Determine the lower- and upper-tail critical values of χ^2 for each of the following two-tail te

Determine the lower- and upper-tail critical values of χ^2...

Key Concepts

Tail Probabilities and Significance Level Splitting

In a two-tailed test, the significance level (?) is divided equally between the two tails of the distribution. The lower-tail critical value is found at a cumulative probability of ?/2, while the upper-tail critical value is found at a cumulative probability of 1 - ?/2. This approach ensures that extreme values on either end of the distribution lead to rejection of the null hypothesis.

Critical Values

Critical values are points on the chi-square distribution that provide thresholds to decide whether to reject the null hypothesis. In a two-tailed test, two critical values are determined: one for the lower tail and one for the upper tail. These values are derived from the cumulative probabilities assigned to the tails based on the significance level, ensuring that the overall Type I error rate is maintained.

Chi-Square Distribution

The chi-square distribution is a continuous probability distribution used extensively in inferential statistics, especially in tests of variance and goodness?of?fit. It is defined only for positive values and is skewed to the right; its shape depends on the degrees of freedom. In many problems, including tests involving variances, the chi-square statistic under the null hypothesis follows this distribution.

Degrees of Freedom

Degrees of freedom generally represent the number of independent pieces of information available for estimating a parameter. In the context of chi-square tests, the degrees of freedom often are based on the sample size (typically n - 1 when estimating the variance of a normally distributed population), and they determine the shape of the chi-square distribution from which critical values are computed.

Transcript

00:01 Hi, in this question we're gonna see how to find the critical value so let me explain something here then we're gonna go how to find the critical value from the table so in both cases for the left tailed and right tailed we're gonna call it one tailed okay and for the right tailed okay it's gonna be the value from here and it's always gonna be a positive value and for the left tail that's gonna be the area from here and it will be a negative value so whatever the value of here in the table it's gonna be positive value so we're gonna keep the sign for the right tailed and for the left tail we're gonna take that the same absolute value and put a negative sign in front of it and in case of two tail okay you're gonna get a positive value from the so the negative value is going to be here this area and the positive one is going to be from here and i'm going to show you in seconds how to find these also in microsoft excel so the first one is asking about the degree freedom of 11 because the sample size or n is 12 and alpha is point on one and it's asking about the left tilt so we get that 11 here okay and degree of freedom of 0 .01 for one tail this one and this one so you're gonna go across and you find it 2 .718 and the second one is gonna be 1 .753 because n is 16 and the alpha is 0 .05 and it's asking about the right tailed and the third one is gonna be here because n is 7 alpha is 0 .1 and it is 2 tails so it's gonna be 1 in that case 1 .9 4 3 and he minus 1 .9 for 3 and the 4th 1 .1 is here and the 4th one is here because n is 11 so the degree of freedom is 10 and it's asking about 0 .025 1 tail the right one so going to be a positive value of 2 .228 and the last one is about two -tailed so it's asking about alpha is 0 .05 and n is 10 which is equivalently to degree of freedom of 9 and it's going to be also two values the positive one is going to be the upper one or 2 .262 and the lower 1 is minus 2 .262 and as you can see here it's matching that values that i compute from microsoft excel.

03:12 And here's the function you are using.

03:15 So if it is one tail, you're going to use t.

03:19 .

03:19 Then you just put the c3, which is here...

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