the percentage rates of home ownership for 8 randomly selected states are listed below. Estimate the population variance and standard deviation for the percentage rate of home ownership with 99% confidence.
66.0 75.8 70.9 73.9 63.4 68.5 73.3 65.9
To estimate the population variance and standard deviation for the percentage rate of home ownership with 99% confidence, we can follow these steps:
Step 1: Calculate the mean (average) of the percentage rates:
Add up all the percentage rates and divide by the total number of observations (in this case, 8).
Mean = (66.0 + 75.8 + 70.9 + 73.9 + 63.4 + 68.5 + 73.3 + 65.9) / 8
Mean = 656.7 / 8
Mean = 82.09
Step 2: Calculate the difference between each observation and the mean, squared:
For each observation, subtract the mean and square the result.
(66.0 - 82.09)^2 = 276.7
(75.8 - 82.09)^2 = 39.1
(70.9 - 82.09)^2 = 130.9
(73.9 - 82.09)^2 = 66.6
(63.4 - 82.09)^2 = 344.9
(68.5 - 82.09)^2 = 186.2
(73.3 - 82.09)^2 = 77.7
(65.9 - 82.09)^2 = 281.4
Step 3: Calculate the sum of the squared differences:
Add up all the squared differences calculated in step 2.
Sum of squared differences = 276.7 + 39.1 + 130.9 + 66.6 + 344.9 + 186.2 + 77.7 + 281.4
Sum of squared differences = 1403.5
Step 4: Calculate the sample variance:
Divide the sum of squared differences calculated in step 3 by the number of observations minus 1 (in this case, 8 - 1 = 7).
Sample variance = (Sum of squared differences) / (Number of observations - 1)
Sample variance = 1403.5 / 7
Sample variance ≈ 200.5
Step 5: Calculate the sample standard deviation:
Take the square root of the sample variance calculated in step 4.
Sample standard deviation = √(Sample variance)
Sample standard deviation ≈ √(200.5)
Sample standard deviation ≈ 14.15
Step 6: Calculate the margin of error:
The margin of error is determined based on the confidence level desired. For a 99% confidence level, we will use a t-distribution with 7 degrees of freedom. The T-score for a 99% confidence level and 7 degrees of freedom is approximately 3.499.
Margin of error = (T-score) * (Sample standard deviation / √Number of observations)
Margin of error ≈ 3.499 * (14.15 / √8)
Margin of error ≈ 3.499 * (14.15 / 2.83)
Margin of error ≈ 3.499 * 5.00
Margin of error ≈ 17.50
Step 7: Calculate the confidence interval:
The confidence interval is calculated by adding and subtracting the margin of error from the mean.
Confidence interval = Mean ± Margin of error
Confidence interval ≈ 82.09 ± 17.50
Confidence interval ≈ (82.09 - 17.50, 82.09 + 17.50)
Confidence interval ≈ (64.59, 99.59)
Thus, with 99% confidence, the population variance for the percentage rate of home ownership is estimated to be approximately 200.5 and the population standard deviation is estimated to be approximately 14.15.
To estimate the population variance and standard deviation with 99% confidence, we can use the following steps:
Step 1: Calculate the mean (average) of the data set. We can do this by adding up all the values and dividing the sum by the number of data points (in this case, 8):
Mean = (66.0 + 75.8 + 70.9 + 73.9 + 63.4 + 68.5 + 73.3 + 65.9) / 8 = 68.725
Step 2: Calculate the deviation of each data point from the mean. This is done by subtracting the mean from each data point:
(66.0 - 68.725) = -2.725
(75.8 - 68.725) = 7.075
(70.9 - 68.725) = 2.175
(73.9 - 68.725) = 5.175
(63.4 - 68.725) = -5.325
(68.5 - 68.725) = -0.225
(73.3 - 68.725) = 4.575
(65.9 - 68.725) = -2.825
Step 3: Square each deviation:
(-2.725)^2 = 7.40625
(7.075)^2 = 49.985625
(2.175)^2 = 4.726875
(5.175)^2 = 26.737625
(-5.325)^2 = 28.330625
(-0.225)^2 = 0.050625
(4.575)^2 = 20.906875
(-2.825)^2 = 7.99625
Step 4: Calculate the sum of squared deviations:
7.40625 + 49.985625 + 4.726875 + 26.737625 + 28.330625 + 0.050625 + 20.906875 + 7.99625 = 146.14075
Step 5: Divide the sum of squared deviations by the number of data points minus 1 (known as degrees of freedom):
Population Variance = Sum of squared deviations / (n - 1) = 146.14075 / (8 - 1) = 146.14075 / 7 = 20.87725
Step 6: Calculate the population standard deviation by taking the square root of the population variance:
Population Standard Deviation = sqrt(Population Variance) = sqrt(20.87725) = 4.572941971
Since we want to estimate the population variance and standard deviation with 99% confidence, we will use a t-distribution instead of a normal distribution. However, since the sample size is small (<30), we need to use a modified formula known as Bessel's correction to adjust for bias.
Bessel's Correction = Square Root((n / (n-1)) * Sample Variance)
Using Bessel's Correction, the estimated population variance and standard deviation with 99% confidence are as follows:
Estimated Population Variance = sqrt((8 / (8-1)) * 20.87725) = sqrt(2.875 * 20.87725) = sqrt(59.9149875) = 7.739054347
Estimated Population Standard Deviation = sqrt(Estimated Population Variance) = sqrt(7.739054347) = 2.783563497
32
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
99% = mean ± 2.575 SD
I'll let you do the calculations.