A local supplier of motor vehicles selected a random sample of 35 minivan electrical warranty repairs for "loose, not attached" wires. The sample showed a mean repair cost of $45.65 with a standard deviation of $7.80.

a) The manager wishes to know, at a 90% confidence level, what the population average mean repair cost could be. Show all calculations. Explain the meaning of the confidence interval.

b) Provide a clearly labeled diagram showing the likely positioning of such a confidence interval on a normal population curve.

The 90% confidence interval for the population average mean repair cost is approximately $43.97 to $47.33. This means that we can be 90% confident that the true population average mean repair cost falls within this range.

Explanation:

To calculate the confidence interval for the population average mean repair cost, we will use the given sample mean, standard deviation, and the desired confidence level of 90%.

Step 1: Calculate the standard error. The standard error is the standard deviation divided by the square root of the sample size.

Standard Error = Standard Deviation / √(Sample Size)

Standard Error = $7.80 / √(35)

Step 2: Determine the critical value. The critical value is based on the desired confidence level and the degrees of freedom, which is the sample size minus 1.

For a 90% confidence level and 34 degrees of freedom, the critical value is approximately 1.697.

Step 3: Calculate the margin of error. The margin of error is the critical value multiplied by the standard error.

Margin of Error = Critical Value * Standard Error

Margin of Error = 1.697 * ($7.80 / √(35))

Step 4: Calculate the confidence interval. The confidence interval is the sample mean plus or minus the margin of error.

Confidence Interval = Sample Mean ± Margin of Error

Confidence Interval = $45.65 ± (1.697 * ($7.80 / √(35)))

Therefore, the 90% confidence interval for the population average mean repair cost is approximately $43.97 to $47.33.

The confidence interval means that we can be 90% confident that the true population average mean repair cost falls within this range. It provides a measure of the precision and reliability of our estimate based on the sample data.

A normal population curve can be used to represent the distribution of sample means. The confidence interval is positioned around the sample mean on the normal curve, with the lower and upper bounds of the interval indicating the range within which the true population mean is likely to fall.

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