Question

A random sample of 40 home theater systems has a mean price of $136.00. Assume the population standard deviation is $15.30.

Construct a 90% confidence interval for the population mean.
The 90% confidence interval is___.
(Round to two decimal places as needed.)
Construct a 95% confidence interval for the population mean.
The 95% confidence interval is ____.
(Round to two decimal places as needed.) Interpret the results.

Answer 1

The 90% confidence interval for the population mean is $130.94 to $141.06. The 95% confidence interval is $129.29 to $142.71.

Step-by-step explanation:

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

`\[ \bar{X} \pm Z \left( (\sigma)/(√(n)) \right) \]`

where:

`- \(\bar{X}\)` is the sample mean,

`- \(Z\)` is the Z-score corresponding to the desired confidence level,

`- \(\sigma\)` is the population standard deviation, and

`- \(n\)` is the sample size.

For the 90% confidence interval, the Z-score is approximately 1.645, and for the 95% confidence interval, it's approximately 1.96.

Using the given values
`(\(\bar{X} = $136.00\), \(\sigma = $15.30\), \(n = 40\)),` we calculate the margin of error and construct the intervals. The 90% confidence interval is narrower than the 95% interval due to the smaller margin of error.

Interpreting the results, we can say with 90% confidence that the true population mean falls within the range of $130.94 to $141.06. Similarly, with 95% confidence, the population mean is estimated to be between $129.29 and $142.71. The wider interval at 95% reflects a higher level of confidence but comes at the cost of increased uncertainty.