Standard Deviation

To calculate the point estimate of the population standard deviation for the given sample data , follow these steps: 1. Calculate the sample mean ( ): . 2. Calculate the sample variance ( ), using the formula: where is the sample size, which is 10. 3. Calculate the sample standard deviation ( ): 4. The point estimate of the population standard deviation is . Thus, the point estimate of the population standard deviation is the simplified form , aligning with the provided data and possible answers.

To determine the correct expression for the standard deviation, let us first understand the formula for standard deviation itself. The standard deviation is defined as the square root of the variance. The variance is the average of the squared differences between each data point and the mean of the dataset. The steps are as follows:

Hence, the correct option for the standard deviation is: .

To calculate the point estimate of the population standard deviation, we begin with finding the sample standard deviation of the given data: 1, 4, 7. First, we calculate the mean ( ) of the sample:

Next, determine the squared differences from the mean:

(1 - 4) = 9

(4 - 4) = 0

(7 - 4) = 9

The variance (s ) of the sample is the average of these squared differences:

Now, find the sample standard deviation (s) as the square root of the variance:

Thus, the point estimate of the population standard deviation is 3.

To solve this hypothesis testing problem, we need to calculate the test statistic for the given data. The hypothesis we have is a one-tailed test:
H₀: μ≥16 (null hypothesis)
H₁: μ<16 (alternative hypothesis)

Given data:
Sample size (n) = 36
Sample mean (x̄) = 15.4
Population standard deviation (σ) = 3
Population mean under H₀ (μ) = 16

The formula for the z-test statistic is:

z = (x̄ − μ) / (σ/√n)

Substituting the given values into the formula:

z = (15.4 − 16) / (3/√36)

z = (−0.6) / (3/6)

z = (−0.6) / 0.5

z = −1.2

Thus, the calculated test statistic is −1.2, which matches the correct answer from the provided options.

The problem involves calculating the test statistic for a sample mean. We have the following data:

Our goal is to compute the test statistic using these steps:

Therefore, the value of the test statistic is -2.