View an example | 5 parts remaining X A government bureau publishes annual price figures for new mobile homes. A simple random sample of 36 new mobile homes yielded the following prices, in thousands of dollars. Assume that the population standard deviation of all such prices is $10.5 thousand, that is, $10,500. Use the data to obtain a 99.7% confidence interval for the mean price of all new mobile homes. Click the icon to view a table of prices of new mobile homes. . . . First, identify the distribution of the variable x, that is, the sampling distribution of the sample mean for samples of prices of new mobile homes of size 36. Determine whether the prices of new mobile homes are normally distributed by constructing a normal probability plot. The normal probability plot is shown to the right. Because the plot is roughly linear, it can be assumed that the prices are normally Normal score distributed. 50 70 80 90 Price (Sthousands)Because n = 36, o = 10.5, and prices of new mobile homes are normally distributed, the properties of the sampling distribution of the sample mean imply that the population mean of x is equal to the unknown population mean, that the population standard deviation of x is equal to the population standard deviation divided by the square root of the sample size, and that x is normally distributed. O- E X Calculate the population standard deviation of x, using o = 10.5 and n = 36. 0- X ECalculate the population standard deviation of x, using o = 10.5 and n = 36. = 10.5 = 1.75 Therefore, the variable x is normally distributed with mean p and standard deviation 1.75, or $1750. According to the empirical rule for variables, 99.7% of all possible observations lie within 3 standard deviations to either side of the mean.0- E 10.5 36 = 1.75 Therefore, the variable x is normally distributed with mean p and standard deviation 1.75, or $1750. According to the empirical rule for variables, 99.7% of all possible observations lie within 3 standard deviations to either side of the mean. Thus, approximately 99.7% of all samples of size 36 have the property that the interval x - 3 . 1.75 to x+ 3 . 1.75 contains the population mean .Therefore, the variable x is normally distributed with mean p and standard deviation 1.75, or $1750. According to the empirical rule for variables, 99.7% of all possible observations lie within 3 standard deviations to either side of the mean. Thus, approximately 99.7% of all samples of size 36 have the property that the interval x - 3 . 1.75 to x+ 3 . 1.75 contains the population mean J. Calculate the sample mean x for this sample, using the formula x = - , where x; is the ith mobile home price. n Round to three decimal places. x = 63.911Thus, approximately 99.7% of all samples of size 36 have the property that the interval x - 3 . 1.75 to x+ 3 . 1.75 contains the population mean J. Exi Calculate the sample mean x for this sample, using the formula x = - n - where x; is the ith mobile home price. Round to three decimal places. x = 63.911 Calculate the lower bound of the 99.7% confidence interval, rounding to three decimal places. x - 3 . 1.75 = 63.911-3 . 1.75 = 58.661Calculate the sample mean x for this sample, using the formula x = n where x; is the ith mobile home price. Round to three decimal places. x = 63.911 Calculate the lower bound of the 99.7% confidence interval, rounding to three decimal places. x - 3 . 1.75 = 63.911-3.1.75 = 58.661 Calculate the upper bound of the 99.7% confidence interval, rounding to three decimal places. x + 3 - 1.75 = 63.911+3 . 1.75 = 69.161x = 63.911 Calculate the lower bound of the 99.7% confidence interval, rounding to three decimal places. x - 3 . 1.75 = 63.911-3 . 1.75 = 58.661 Calculate the upper bound of the 99.7% confidence interval, rounding to three decimal places. x +3 - 1.75 = 63.911+3 - 1.75 = 69.161 Thus, the 99.7% confidence interval for the mean price of all new mobile homes is from 58.661 thousand dollars to 69.161 thousand dollars, or $58,661 to $69, 161