Question

If n=21, (x− bar )=41, and s=16, construct a confidence interval at a 90% confidence level. Assume the data came from a normally distributed population. Give your answers to one decimal place. <μ

Answer 1

The 90% confidence interval for the population mean μ, given n=21,
`\bar{x}`=41, and s=16, is 39.1, 42.9.

Step-by-step explanation:

To construct a confidence interval for the population mean μ, we use the formula
`\(\bar{x} \pm z * (s)/(√(n))\)`, where
`\(\bar{x}\)` is the sample mean, s is the sample standard deviation, n is the sample size, and z is the critical value from the standard normal distribution corresponding to the desired confidence level.

Given n=21,
`\((\bar{x})` =41, and s=16, we first find the critical value. For a 90% confidence interval, the critical value is approximately 1.645. Now, plug these values into the formula:

`\[\begin{equation}\bar{x} \pm 1.645 * (16)/(√(21))\end{equation}\]`

Calculating the expression inside the parentheses, we get
`\((16)/(√(21)) \approx 3.477\)`. Multiplying this by 1.645 gives approximately 5.72. So, the margin of error is 5.72. Now, subtract and add this margin of error to the sample mean:

`\[\begin{equation}41 - 5.72 \approx 39.1 \quad \text{and} \quad 41 + 5.72 \approx 42.9\end{equation}\]`

Therefore, the 90% confidence interval for μ is 39.1, 42.9. This means we can be 90% confident that the true population mean falls within this interval.