(a) Determine the critical value for a right-tailed test of...

(a) Determine the critical value for a right-tailed test of a population standard deviation with 16 degrees of freedom at the $\alpha=0.01$ level of significance. (b) Determine the critical value for a left-tailed test of a population standard deviation for a sample of size $n=14$ at the $\alpha=0.01$ level of significance. (c) Determine the critical values for a two-tailed test of a population standard deviation for a sample of size $n=61$ at the $\alpha=0.05$ level of significance.

(a) Determine the critical value for a right-tailed test of a population standard deviation with 16 degrees of freedom at the $\alpha=0.01$ level of significance.
(b) Determine the critical value for a left-tailed test of a population standard deviation for a sample of size $n=14$ at the $\alpha=0.01$ level of significance.
(c) Determine the critical values for a two-tailed test of a population standard deviation for a sample of size $n=61$ at the $\alpha=0.05$ level of significance.

Key Concepts

Chi-Square Distribution

The chi-square distribution is a continuous probability distribution that is used extensively in inferential statistics, particularly for hypothesis testing regarding population variances. In the context of testing a population standard deviation (or variance), the test statistic follows a chi-square distribution, allowing researchers to determine the probability of observing a particular value under the null hypothesis.

Degrees of Freedom

Degrees of freedom, typically calculated as the sample size minus one, are a critical parameter in many statistical tests, including the chi-square tests for variances. They indicate the number of independent values that can vary in the analysis and directly impact the shape of the chi-square distribution, thereby influencing the critical values for hypothesis testing.

Significance Level

The significance level, denoted by alpha (?), represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). It is a threshold that determines the critical region of the distribution in hypothesis testing, with smaller values (e.g., 0.01, 0.05) indicating more stringent criteria for rejection.

One-Tailed Test

A one-tailed test focuses on deviations in one direction (either greater than or less than the hypothesized value). When testing standard deviation or variance, a right-tailed test checks for significantly higher variance values and a left-tailed test checks for significantly lower variance values than those specified under the null hypothesis. The direction of the test affects the location of the critical value along the chi-square distribution.

Two-Tailed Test

A two-tailed test considers deviations in both directions away from the hypothesized parameter value. In the context of testing a population standard deviation or variance, this test sets two critical values, one for the lower tail and one for the upper tail of the chi-square distribution. The significance level is split between the two tails, and values falling outside these bounds would lead to rejection of the null hypothesis.

Transcript

00:01 I'm not a right -tailed value for chichy square here we have for degree of freedom 16 alpha 0 .01.

00:06 16 here we have 16 and this is 0 .01 over which is here 16.

00:16 Alpha 0 .01, degree of freedom 16.

00:18 This is giving to the right, here to the right of critical value we have.

00:21 So we just dynamically seen the right here we have.

00:26 This is the critical value chichie square here we have.

00:29 So it's given to the right this value.

00:35 So i'm following the graph.

00:36 We talk later on because in detail okay let it be just to understand like this here we have the right value here we have like this vital value to the right here we have beyond the critical value here it is c the critical value so uh now it is given as we have for a negative 11 16 16 and it is 0 .01 side 2 .000.

01:06 Next for the b part determine a critical value for left tail over here on a deviation from sample of size n equal to 14 alpha is 0 .01 so the b part n equal to 14 alpha is 0 .01 so n equal to 14 we have an alpha is 0 .0 .0 .0 .0.

01:31 Let's check it out for that.

01:37 The value for we have n negative 1 that equal to 13 so 180 of alpha.

01:42 So left we do 180 alpha that's coming up to be 0 .99 13 0 .99.

01:47 13 .0 .99.

01:47 And 30 near we have and 0 .99.

01:51 There we go.

01:52 4 .107...