Russell is doing some research before buying his first house. He is looking at two different areas of the city, and he wants to know if there is a significant difference between the mean prices of homes in the two areas. For the 33 homes he samples in the first area, the mean home price is $168,300. Public records indicate that home prices in the first area have a population standard deviation of $37,825. For the 32 homes he samples in the second area, the mean home price is $181,900. Again, public records show that home prices in the second area have a population standard deviation of $25,070. Let Population 1 be homes in the first area and Population 2 be homes in the second area. Construct a 95% confidence interval for the true difference between the mean home prices in the two areas.

The 95% confidence interval for the true difference between the mean home prices in the two areas is (-$29156.52, $1956.52).

Step-by-step explanation:

Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Subtraction between normal variables:

When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.

First area:

33 homes, mean of $168,300, standard deviation of $37,825. Thus:

Second area:

33 homes, mean of $181,900, standard deviation of $25,070. Thus:

Distribution of the difference:

Confidence interval:

In which

z is the z-score that has a p-value of .

95% confidence level

So , z is the value of Z that has a p-value of , so .  

The lower bound of the interval is:

The upper bound of the interval is:

The 95% confidence interval for the true difference between the mean home prices in the two areas is (-$29156.52, $1956.52).

step-by-step explanation: