Consider the following data:
23 17 20 29 21 14 19 24
Develop and test a hypothesis to determine if the variance is greater than 4. Use the critical value approach and α = 0.05
To determine if the variance is greater than 4, we can use a hypothesis test. Here is how to conduct the test using the critical value approach:
Step 1: State the null and alternative hypotheses.
The null hypothesis (H0) assumes that the variance is equal to 4.
H0: σ^2 = 4
The alternative hypothesis (H1) assumes that the variance is greater than 4.
H1: σ^2 > 4
Step 2: Set the significance level (α).
The significance level (α) is given as 0.05.
Step 3: Calculate the test statistic.
To calculate the test statistic, we can use the chi-square distribution with n-1 degrees of freedom, where n is the sample size. In this case, n = 8 (the number of data points).
The test statistic (chi-square) can be calculated using the following formula:
(X-bar - μ) / (σ / √n)
Step 4: Determine the critical value.
Since the alternative hypothesis is one-sided (greater than), we need to find the critical value from the chi-square distribution table. With α = 0.05 and degrees of freedom = n - 1 = 7, the critical value is 14.067.
Step 5: Compare the test statistic with the critical value.
If the test statistic is greater than the critical value, we reject the null hypothesis.
Step 6: Make a decision.
If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the variance is greater than 4. Otherwise, we fail to reject the null hypothesis.
To calculate the test statistic and make a decision, we need the sample variance. The sample variance (s^2) can be calculated using the following formula:
s^2 = Σ(x - x-bar)^2 / (n - 1)
Using the provided data, we can calculate the sample variance as follows:
(23 - x-bar)^2 + (17 - x-bar)^2 + (20 - x-bar)^2 + (29 - x-bar)^2 + (21 - x-bar)^2 + (14 - x-bar)^2 + (19 - x-bar)^2 + (24 - x-bar)^2 / (n - 1)
In this case, we have n = 8, and x-bar can be calculated as the sum of the data points divided by n. After calculating the sample variance, we can compare it with the critical value to make a decision.