Question

The average price for new mobile homes is known to be distributed with a population standard deviation of 400. If a random sample of 25 homes resulted in a sample mean of 47,500, obtain a 90% confidence interval.

Answer 1

The 90% confidence interval for the average price of new mobile homes is $47,367.40 to $47,632.60, using a known population standard deviation and the sample mean.

Step-by-step explanation:

To construct a 90% confidence interval for the average price of new mobile homes, we will use the given population standard deviation and the sample mean. Since the population standard deviation is known, we apply the z-distribution for our calculations. The steps are as follows:

• Determine the z-score corresponding to a 90% confidence level. For a 90% confidence interval, the z-score is typically around 1.645.
• Calculate the standard error of the mean (SEM) by dividing the population standard deviation by the square root of the sample size: SEM = 400 / sqrt(25) = 400 / 5 = 80.
• Multiply the z-score by the SEM to find the margin of error: Margin of Error = 1.645 × 80 = 132.60.
• Finally, construct the confidence interval by adding and subtracting the margin of error from the sample mean: Lower Limit = 47,500 - 132.60 = 47,367.40 and Upper Limit = 47,500 + 132.60 = 47,632.60.

Therefore, the 90% confidence interval for the average price of new mobile homes is $47,367.40 to $47,632.60.