Question

In Exercise \(5.4 .14\) we found a confidence interval for the variance \(\sigma^{2}\) using the variance \(S^{2}\) of a random sample of size \(n\) arising from \(N\left(\mu, \sigma^{2}\right)\), where the mean \(\mu\) is unknown. In testing \(H_{0}: \sigma^{2}=\sigma_{0}^{2}\) against \(H_{1}: \sigma^{2}>\sigma_{0}^{2}\), use the critical region defined by \((n-1) S^{2} / \sigma_{0}^{2} \geq c .\) That is, reject \(H_{0}\) and accept \(H_{1}\) if \(S^{2} \geq c \sigma_{0}^{2} /(n-1)\). If \(n=13\) and the significance level \(\alpha=0.025\), determine \(c .\)

Short answer

The critical value \(c\) equals \(21.026 * \sigma_{0}^{2} / 12\)

Step-by-step solution

Define the problem

We are given a random sample of size \(n=13\). We want to test a hypothesis, \(H_0\) that \(\sigma^{2} = \sigma_{0}^{2}\) against \(H_1\): \(\sigma^{2} > \sigma_{0}^{2}\). We also have a critical region defined by \((n-1)S^2 / \sigma_{0}^2 \geq c\), \(\alpha = 0.025\), and we need to determine \(c\). The test statistic \((n-1)S^{2} / \sigma_{0}^{2}\) has a chi-square distribution with \((n-1)\) degrees of freedom.

Apply inverse chi-square distribution

The critical value \(c\) is determined by the requirement that the probability in the right tail is equal to the significance level \(\alpha = 0.025\). Therefore, it is the 97.5 percentile, \(P_{0.025}\), of the chi-square distribution with \((n-1)\) degrees of freedom. We can find this by using chi-square distribution table.

Refer to the chi-square table

Upon referring to the chi-square table with (n-1) degrees of freedom which equals 12 (since n=13), we find P_{0.025} equals to 21.026.

Calculate c

Now we substitute this value back into the equation for finding \(c = (n-1) S^{2} / \sigma_{0}^{2} \geq P_{0.025}\) and rearrange to solve for \(c\), we find that \(c = P_{0.025} * \sigma_{0}^{2} /(n-1)\) or \(c = 21.026 * \sigma_{0}^{2} / 12.\)

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide if there is enough evidence to support a specific claim about a population parameter. In this context, we're focusing on hypotheses related to variance.Here's a breakdown of how hypothesis testing works:- **Null Hypothesis (H_0):** This is a statement that there is no effect or change, often represented as an equality. In our exercise, it's \(H_0: \sigma^2 = \sigma_0^2\), assuming the population variance equals a specific value.- **Alternative Hypothesis (H_1):** This represents a new claim or the effect we suspect is true. Here, it's \(H_1: \sigma^2 > \sigma_0^2\), suggesting the population variance is greater than the stated value.- **Test Statistic:** A standardized value calculated from sample data, which is used to decide whether to reject H_0. In this case, the test statistic is \((n-1)S^2 / \sigma_0^2\), which follows a Chi-Square distribution.Once the hypotheses are set, analyze sample data and assess them through the prism of these hypotheses to reach meaningful conclusions. Hypothesis testing boils down to figuring out if the sample supports the null or the alternative perspective.

Significance Level

The significance level, denoted as \(\alpha\), is a critical aspect of hypothesis testing. This value represents the probability of rejecting the null hypothesis, H_0, when it is actually true. Essentially, it measures the risk of making a type I error.- **Common Significance Levels:** Typical values for \(\alpha\) include 0.05, 0.01, and 0.10. For this exercise, we've chosen \(\alpha = 0.025\), which is a more conservative threshold.- **Decision Making:** If the calculated p-value in a test is less than \(\alpha\), we reject H_0. Otherwise, we do not reject H_0.- **Critical Region:** In our problem, the critical region is determined by finding a cutoff value \(c\) such that \((n-1)S^2 / \sigma_0^2 \geq c\) if the null hypothesis is false.Choosing an appropriate significance level is crucial as it influences how strict or lenient our hypothesis test is, affecting conclusions drawn from data results.

Confidence Interval

A confidence interval provides a range of values within which we can be certain a parameter lies, based on sample data. While this exercise focuses on hypothesis testing, understanding confidence intervals is key to interpreting tests related to variance. - **Relation to Hypothesis Testing:** If a null hypothesis value falls outside the confidence interval, the hypothesis can often be rejected at the corresponding significance level. - **Variance Confidence Interval:** For variance, specifically, you might use a Chi-Square distribution when constructing the interval. Variance implies how much the data points in a sample differ from the mean, and understanding this spread is essential. Suppose you calculate a confidence interval saying the true variance might be between two values. If your null hypothesis value does not fit in this range, this provides more evidence against H_0 . Remember, confidence intervals complement hypothesis tests by giving a visual representation of what values are plausible based on the sample. In studying these, you harness more power to inspect claims and draw informed conclusions.