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:
(pick one from below)
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.
To construct the confidence intervals, we first need to calculate the sample mean and sample standard deviation.
The given data points are:
$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:
The sample mean is calculated by summing all the values and dividing by the number of values:
Mean = (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 = $236,966.67 (rounded to nearest cent).
The sample standard deviation is calculated using the formula:
Standard deviation = sqrt((sum of squares of values - (n * mean squared)) / (n - 1))
Squares of values = (231,500^2 + 279,900^2 + 219,900^2 + 143,000^2 + 205,800^2 + 253,500^2 + 459,900^2 + 273,500^2 + 187,500^2 + 167,500^2 + 147,800^2 + 264,900^2) = 1,300,094,120,500
Standard deviation = sqrt((1,300,094,120,500 - (12 * 236,966.67^2)) / (12 - 1)) = $97,745.37 (rounded to nearest cent).
Now we can calculate the margin of error using the formula:
Margin of error = critical value * (standard deviation / sqrt(sample size))
The critical value for a 99% confidence level with a sample size of 12 is approximately 2.68 (from the t-distribution table).
Margin of error = 2.68 * (97,745.37 / sqrt(12)) = $86,716.64 (rounded to nearest cent).
Finally, we can construct the confidence interval:
Lower bound = mean - margin of error = 236,966.67 - 86,716.64 = $150,250.03 (rounded to nearest cent).
Upper bound = mean + margin of error = 236,966.67 + 86,716.64 = $323,182.71 (rounded to nearest cent).
Therefore, the 99% confidence interval with the outlier included is ($150,250.03 , $323,182.71).
B) Construct a 99% confidence interval with the outlier removed:
If we remove the outlier ($459,900) from the dataset, we will now have a sample size of 11.
The new mean can be calculated as:
Mean = (231,500 + 279,900 + 219,900 + 143,000 + 205,800 + 253,500 + 273,500 + 187,500 + 167,500 + 147,800 + 264,900) / 11 = $223,590.91 (rounded to nearest cent).
The new standard deviation can be calculated using the same formula as before:
Standard deviation = sqrt((sum of squares of values - (n * mean squared)) / (n - 1))
Squares of values = (231,500^2 + 279,900^2 + 219,900^2 + 143,000^2 + 205,800^2 + 253,500^2 + 273,500^2 + 187,500^2 + 167,500^2 + 147,800^2 + 264,900^2) = 1,300,094,120,500
Standard deviation = sqrt((1,300,094,120,500 - (11 * 223,590.91^2)) / (11 - 1)) = $96,463.33 (rounded to nearest cent).
The critical value remains the same at 2.68 (from the t-distribution table).
The margin of error can now be calculated:
Margin of error = 2.68 * (96,463.33 / sqrt(11)) = $85,926.02 (rounded to nearest cent).
The confidence interval without the outlier can be constructed as:
Lower bound = mean - margin of error = 223,590.91 - 85,926.02 = $137,664.89 (rounded to nearest cent).
Upper bound = mean + margin of error = 223,590.91 + 85,926.02 = $309,255.93 (rounded to nearest cent).
Therefore, the 99% confidence interval without the outlier is ($137,664.89 , $309,255.93).
C) Comment on the effect the outlier has on the confidence interval:
c) The outlier had no effect on the width of the confidence interval.
The outlier's removal did not significantly impact the confidence interval's width.
To construct a confidence interval, we need to calculate the mean and standard deviation of the sample data.
Step 1: Calculate the mean
We sum up all the values in the sample and divide by the number of values.
Mean = (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 = 233,250
Step 2: Calculate the standard deviation
We need to find the difference between each value and the mean, square it, sum up all the squared differences, divide by the number of values, and take the square root of the result.
Standard deviation = sqrt(((231,500 - 233,250)^2 + (279,900 - 233,250)^2 + (219,900 - 233,250)^2 + (143,000 - 233,250)^2 + (205,800 - 233,250)^2 + (253,500 - 233,250)^2 + (459,900 - 233,250)^2 + (273,500 - 233,250)^2 + (187,500 - 233,250)^2 + (167,500 - 233,250)^2 + (147,800 - 233,250)^2 + (264,900 - 233,250)^2) / 12)
Standard deviation ≈ 92,414
Now we can construct the confidence intervals.
A) Confidence interval with the outlier included
The formula for a confidence interval is: mean ± (critical value) * (standard deviation / sqrt(sample size))
Since we want a 99% confidence interval, the critical value for a two-tailed test is approximately 2.821.
Confidence interval = 233,250 ± (2.821 * (92,414 / sqrt(12)))
Confidence interval ≈ (180,997, 285,503)
B) Confidence interval with outlier removed
We remove the outlier, which is the value 459,900, from the sample.
Now we have a sample size of 11.
Mean = (231,500 + 279,900 + 219,900 + 143,000 + 205,800 + 253,500 + 273,500 + 187,500 + 167,500 + 147,800 + 264,900) / 11 ≈ 226,781
Standard deviation ≈ 88,707
Using the same formula as above, with a critical value of 2.821:
Confidence interval = 226,781 ± (2.821 * (88,707 / sqrt(11)))
Confidence interval ≈ (165,134, 288,428)
C) Comment on the effect of the outlier on the confidence interval
c) The outlier had no effect on the width of the confidence interval.
The removal of the outlier did not significantly change the confidence interval width because its value was relatively large and similar to other values in the sample. The impact of an outlier on the confidence interval width is more significant when the outlier is drastically different from the other data points.