Refer to exercise 2. a. What hypotheses are implied in this problem? b. At the α=.05 level of si

step 1 So this problem is based on question number 29. So what I'm going to do is basically give you th

Refer to exercise 2. a. What hypotheses are implied in this...

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population based on sample data. It involves setting up a null hypothesis (a default position or statement of no effect) and an alternative hypothesis (what you want to test), and then using evidence from the data to decide whether to reject the null hypothesis in favor of the alternative.

Null Hypothesis

The null hypothesis is a statement that there is no effect, no difference, or no relationship between variables in the context of the investigation. It serves as the default assumption that any observed effect in the sample data is due to chance, and it is the hypothesis that the test seeks to reject if sufficient evidence is found.

Alternative Hypothesis

The alternative hypothesis is the statement that contradicts the null hypothesis. It reflects the outcome that the researcher suspects or wants to prove, suggesting that there is a real effect, difference, or association in the population. The alternative hypothesis is accepted if the evidence from the data is strong enough to reject the null hypothesis.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used to determine whether the observed data are sufficiently unlikely under the null hypothesis. A common value is 0.05. If the probability (p-value) of obtaining the observed data (or more extreme) is less than ?, the null hypothesis is rejected, indicating that the result is statistically significant.

p-value and Decision Rule

The p-value is the probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true. In hypothesis testing, the decision rule is to reject the null hypothesis if the p-value is less than the significance level (?). This process helps assess the strength of the evidence against the null hypothesis and guides the conclusion on whether to support the alternative hypothesis.

Transcript

00:01 So this problem is based on question number 29.

00:03 So what i'm going to do is basically give you the answer and the methodology for figuring out the answers to question number 29 and then go into how we're going to do this problem as well.

00:13 So if you just want the answer to question 30, go ahead and fast forward and figure out where that occurs.

00:23 So the first thing we're going to do is find a sum of squares due to error.

00:28 So we have a sum of squares of treatments of 300 and a sum of squares of the total of 460.

00:38 So in order to find the sum of squares due to error, we simply have to take the sum of squares of the total.

00:49 We also could write this as the variance of the total.

00:58 So let me just finish this, minus the sum of squares of the treatment.

01:06 We could also write that as variance of the treatment.

01:12 So we have a sum of squares of the total of 460, and sum of square of the treatment of 300, which is equal to 160.

01:22 So this sum of squares of the error, sorry, is 160.

01:28 Now the next thing we need to do is figure out mean of squares due to treatment.

01:34 So the mean of squares due to treatment is equal to the sum of squares due to treatment over the degrees of freedom.

01:42 And for mean of squares due to freedom or due to treatment, the degrees of freedom is equal to the number of factor levels minus one.

01:53 And we have a number of factor levels of five.

01:56 So this is equal to 5 minus 1, which is equal to 4.

02:02 So this over here is equal to 300 over 4, which is equal to 75.

02:16 So this is equal to 75, and this is equal to 5.

02:19 The degrees of freedom is equal to 5.

02:21 Oh, sorry, 4.

02:22 Degrees of freedom is equal to 4.

02:25 And now to find the mean of squares due to error, we're basically going to do the same thing.

02:29 We take the sum of squares due to error over the degrees of freedom.

02:35 The degrees of freedom for error is equal to the number of total elements that we have altogether, which is equal to the number of factor levels times the number of experimental units.

03:01 And then this minus the number of factor levels.

03:10 Factor wrong color minus number of factor levels which we denote with n.

03:20 So the number of elements that we have is equal to, oh sorry, sorry.

03:31 It is the number of factor levels times experimental units minus one.

03:37 I got that wrong.

03:37 I'm so sorry...

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