# The following data represent the asking price of a simple random sample of homes for sale. Construct a 99% confidence interval with and without the outlier included. Comment on the effect the outlier has on the confidence interval.

Here is the information: \$231,500 \$279,900 \$219,900 \$143,000 \$205,800 \$253,500 \$459,900 \$273,500 \$187,500 \$167,500 \$147,800 \$264,900

A) Construct a 99% confidence interval with the outlier included: (\$_____ , \$ _____)

B) Construct a 99% confidence interval with the outlier removed: (\$_____ , \$ _____)

C) Comment on the effect the outlier has on the confidence interval:

a) The outlier caused the width of the confidence interval to increase.

b) The outlier caused the width of the confidence interval to decrease.

c) The outlier had no effect on the width of the confidence interval.

a)

## To construct a confidence interval, we first need to calculate the sample mean and the standard deviation of the sample.

Step 1: Calculate the sample mean (x̄):
Add up all the house prices and divide by the number of prices in the sample.
x̄ = (231,500 + 279,900 + 219,900 + 143,000 + 205,800 + 253,500 + 459,900 + 273,500 + 187,500 + 167,500 + 147,800 + 264,900) / 12

Step 2: Calculate the sample standard deviation (s):
Subtract the mean from each value, square the differences, and calculate the average of these squared differences. Finally, take the square root of this average.
s = sqrt(((231,500 - x̄)² + (279,900 - x̄)² + ... + (264,900 - x̄)²) / (n - 1))

Step 3: Calculate the margin of error (E):
Multiply the standard deviation by the appropriate t-value. The t-value is based on the desired confidence level and the sample size.
E = t * (s / sqrt(n))
For a 99% confidence level with a sample size of 12, the t-value can be found in a t-distribution table or by using software like R or Excel.

Step 4: Calculate the confidence interval:
The confidence interval is constructed by subtracting and adding the margin of error to the mean.
Lower bound = x̄ - E
Upper bound = x̄ + E

Now, let's apply these steps to answer the question.

A) Construct a 99% confidence interval with the outlier included:
First, calculate x̄ and s using the given data. Then find the t-value for a 99% confidence level with a sample size of 12. Finally, substitute the values into the confidence interval formula.
Lower bound = x̄ - E
Upper bound = x̄ + E

B) Construct a 99% confidence interval with the outlier removed:
Repeat the same steps as in part A, but this time, exclude the outlier (\$459,900) from the sample. Calculate the new x̄, s, and t-value, and substitute them into the confidence interval formula.

C) Comment on the effect the outlier has on the confidence interval:
Compare the width of the confidence intervals from parts A and B. Note the difference in range between the two intervals, and determine whether the outlier had an effect on the width of the confidence interval.