In 2015, the average price of new homes in a certain suburb was $145,000. Assume that this
mean is based on a random sample of 1000 new home sales and that the sample standard
deviation is $24,000. Construct a 99% confidence interval for the 2002 mean price of all such
homes.
99% = mean ± Z(SEm)
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.005) and its Z score. Insert data into above equation.
To construct a confidence interval, we need three key pieces of information: the sample mean, the sample standard deviation, and the desired level of confidence. In this case, we have the sample mean of $145,000 and the sample standard deviation of $24,000.
The desired level of confidence is 99%, which means that we want to be 99% confident that the true population mean falls within the calculated interval.
To construct the confidence interval, we can use the formula:
Confidence interval = sample mean ± (critical value * standard error)
Step 1: Determine the critical value
Since we want a 99% confidence interval, we need to find the critical value associated with a 99% confidence level. This can be obtained from a t-distribution table or using a statistical software. For a large sample size like 1000, we can use a Z-value instead of a t-value. The Z-value for a 99% confidence level is approximately 2.576.
Step 2: Calculate the standard error
The standard error represents the standard deviation of the sampling distribution of the sample mean. In this case, since we have the sample standard deviation, we can calculate the standard error using the formula:
Standard error = sample standard deviation / sqrt(sample size)
The sample size is 1000, so the standard error is $24,000 / sqrt(1000) = $760.68 (rounded to two decimal places).
Step 3: Calculate the confidence interval
Using the formula mentioned earlier, we can calculate the confidence interval as follows:
Confidence interval = $145,000 ± (2.576 * $760.68)
Calculating this expression, we get:
Confidence interval = $145,000 ± $1,959.53
Therefore, the 99% confidence interval for the 2002 mean price of all new homes in the given suburb is approximately $143,040.47 to $146,959.53.