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

27.1, 34.8, 36.3, 15.1, 70.3, 39, 41.5, 13.6, 61.1, 40.3

Assuming the population has a bell-shaped distribution, find the 90% confidence interval about the population mean. Enter your answer as an open interval (i.e., parentheses) accurate to two decimal places.

90% C.I. = ______

Question

High School

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Answer :

tokoyoknows tokoyoknows · Answer 1 · 2024-11-08T22:42:08-05:00

Final Answer:

The 90% confidence interval about the population mean is (28.88, 49.02).

Explanation:

To calculate the confidence interval for the population mean, we use the formula:

[tex]\[ \text{Confidence Interval} = \bar{x} \pm Z \cdot \frac{s}{\sqrt{n}} \][/tex]

Where:

- [tex]$\bar{x}$[/tex] is the sample mean,

- [tex]$Z$[/tex] is the critical value corresponding to the desired confidence level (for a 90% confidence level, [tex]$Z \approx 1.645$)[/tex],

- [tex]$s$[/tex] is the sample standard deviation,

- [tex]$n$[/tex] is the sample size.

From the given sample, we have:

- Sample mean [tex]($\bar{x}$)[/tex]: 34.13

- Sample standard deviation [tex]($s$)[/tex]: 17.96

- Sample size [tex]($n$)[/tex]: 10

Plugging in these values and the critical value, we get:

[tex]\[ \text{Confidence Interval} = 34.13 \pm 1.645 \cdot \frac{17.96}{\sqrt{10}} \][/tex]

Calculating this gives us a confidence interval of (28.88, 49.02), rounded to two decimal places.

This means that we can be 90% confident that the true population mean falls within the interval (28.88, 49.02).

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