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^2>19, α=0.1. Sample statistics: s^2=28, n=17

Verified step by step guidance

1. Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis is H₀: σ² ≤ 19, and the alternative hypothesis is H₁: σ² > 19. This is a right-tailed test because the claim is that the population variance is greater than 19.
2. Step 2: Determine the test statistic formula for a chi-square test for variance. The formula is χ² = ((n - 1) * s²) / σ₀², where n is the sample size, s² is the sample variance, and σ₀² is the hypothesized population variance.
3. Step 3: Substitute the given values into the formula. Here, n = 17, s² = 28, and σ₀² = 19. Compute the test statistic χ² using these values.
4. Step 4: Determine the critical value for the chi-square distribution at the given significance level α = 0.1 and degrees of freedom (df = n - 1). Use a chi-square distribution table or statistical software to find the critical value for df = 16.
5. Step 5: Compare the calculated test statistic to the critical value. If the test statistic is greater than the critical value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the claim.

Key Concepts

Population Variance and Standard Deviation

Population variance (σ²) measures the spread of a set of values in a population, while the standard deviation (σ) is the square root of the variance. These metrics are crucial for understanding the variability within a dataset. In hypothesis testing, they help determine if observed data significantly deviates from a hypothesized value.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters 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 this case, the null hypothesis would state that the population variance is less than or equal to 19.

Level of Significance (α)

The level of significance (α) is the probability of rejecting the null hypothesis when it is actually true, commonly set at 0.05 or 0.1. In this scenario, α=0.1 indicates a 10% risk of concluding that the population variance exceeds 19 when it does not. This threshold helps determine the critical value for making decisions in hypothesis testing.