Question

In Exercise \(4.2 .18\) 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 \(c\) is calculated by using the given sample size, the defined significance level, and using a Chi-square distribution table. This computed critical value is then used to define the critical region for this specific hypothesis testing problem.

Step-by-step solution

Define the Hypothesis

We have the null hypothesis \(H_0: \sigma^2=\sigma_0^2\) and our alternate hypothesis \(H_1: \sigma^2>\sigma_0^2\)

Determine the Significance Level

The significance level is given as \(\alpha=0.025\)

Refer to the Chi-Square Distribution

The test statistic for variance follows a Chi-square distribution. In this case, with sample size \(n=13\), the degrees of freedom for the Chi-square distribution is \(n-1=12\)

Find the Critical Value

We find the critical value \(c\) from the Chi-square distribution table value that corresponds to the given significance level \(\alpha\) and degrees of freedom \(n - 1 = 12\). Here, because it's a one-tailed test and the right tail is of interest, we look up the Chi-square value corresponding to \(\alpha\) in tail.

Calculate the Critical Region Critical Value

To find the Critical Region we use the provided formula \((n-1)S^2 /\sigma_0^2 >= c\) and solve for \(c\). That is \(c = (n-1)\cdot S^2 / \sigma_0^2\), given that the significance level \(\alpha=0.025\), \(n=13\) and \(\sigma_0^2\) is derived from the null hypothesis.

Key concepts

Chi-Square Distribution

The chi-square distribution is fundamental when dealing with statistical methods related to variance. It emerges quite naturally when working with samples from a normally distributed population. In hypothesis testing, the chi-square distribution is particularly useful to determine whether there is a significant difference between the expected and observed values.

When we describe the chi-square distribution, we focus on its degrees of freedom, typically denoted as \(df\). The degrees of freedom for a chi-square distribution related to variance testing is usually \(n-1\), where \(n\) is the sample size. This is because the estimate of the variance relies on \(n-1\) independent comparisons (each data point compared to the sample mean), rather than \(n\) comparisons, to avoid the bias that would result from including the mean itself in the comparisons.

To use this distribution in hypothesis testing, a test statistic is calculated. If the null hypothesis is true, this statistic follows a chi-square distribution with the relevant degrees of freedom. Critical values obtained from chi-square tables help determine whether to reject the null hypothesis, based on whether the test statistic falls within a specified critical region.

Confidence Interval for Variance

Creating a confidence interval for the variance \(\sigma^2\) involves estimating the range within which the true population variance is likely to fall, with a specific level of confidence. Confidence intervals are a way of quantifying the uncertainty inherent in the sample statistics – they reflect the precision of the sample variance as an estimate of the population variance.

To construct this interval, we need the sample variance \(S^2\), sample size \(n\), and the desired level of confidence. The confidence interval is then derived from the sample statistics and the chi-square distribution. The bounds of the interval are found using the critical values of the chi-square distribution that correspond to the chosen confidence level. This interval communicates that we believe, with a certain level of confidence, that the true variance lies within this range.

A key point to note is that because we're working with variance, which is always positive, these intervals are not symmetrical around the sample variance. This asymmetry is inherent to the nature of the chi-square distribution.

Null Hypothesis in Statistics

The concept of the null hypothesis \(H_0\) is a cornerstone of statistical hypothesis testing. It represents the default or baseline assumption that there is no effect or no difference – in the context of variance, it typically asserts that the population variance is equal to a specified value \(\sigma_0^2\).

The null hypothesis serves as a contradiction to the alternative hypothesis \(H_1\), which asserts the presence of an actual effect or difference. In variance testing, for example, the alternative might claim that the population variance is greater than (or less than or different from) \(\sigma_0^2\). The decision to reject the null hypothesis, and thus support the alternative, is made based on whether the calculated test statistic falls within the critical region previously determined by the significance level and the chi-square distribution for the given degrees of freedom.

In practice, we never prove a null hypothesis; we only reject it or fail to reject it based on the evidence provided by sample data. If the null hypothesis is rejected, it suggests that our sample provides enough evidence to support the alternate hypothesis, within the chosen level of significance. The level of significance \(\alpha\) reflects the probability of rejecting the null hypothesis when it is actually true, known as a Type I error.