text{sample mean} + \text{margin of error} \approx 74.28
\][/tex]

Putting all this together, the 99.9% confidence interval for the population mean [tex]\(\mu\)[/tex] is approximately:
[tex]\[ 61.07 < \mu < 74.28 \][/tex]

This interval suggests that we are 99.9% confident that the true population mean lies between 61.07 and 74.28.