A simple random sample of 27 adult mice has a length standard deviation of 0.66 mm. Using a 0.10 level of significance, does it appear that these mice come from a population with a length standard deviation of 0.87 mm?
To determine if there is a significant difference between the sample standard deviation (0.66 mm) and the population standard deviation (0.87 mm), we can use a statistical test called the F-test for equality of variances.
The null hypothesis (H₀) assumes that the sample and population standard deviations are equal, while the alternative hypothesis (H₁) assumes they are not equal.
Here's the step-by-step process to perform the F-test:
Step 1: Set up the hypotheses:
- Null hypothesis (H₀): The population standard deviation is 0.87 mm.
- Alternative hypothesis (H₁): The population standard deviation is not equal to 0.87 mm.
Step 2: Choose the significance level:
- The significance level, denoted as α, is given as 0.10 (or 10%).
Step 3: Define the test statistic:
- The test statistic for the F-test is calculated using the formula F = (s₁² / s₂²), where s₁ and s₂ are the sample and population standard deviations, respectively.
Step 4: Determine the critical value:
- The critical value for the F-test can be obtained from an F-distribution table or by using statistical software. The degrees of freedom are calculated as (n₁ - 1) and (n₂ - 1), where n₁ and n₂ are the sample sizes of the two groups being compared. In this case, both samples come from the same group, so the degrees of freedom are (27 - 1) = 26 for both the numerator and denominator.
Step 5: Calculate the test statistic:
- Plug in the values into the formula for the test statistic: F = (0.66² / 0.87²). This gives us the value of F.
Step 6: Determine the p-value:
- Using the test statistic (F) and degrees of freedom, calculate the p-value associated with the test statistic using an F-distribution table or statistical software.
Step 7: Make a decision:
- Compare the p-value to the chosen significance level (α). If the p-value is less than α, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.
By going through these steps, you will be able to determine if the mice come from a population with a length standard deviation of 0.87 mm.