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.
I wish you would use the same name all the time. Check under LAGO.
To construct a confidence interval, we need to calculate the sample mean and the standard error.
First, let's find the sample mean. We sum up all the values in the sample and divide it by the total number of data points:
Sample mean (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
x̄ ≈ $237,308.33
Next, we need to calculate the standard deviation for the sample. The formula for the standard deviation is:
s = √[ Σ(xi - x̄)² / (n - 1) ]
where xi represents each individual value in the sample, x̄ is the sample mean, and n is the sample size.
Using this formula, we calculate the standard deviation:
s = √[ (231,500 - 237,308.33)² + (279,900 - 237,308.33)² + ... + (264,900 - 237,308.33)² / (12 - 1) ]
s ≈ $104,991.60
Now, we can calculate the standard error, which is the standard deviation divided by the square root of the sample size:
Standard error (SE) = s / √n
SE ≈ $104,991.60 / √12 ≈ $30,305.20
To construct the confidence interval, we use the formula:
Confidence interval = sample mean ± (critical value * standard error)
The critical value depends on the desired level of confidence. Since we want a 99% confidence interval, the critical value is found using a t-distribution table with 12 - 1 = 11 degrees of freedom. The critical value for a 99% confidence level is approximately 3.106.
A) Including the outlier:
Lower bound = x̄ - (critical value * SE) = $237,308.33 - (3.106 * $30,305.20)
Upper bound = x̄ + (critical value * SE) = $237,308.33 + (3.106 * $30,305.20)
A 99% confidence interval including the outlier is ($126,306.56, $348,310.10).
B) Removing the outlier:
Let's remove the outlier, $459,900, from the sample.
New sample mean (x̄) = (231,500 + 279,900 + 219,900 + 143,000 + 205,800 + 253,500 + 273,500 + 187,500 + 167,500 + 147,800 + 264,900) / 11
x̄ ≈ $227,545.45
New standard deviation (s) and standard error (SE) are calculated using the same formulas as before.
New s ≈ $50,475.33
New SE ≈ $50,475.33 / √11 ≈ $15,220.25
Using the same critical value of 3.106, we can calculate the confidence interval:
Lower bound = x̄ - (critical value * SE) = $227,545.45 - (3.106 * $15,220.25)
Upper bound = x̄ + (critical value * SE) = $227,545.45 + (3.106 * $15,220.25)
A 99% confidence interval without the outlier is ($181,673.53, $273,417.37).
C) Comment on the effect the outlier has on the confidence interval:
a) The outlier caused the width of the confidence interval to increase.