To find the range of values for the 90% confidence interval of the population mean, a few values must first be calculated: the standard error and the margin of error.
The standard error is calculated using the formula:
`Standard Error = Population Standard Deviation / √Sample Size`
Here, the population standard deviation is $15.30 and the sample size is 55.
Using these values, the standard error is calculated as 2.06 (rounded to two decimal places).
The standard error gauges the fluctuation in the sample mean from the population mean. So, the standard error of 2.06 suggests that the sample mean varies by approximately $2.06 from the population mean.
The next step is to compute the margin of error. At a 90% confidence level, the z-score is approximately 1.645.
The margin of error is the z-score multiplied by the standard error:
`Margin of Error = Z-score * Standard Error`
So, the margin of error is approximately 3.39 (rounded to two decimal places).
Now, let's compute the 90% confidence interval. The formula to determine the confidence interval is:
`Confidence Interval = Sample Mean ± Margin of Error`
The sample mean is $146.00. So, the confidence interval is ($146.00 - 3.39, $146.00 + 3.39), which results in the interval (142.61, 149.39).
These computations suggest that we can be 90% confident that the average price of home theater systems is between $142.61 and $149.39.