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 whether the mice come from a population with a length standard deviation of 0.87 mm, we can perform a hypothesis test. The null hypothesis (H₀) is that the population has a standard deviation of 0.87 mm, and the alternative hypothesis (H₁) is that the population does not have a standard deviation of 0.87 mm.
To conduct this hypothesis test, we will use a chi-square test statistic. The formula for the chi-square test statistic for testing the population standard deviation is:
χ² = (n-1) * s² / σ₀²
where:
χ² = chi-square test statistic
n = sample size
s = sample standard deviation
σ₀ = hypothesized population standard deviation
In this case, the sample size (n) is 27, the sample standard deviation (s) is 0.66 mm, and the hypothesized population standard deviation (σ₀) is 0.87 mm.
Substituting these values into the formula, we get:
χ² = (27-1) * (0.66^2) / (0.87^2)
Calculating this expression, we find that χ² ≈ 13.67.
To find the critical value for a 0.10 level of significance, we need to look up the corresponding chi-square value for a 0.10 right-tailed test with (27-1) = 26 degrees of freedom. The critical chi-square value is approximately 36.4148.
Since the test statistic (13.67) is less than the critical value (36.4148), we do not have enough evidence to reject the null hypothesis. Thus, it does not appear that these mice come from a population with a length standard deviation of 0.87 mm.
In summary, based on the given information, we performed a chi-square test using the formula to determine the test statistic and found that it was less than the critical value. Therefore, we failed to reject the null hypothesis and cannot conclude that the mice come from a population with a length standard deviation of 0.87 mm.