Answer :
The answer of the given question based on the standard deviation is the point estimate of the population standard deviation is approximately 7.688. The answer choice is A.
What is Standard deviation?
Standard deviation is a measure of the variability or dispersion of a set of data points. It tells us how much the data deviates from the mean or average value. The standard deviation is calculated by taking the square root of the variance. The variance is calculated by taking the sum of the squared differences between each data point and the mean, and dividing by the total number of data points.
To estimate the population standard deviation from a sample, we can use the formula:
s = √[Σ(x i - ₓ⁻)² / (n - 1)]
where s is the sample standard deviation, Σ(x i - ₓ⁻)² is the sum of the squared differences between each sample value and the sample mean, n is the sample size, and ₓ⁻ is the sample mean.
Using the given data, we have:
ₓ⁻ = (9 + 13 + 15 + 15 + 21 + 24) / 6 = 15.5
Σ(x i - ₓ⁻)² = (9 - 15.5)² + (13 - 15.5)² + (15 - 15.5)² + (15 - 15.5)² + (21 - 15.5)² + (24 - 15.5)² = 611
n = 6
Substituting the values into formula, we will get:
s = √[Σ(x i - ₓ⁻)² / (n - 1)] = √[611 / 5] ≈ 7.688
Therefore, the point estimate of the population standard deviation is approximately 7.688. The answer choice is A.
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Finally, to get the sample standard deviation, we take the square root of the sample variance: [tex]s \sqrt(49.27) \approx 7.02[/tex] (rounded to two decimal places)Thus, option D is correct.
What is the sample standard deviation?
To calculate the point estimate of the population standard deviation, we can use the sample standard deviation formula. The sample standard deviation (denoted as s) is given by:
[tex]s = \sqrt(Σ(x - xx_1)^2 / (n - 1))[/tex]
where:
x = individual data points in the sample
[tex]x_1 =[/tex]mean of the sample
n = number of data points in the sample
Given the data points in the simple random sample: [tex]9, 13, 15, 15, 21, 24[/tex]
First, we need to calculate the sample mean (x):
[tex]x = (9 + 13 + 15 + 15 + 21 + 24) / 6 = 97 / 6 \approx 16.17[/tex](rounded to two decimal places)
Next, we can plug the sample mean (x) into the formula and calculate the sum of squared differences:
[tex]Σ(x - xx_1)^2 = (9 - 16.17)^2 + (13 - 16.17)^2 + (15 - 16.17)^2 + (15 - 16.17)^2 + (21 - 16.17)^2 + (24 - 16.17)^2 \approx 246.33[/tex] (rounded to two decimal places)
Then, we divide the sum of squared differences by (n - 1) to get the sample variance:
[tex]s^2 = Σ(x - xx)^2 / (n - 1) = 246.33 / 5 \approx 49.27[/tex] (rounded to two decimal places)
Finally, to get the sample standard deviation, we take the square root of the sample variance:
[tex]s \approx \sqrt(49.27) ≈ 7.02[/tex] (rounded to two decimal places)
Therefore, Finally, to get the sample standard deviation, we take the square root of the sample variance: [tex]s \sqrt(49.27) \approx 7.02[/tex] (rounded to two decimal places)
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