Use the following information to answer the next three...

Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses. TABLE CANNOT COPY. $$\begin{array}{l}{H_{0} : \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}=\mu_{5}} \\ {H \alpha : \text { At least any two of the group means } \mu_{1}, \mu_{2}, \ldots, \mu_{5} \text { are not equal. }}\end{array}$$ degrees of freedom – denominator: df(denom) = ________

Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which
teenagers obtained their drivers licenses.
TABLE CANNOT COPY.
$$\begin{array}{l}{H_{0} : \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}=\mu_{5}} \\ {H \alpha : \text { At least any two of the group means } \mu_{1}, \mu_{2}, \ldots, \mu_{5} \text { are not equal. }}\end{array}$$

degrees of freedom – denominator: df(denom) = ________

Key Concepts

Analysis of Variance (ANOVA)

ANOVA is a statistical technique used to compare the means of three or more groups to determine if at least one group mean is significantly different from the others. It works by analyzing the variation within each group and between groups to assess the overall significance of differences.

Degrees of Freedom

Degrees of freedom refer to the number of independent pieces of information available to estimate statistical parameters. In ANOVA, they are critical for assessing the variability and for constructing the F statistic.

Within-Group (Denominator) Degrees of Freedom

In ANOVA, the within-group or error degrees of freedom, which form the denominator of the F statistic, is calculated by subtracting the number of groups from the total number of observations (N - k). This value reflects the independent variability within the groups and is essential for testing the null hypothesis.

Transcript

00:01 So for this question, we first start to answer what is our df numerator and what is our df denominator? so df numerator is k minus 1.

00:20 K is the total number of different group.

00:22 In this case, we have five different group.

00:25 So therefore, our df numerator is 5 minus 1 is 4.

00:29 For our df denominator, we have a number number.

00:32 Have in total 20 values minus k which is a total number of different group.

00:40 So n minus k equals to 20 minus five different group equals to 15.

00:48 So next we would find what is our s1, s2, s3, s4, s5.

01:01 So as 1 as 2 s 3 s 4 is 5 they are the sum of each column.

01:13 So s 1 it would be equals to 65 .3, s2 equals to 66, s3 equals to 606, s3 equals to 1075 .84, s4 equals, oh, sorry, so s3 is equals to 65 .6.

01:43 That's why i was like, why the number is so big.

01:46 S4 is equals to 65 .7.

01:52 S5 equals to 67 .7.

01:57 Okay.

01:59 So now, in order for us to calculate the p value, we have to know what is our f statistic is.

02:08 So to calculate f statistic, we would calculate the mean square of between and mean square of within.

02:27 So for calculating mean square between, it would follow the following formula.

02:36 So the total sum of, let me double check.

02:48 Square over nj and j is the number of value per column over the total of s j that sum would be square and over n so to write it as in terms of s1 to s5 we would have s1 square over in our case over 4 plus s2 square over 4 plus dot dot dot to s5 square over 4 minus s1 plus s2 plus dot dot dot to s5 that sum would then square over the total value the total elements in the data is 20 so we know that the calculations you can do it yourself by plugging s1 in this equation and then the final answer would be 0 .903.

04:03 So to calculate, ooh this is not ms, this is some square value and now we need, now we calculate mean square.

04:15 So mean square would be s s over k minus 1.

04:22 So in our case, is df numerator.

04:26 It would be 0 .903 over 4...

Please give Ace some feedback