Find the test statistic for the hypothesis test: a. Ho: σ^2=532 versus Ha: σ^2>532 using sample

step 1 Hi, to find that this statistics simply follow this formula, which is simply n minus 1, where n

Find the test statistic for the hypothesis test: a. Ho:...

Find the test statistic for the hypothesis test:
a. $H_{o}: \sigma^{2}=532$ versus $H_{a}: \sigma^{2}>532$ using sample information $n=18$ and $s^{2}=785$
b. $H_{o}: \sigma^{2}=52$ versus $H_{a}: \sigma^{2} \neq 52$ using sample information $n=41$ and $s^{2}=78.2$

Key Concepts

Test Statistic Formula

The test statistic used in these variance tests is given by the formula: (n-1)*s²/??², where s² is the sample variance, n is the sample size, and ??² is the hypothesized population variance. This formula standardizes the sample variance under the assumption of normality, allowing the use of the chi-square distribution to draw conclusions about the population variance based on the sample data.

Hypothesis Testing for Variance

Hypothesis testing for the variance involves setting up a null hypothesis that the population variance is equal to a specific value versus an alternative hypothesis that it differs (either being greater than, less than, or simply not equal to the specified value). The outcomes of the test determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative.

Chi-Square Distribution

The chi-square distribution plays a central role in hypothesis testing for a population variance. When sampling from a normally distributed population, the test statistic formed by multiplying the sample variance by its degrees of freedom and dividing by the hypothesized variance follows a chi-square distribution. This property is used to assess how well the sample variance agrees with the hypothesized population variance.

Degrees of Freedom

Degrees of freedom in the context of variance estimation are determined by the sample size minus one (n-1). They represent the number of independent values that can vary in the calculation of a statistic such as the sample variance. This factor is crucial for accurately characterizing the chi-square distribution used in testing hypotheses about the variance.

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