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

Significance Level and Two-Tailed Test

The significance level, denoted as alpha (?), represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). In a two-tailed test, this level is divided equally between the two tails of the distribution. This means that for a two-tailed test, the lower-tail critical value corresponds to the ?/2 percentile, while the upper-tail critical value corresponds to the 1-?/2 percentile, ensuring that the total probability of a Type I error is correctly allocated to both extremes of the distribution.

Critical Values Determination

Critical values are specific points on the chi-square distribution that delineate the rejection regions for a hypothesis test. For a given significance level and degrees of freedom, these values are looked up in chi-square distribution tables or computed using statistical software. In a two-tailed test, separate critical values are obtained for the lower and upper tails, ensuring that the combined area under the tails equals the total significance level. This determination is essential for making decisions about rejecting or failing to reject the null hypothesis.

Chi-Square Distribution

The chi-square distribution is a probability distribution that is widely used in statistics, particularly for hypothesis testing and constructing confidence intervals concerning variance. It is a non-symmetric, positively skewed distribution that depends on the degrees of freedom, and is fundamental in tests such as the goodness-of-fit test and tests for independence in contingency tables.

Degrees of Freedom

Degrees of freedom in the context of a chi-square distribution refer to the number of independent values or quantities that can vary in an analysis without violating any given constraints. In many statistical tests, the degrees of freedom are determined by the sample size or by the number of categories being compared, and they affect the shape and critical values of the chi-square distribution.

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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