the percentage rates of home ownership for 8 randomdomly 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 with 99% confidence, we can use the sample data provided.
First, let's calculate the sample mean, which is the average of the percentage rates:
Sample Mean (x̄) = (66.0 + 75.8 + 70.9 + 73.9 + 63.4 + 68.5 + 73.3 + 65.9) / 8
= 556.7 / 8
= 69.5875
Next, we need to calculate the sample variance, which measures the spread of the data:
Sample Variance (s^2) = ∑(x - x̄)^2 / (n - 1)
where x is the individual data point, x̄ is the sample mean, and n is the sample size.
Using the provided data, we can calculate the sample variance as follows:
s^2 = [(66.0 - 69.5875)^2 + (75.8 - 69.5875)^2 + (70.9 - 69.5875)^2 + (73.9 - 69.5875)^2 + (63.4 - 69.5875)^2 + (68.5 - 69.5875)^2 + (73.3 - 69.5875)^2 + (65.9 - 69.5875)^2] / (8 - 1)
= [(-3.5875)^2 + (6.2125)^2 + (1.3125)^2 + (4.3125)^2 + (-6.1875)^2 + (-1.0875)^2 + (3.7125)^2 + (-3.6875)^2] / 7
= [12.8925 + 38.6772 + 1.7216 + 18.4976 + 38.2862 + 1.1811 + 13.7534 + 13.6141] / 7
= 13.9107
Lastly, to estimate the population standard deviation, we take the square root of the sample variance:
Sample Standard Deviation (s) = √s^2
= √13.9107
≈ 3.7317
Since we want a 99% confidence interval, we need to consider the critical value (t*) associated with the degrees of freedom (df = n - 1 = 7). Using a t-distribution table or calculator, the critical value for a 99% confidence interval with 7 degrees of freedom is approximately 3.4995.
Now, we can calculate the margin of error (E) using the formula:
E = t* * (s / √n)
where t* is the critical value, s is the sample standard deviation, and n is the sample size.
E = 3.4995 * (3.7317 / √8)
≈ 4.5266
Finally, we can estimate the population variance (σ^2) and standard deviation (σ) using the following formulas:
Lower Bound = s^2 / (1 + E^2 / (2 * (n - 1)))
Upper Bound = s^2 / (1 - E^2 / (2 * (n - 1)))
Estimated Population Variance (σ^2) = (Lower Bound + Upper Bound) / 2
Estimated Population Standard Deviation (σ) = √σ^2
Lower Bound = 13.9107 / (1 + 4.5266^2 / (2 * 7))
≈ 7.6727
Upper Bound = 13.9107 / (1 - 4.5266^2 / (2 * 7))
≈ 27.6288
Estimated Population Variance (σ^2) = (7.6727 + 27.6288) / 2
≈ 17.6508
Estimated Population Standard Deviation (σ) = √17.6508
≈ 4.1981
Therefore, the estimated population variance for the percentage rate of home ownership with 99% confidence is approximately 17.6508, and the estimated population standard deviation is approximately 4.1981.
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 SEm
SEm = SD/√n
I'll let you do the calculations.