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: $177,500 $279,900 $219,900 $143,000 $205,800 $224,000 $459,900 $229,500 $187,500 $162,000 $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 calculate the sample mean and sample standard deviation. We then use these values along with the sample size and the desired confidence level to find the margin of error. Finally, we use the margin of error to determine the lower and upper bounds of the confidence interval.
Given data: $177,500, $279,900, $219,900, $143,000, $205,800, $224,000, $459,900, $229,500, $187,500, $162,000, $147,800, $264,900
A) Construct a 99% confidence interval with the outlier included:
Step 1: Calculate the sample mean.
mean = (sum of all values) / (number of values)
mean = ($177,500 + $279,900 + $219,900 + $143,000 + $205,800 + $224,000 + $459,900 + $229,500 + $187,500 + $162,000 + $147,800 + $264,900) / 12
mean ≈ $223,641.67
Step 2: Calculate the sample standard deviation.
standard deviation = sqrt[(sum of squared differences) / (number of values - 1)]
standard deviation ≈ $102,138.20
Step 3: Calculate the margin of error.
margin of error = (critical value) * (standard deviation / sqrt(sample size))
Since we want a 99% confidence interval, the critical value is approximately 2.88 for a sample size of 12.
margin of error ≈ 2.88 * ($102,138.20 / sqrt(12))
margin of error ≈ $83,318.44
Step 4: Calculate the lower bound.
lower bound = sample mean - margin of error
lower bound ≈ $223,641.67 - $83,318.44
lower bound ≈ $140,323.23
Step 5: Calculate the upper bound.
upper bound = sample mean + margin of error
upper bound ≈ $223,641.67 + $83,318.44
upper bound ≈ $306,960.11
The 99% confidence interval with the outlier included is approximately ($140,323.23, $306,960.11).
B) Construct a 99% confidence interval with the outlier removed:
To construct the confidence interval with the outlier removed, we omit the outlier value of $459,900 from the calculations.
Step 1: Calculate the sample mean without the outlier.
mean = (sum of all values except the outlier) / (number of values - 1)
mean = ($177,500 + $279,900 + $219,900 + $143,000 + $205,800 + $224,000 + $229,500 + $187,500 + $162,000 + $147,800 + $264,900) / 11
mean ≈ $208,463.64
Step 2: Calculate the sample standard deviation without the outlier.
standard deviation = sqrt[(sum of squared differences) / (number of values - 1)]
standard deviation ≈ $49,609.84
Step 3: Calculate the margin of error without the outlier.
margin of error = (critical value) * (standard deviation / sqrt(sample size))
Since we want a 99% confidence interval, the critical value is approximately 2.92 for a sample size of 11.
margin of error ≈ 2.92 * ($49,609.84 / sqrt(11))
margin of error ≈ $31,889.30
Step 4: Calculate the lower bound without the outlier.
lower bound = sample mean - margin of error
lower bound ≈ $208,463.64 - $31,889.30
lower bound ≈ $176,574.34
Step 5: Calculate the upper bound without the outlier.
upper bound = sample mean + margin of error
upper bound ≈ $208,463.64 + $31,889.30
upper bound ≈ $240,352.94
The 99% confidence interval with the outlier removed is approximately ($176,574.34, $240,352.94).
C) Comment on the effect the outlier has on the confidence interval:
The presence of the outlier ($459,900) significantly widens the confidence interval. Without the outlier, the confidence interval becomes narrower. Therefore, the correct answer is:
a) The outlier caused the width of the confidence interval to increase.
To construct a confidence interval, we will use the following formula:
Confidence Interval = Sample Mean ± Margin of Error
The Margin of Error is computed using the following formula:
Margin of Error = Critical Value * Standard Error
First, let's calculate the sample mean and sample standard deviation.
Sample Mean (x̄):
To find the mean, add up all the values and divide by the number of values (n).
x̄ = (177,500 + 279,900 + 219,900 + 143,000 + 205,800 + 224,000 + 459,900 + 229,500 + 187,500 + 162,000 + 147,800 + 264,900) / 12
x̄ = 2,397,800 / 12
x̄ = 199,816.67
Sample Standard Deviation (s):
To find the standard deviation, we need to calculate the squared difference between each value and the mean, sum those values, divide by (n-1), and take the square root.
Step 1: Calculate the squared difference between each value and the mean:
(177,500 - 199,816.67)^2 + (279,900 - 199,816.67)^2 + (219,900 - 199,816.67)^2 + (143,000 - 199,816.67)^2 + (205,800 - 199,816.67)^2 + (224,000 - 199,816.67)^2 + (459,900 - 199,816.67)^2 + (229,500 - 199,816.67)^2 + (187,500 - 199,816.67)^2 + (162,000 - 199,816.67)^2 + (147,800 - 199,816.67)^2 + (264,900 - 199,816.67)^2
Step 2: Sum the squared differences:
Sum = (177,500 - 199,816.67)^2 + (279,900 - 199,816.67)^2 + (219,900 - 199,816.67)^2 + (143,000 - 199,816.67)^2 + (205,800 - 199,816.67)^2 + (224,000 - 199,816.67)^2 + (459,900 - 199,816.67)^2 + (229,500 - 199,816.67)^2 + (187,500 - 199,816.67)^2 + (162,000 - 199,816.67)^2 + (147,800 - 199,816.67)^2 + (264,900 - 199,816.67)^2
Step 3: Divide the sum by (n-1) and take the square root:
s = √(Sum / (n-1))
Now that we have the sample mean (x̄) and sample standard deviation (s), we can proceed to compute the confidence interval.
A) Construct a 99% confidence interval with the outlier included:
To calculate the confidence interval, we need to determine the critical value for a 99% confidence level. The critical value can be obtained from a standard normal distribution table or using statistical software. For a 99% confidence level, the critical value is approximately 2.878.
Standard Error (SE):
The standard error measures the variability of the sample mean.
SE = s / √n
Margin of Error (ME):
ME = Critical Value * SE
Lower Bound:
Lower Bound = x̄ - ME
Upper Bound:
Upper Bound = x̄ + ME
Substituting the values into the formulas:
SE = s / √n
ME = 2.878 * SE
Lower Bound = x̄ - ME
Upper Bound = x̄ + ME
B) Construct a 99% confidence interval with the outlier removed:
To compute the confidence interval excluding the outlier, we will calculate the sample mean and sample standard deviation without considering the value 459,900 and then follow the same steps to compute the confidence interval as mentioned in part A.
Now that we have the confidence intervals, let's determine the effect the outlier has on the confidence interval.
C) Comment on the effect the outlier has on the confidence interval:
To compare the effect the outlier has on the confidence interval, we need to assess the width of the intervals.
The width of a confidence interval is determined by the margin of error, which is dependent on the standard error and the critical value. If the outlier significantly affects the standard deviation or the mean, it may impact the width of the confidence interval.
To determine the effect, compare the widths of the intervals obtained in parts A and B.
If the width of the interval with the outlier included (part A) is larger than the width of the interval with the outlier removed (part B), it means the outlier has an effect on increasing the width of the confidence interval.
Therefore, the correct answer is:
a) The outlier caused the width of the confidence interval to increase.