A researcher wants to investigate claims that a new plant compound is effective in curbing appetite. She randomly assigns 22 rats to a treatment or control group (11 rats in each group). Rats in the treatment group are given the plant compound daily for a week, while rats in the control group are given a placebo. Over the course of the week, the researcher monitors the total number of calories consumed by all 22 rats. All the rats are kept separate from one another, but in their typical environments, for the duration of the week. The researcher’s null hypothesis is that the rats given the plant compound consume the same number of calories as the rats given a placebo. Since the rats in the treatment group are independent of the rats in the control group, the researcher realizes that the design of her study is an independent-measures design. The researcher knows that the total calories consumed by a rat over the course of a week are normally distributed. However, the researcher is not sure that the variance (σ²) of the total calories consumed by a rat over the course of a week is the same regardless of whether the rat is given the plant compound or a placebo. Suppose the mean number of weekly calories consumed by the rats given the plant compound (the treatment group) is 843 with a standard deviation of 111, and the mean number of calories consumed by the rats in the control group is 946 with a standard deviation of 73. The researcher is unsure of which of the following required assumptions for the independent-measures t test? Are the observations within each group independent? Is there homogeneity of variance? Are the calories consumed by the rats in each group normally distributed? Are the control and treatment groups independent of one another? The researcher decides to use a Hartley’s F-max test. The value of F-max is . Using the partial Critical Values for the F-max Statistic table below, the critical value for the researcher’s F-max with α = 0.01 is . Critical Values for the F-max Statistic (Partial) Critical values for α = .05 are in regular type; critical values for α = .01 are bold. k = Number of Samples n – 1 2 3 4 9.60 15.5 23.2 37.0 5 7.15 10.8 14.9 22.0 6 5.82 8.38 11.1 15.5 7 4.99 6.94 8.89 12.1 8 4.43 6.00 7.50 9.9 9 4.03 5.34 6.54 8.5 10 3.72 4.85 5.85 7.4 12 3.28 4.16 4.91 6.1 15 2.86 3.54 4.07 4.9 20 2.46 2.95 3.32 3.8 30 2.07 2.40 2.63 3.0 60 1.67 1.85 1.96 2.2 Given the results of the F-max test, the researcher’s conclusion is that . Can you still use the independent-measures t test if the homogeneity of variance assumption is violated? Yes, you don’t use a pooled variance and you recompute the degrees of freedom so that the resulting df is larger. No, you cannot use the independent-measures t test. Yes, you don’t use a pooled variance and you recompute the degrees of freedom so that the resulting df is smaller.
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
k = Number of Samples | ||
n – 1 | 2 | 3 |
4 | 9.60 | 15.5 |
23.2 | 37.0 | |
5 | 7.15 | 10.8 |
14.9 | 22.0 | |
6 | 5.82 | 8.38 |
11.1 | 15.5 | |
7 | 4.99 | 6.94 |
8.89 | 12.1 | |
8 | 4.43 | 6.00 |
7.50 | 9.9 | |
9 | 4.03 | 5.34 |
6.54 | 8.5 | |
10 | 3.72 | 4.85 |
5.85 | 7.4 | |
12 | 3.28 | 4.16 |
4.91 | 6.1 | |
15 | 2.86 | 3.54 |
4.07 | 4.9 | |
20 | 2.46 | 2.95 |
3.32 | 3.8 | |
30 | 2.07 | 2.40 |
2.63 | 3.0 | |
60 | 1.67 | 1.85 |
1.96 | 2.2 |
Given:
The required assumptions for the independent-measures t test is, Is there homogeneity of variance?
The test hypothesis are,
H0: σ12 = σ22
H1: σ12 ≠ σ22
This is a two-tailed test.
We reject the null hypothesis if F-value > F-critical value.
The critical value for the researcher’s F-max with α = 0.01 is 5.85.
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