The following data are from a completely randomized design. Treatment Treatme

step 1 So in part A, we're doing an ANOVA test with the values. And I have my data stored in list one,

The following data are from a completely randomized...

$$ \begin{aligned} &\text { The following data are from a completely randomized design. }\\ &\begin{array}{lccc} & \text { Treatment } & \text { Treatment } & \text { Treatment } \\ & \text { A } & \text { B } & \text { C } \\ & 32 & 44 & 33 \\ & 30 & 43 & 36 \\ & 30 & 44 & 35 \\ & 26 & 46 & 36 \\ \text { Sample mean } & 32 & 48 & 40 \\ \text { Sample variance } & 30 & 45 & 36 \\ & 6.00 & 4.00 & 6.50 \end{array} \end{aligned} $$

$$
\begin{aligned}
&\text { The following data are from a completely randomized design. }\\
&\begin{array}{lccc}
& \text { Treatment } & \text { Treatment } & \text { Treatment } \\
& \text { A } & \text { B } & \text { C } \\
& 32 & 44 & 33 \\
& 30 & 43 & 36 \\
& 30 & 44 & 35 \\
& 26 & 46 & 36 \\
\text { Sample mean } & 32 & 48 & 40 \\
\text { Sample variance } & 30 & 45 & 36 \\
& 6.00 & 4.00 & 6.50
\end{array}
\end{aligned}
$$

Key Concepts

F-Test

The F-test is a hypothesis test used in the context of ANOVA to determine whether the variance between group means is larger than would be expected by chance. A significant F-test result indicates that not all group means are equal, suggesting that treatment differences exist.

Treatment Effects

Treatment effects refer to the impact or influence of the different conditions or interventions being studied in an experiment. Assessing treatment effects involves comparing group means to determine whether the differences observed are statistically significant, which can indicate the presence of a real effect attributable to the treatments.

Statistical Summary Measures

Statistical summary measures, such as the sample mean and sample variance, are used to summarize the central tendency and variability of the data within each treatment group. These measures provide a basis for comparing groups and are essential inputs in methods like ANOVA for evaluating the consistency and differences of data distributions.

Completely Randomized Design

A completely randomized design is an experimental design in which subjects are randomly allocated to different treatment groups. This design ensures that each experimental unit has an equal chance of receiving any treatment, thereby reducing bias and helping to balance the effects of uncontrolled variables across groups.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to compare the means of three or more groups to determine if at least one group mean differs significantly from the others. It works by partitioning the total variability in the data into components attributable to between-group differences and within-group variability, allowing a formal F-test to assess overall differences.

Transcript

00:01 So in part a, we're doing an anova test with the values.

00:05 And i have my data stored in list one, list two, and list three.

00:10 And we find that our f statistic is equivalent to 51 .8 and 18.

00:19 And the p value for this is equal to 1 .2 times 10 to the negative 6th power.

00:27 And we're using an alpha level of 0 .05.

00:30 And we were assuming that the mean from a is equal to the mean from b is equal to the mean for c, and alternately that not all above are equal.

00:48 And so we definitely have evidence at a 5 % significance level to reject the null.

00:58 Now, part b, we want to use the technique that they show us, that fisher technique, to show if there are differences between a and b.

01:07 So we would start out with whether the mean of a is equal to the mean of b, and alternately that the mean of a is not equal to the mean of b.

01:20 And our test statistic is to take, which will be a t value, and our number of pieces of data, n total, was 15, and we have a k value of 3.

01:33 So our degrees of freedom will be 12.

01:37 And we'll have, and this is a two -tail test because we're doing a not equal to.

01:41 And we take the difference between the two means, and that is 30 minus 45.

01:47 And then we take the square root of, and we need to have what the mean square error is.

01:52 And from my analysis, i have the mean square of the air is 5 .5, and that's times one over the sample size of the first group and one over the sample size of the second.

02:03 Group and so we'll end up having let's see that's going to be negative 5 divided by the square root of and we'll have that 5 .5 times in this 5 .5 and it will be times put in a decimal point rather than times and then this will be two -fifths times two divided by five and that gives us a test statistic of negative 3 .37 and that p value is going to be this like of being in that lower tail and then times two.

02:40 So let me quick get that for use.

02:41 The second distribution, my tcdf, and i can plug in that negative 3 .37.

02:50 And then my degrees of freedom is my 12.

02:53 And we'll paste that down.

02:55 And then i'm going to multiply that by 2.

02:57 And this is going to be a small p value.

02:58 It's only 0 .0056.

03:01 And this is less than 0 .055.

03:03 So therefore, we would reject the null.

03:07 So we do have evidence that these two are different.

03:11 Now we can carry on and do the same thing for doing the comparison between, and i'm just going to put the between b and c...