Consider the hypothesis test of H0: σ^2=7 against H1: σ^2 ≠7. Approximate the P -value for each

step 1 Hello, welcome to this lesson. In this lesson we have the test statistic that is 25 .2. We have

Consider the hypothesis test of H0: σ^2=7 against H1: σ^2...

Consider the hypothesis test of $H_{0}: \sigma^{2}=7$ against $H_{1}: \sigma^{2} \neq 7$. Approximate the $P$ -value for each of the following test statistics.
(a) $x_{0}^{2}=25.2$ and $n=20$
(b) $x_{0}^{2}=15.2$ and $n=12$
(c) $x_{0}^{2}=23.0$ and $n=15$

Key Concepts

P-Value Calculation

The p-value in hypothesis testing quantifies the probability of obtaining a test statistic at least as extreme as the one observed, under the assumption that the null hypothesis is true. In variance tests using the chi-square distribution, the p-value is calculated by determining the tail probability of the test statistic and, in the case of a two-tailed test, combining the probabilities from both tails. This value helps in deciding whether to reject or fail to reject the null hypothesis based on a predetermined significance level.

Hypothesis Testing for Variance

Hypothesis testing for variance involves comparing the observed sample variance to the hypothesized population variance. The null hypothesis usually postulates that the population variance is a specific value, against an alternative hypothesis that the population variance is different. A test statistic is computed, which, under the null hypothesis, follows a chi-square distribution. This framework allows statisticians to determine if observed discrepancies are statistically significant.

Chi-Square Distribution

The chi-square distribution is commonly used in statistics to model the distribution of a sum of squared standard normal random variables. It plays a central role in tests concerning variances, where the test statistic (often involving a scaled sample variance) follows a chi-square distribution under the null hypothesis. The shape of the chi-square distribution depends on the degrees of freedom, making it suitable for modeling variability in sample variance estimates.

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available for estimating a parameter. In the context of testing the variance of a normally distributed population, the degrees of freedom are typically the sample size minus one (n - 1). This adjustment is necessary because one parameter (the mean) is estimated from the data, and it directly affects the shape of the chi-square distribution used in the test.

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