In Exercises 15–22, test the claim about the population variance or standard deviation at the level of significance Assume the population is normally distributed.
Claim: σ<40, α=0.01 . Sample statistics: s=40.8, n=12

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: σ ≥ 40, and the alternative hypothesis is H₁: σ < 40. This is a left-tailed test since the claim is that the population standard deviation is less than 40.
2. Step 2: Determine the test statistic to use. Since this is a test about the population standard deviation, use the chi-square test statistic formula: χ² = ((n - 1) * s²) / σ₀², where n is the sample size, s is the sample standard deviation, and σ₀ is the hypothesized population standard deviation.
3. Step 3: Substitute the given values into the formula. Here, n = 12, s = 40.8, and σ₀ = 40. Calculate s² (the sample variance) as s² = (40.8)², and then compute χ² using the formula.
4. Step 4: Determine the critical value for the chi-square distribution. Use the chi-square table with degrees of freedom (df) = n - 1 = 12 - 1 = 11 and a significance level of α = 0.01 for a left-tailed test. Find the critical value χ²_critical corresponding to these parameters.
5. Step 5: Compare the calculated χ² value to the critical value χ²_critical. If χ² < χ²_critical, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the claim about the population standard deviation.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0 in favor of H1. In this case, the null hypothesis would state that the population standard deviation is greater than or equal to 40, while the alternative claims it is less than 40.

Significance Level (α)

The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error. In this scenario, α is set at 0.01, indicating a 1% risk of concluding that the population standard deviation is less than 40 when it is not. This level of significance helps determine the threshold for making statistical inferences based on the sample data.

Chi-Square Test for Variance

The Chi-Square test for variance is a statistical test used to determine if the variance of a population is equal to a specified value. It is particularly useful when the population is normally distributed. In this case, the test will compare the sample variance (derived from the sample standard deviation) to the hypothesized population variance to assess whether the claim about the population standard deviation being less than 40 holds true.