Question

1. A medical research team is interested in determining whether a new drug has an effect on creatine kinase (CK), which is often assayed in blood tests as an indicator of myocardial infarction. A random selection of 20 patients from a pool of possible subjects is selected, and each subject is given the medication. The subjects’ CK levels are observed initially, after 3 weeks, and again after 6 weeks. The purpose is to study the CK levels over time. Here is a summary of the findings:

Time (weeks)	Mean CK Level (U/L)	Standard Deviation (U/L)
0	121	20.37
3	106	16.09
6	100	10.21

In this example, what should we notice?

1. The data show very strong evidence of a violation of the assumption that the three populations have the same standard deviation.
2. ANOVA cannot be used on these data because the sample sizes are much too small.
3. The assumption that the data are independent for the three time points is unreasonable because the same subjects were observed each time.
4. There is no reason not to use ANOVA in this situation.

2. An experiment examined the psychophysiological effect of THC, the active ingredient in marijuana. The study recruited a sample of 18 young adults who were habitual marijuana smokers. Subjects came to the lab three times, each time completing the same questionnaire, but each time smoking a different marijuana cigarette: one with 3.9% THC, one with 1.8% THC, and one with no THC (a placebo). The order of the conditions was randomized in a double-blind design. Why can we not use a one-way ANOVA procedure here to test whether the mean “feeling of high” is the same for all three THC amounts?

1. The assumption of independent random samples is not met.
2. The assumption of Normality is not met.
3. The assumption of equal population standard deviations is not met.
4. The explanatory variable is categorical.

3. Researchers wish to examine the effectiveness of a new weight-loss pill. A total of 200 obese adults are randomly assigned to one of four conditions: weight-loss pill alone, weight-loss pill with a low-fat diet, placebo pill alone, or placebo pill with a low-fat diet. The weight loss after six months of treatment is recorded in pounds for each subject. To analyze these data, which inference procedure would you use?

1. ztest
2. ttest
3. ANOVA Ftest
4. Chi-square test

4. A study of the effects of sedating and nonsedating antihistamines on driving impairment was done in a driving simulator. Volunteers were randomly assigned to take either a sedating antihistamine, a nonsedating antihistamine, or a placebo. Their steering instability in the simulator was recorded on a quantitative scale. To analyze these data, which inference procedure would you use?

1. ztest
2. ttest
3. ANOVA Ftest
4. Chi-square test

5. Heliconiais a genus of tropical plant with different varieties often fertilized by distinct species of hummingbirds. Researchers measured the flower length (in millimeters) of independent random samples of three varieties of Heliconiato see if the three varieties differ significantly in flower length. To analyze these data, which inference procedure would you use?

1. ztest
2. ttest
3. ANOVA Ftest
4. Chi-square test

6. True/False: The F distributions are a family of distributions that take on only positive values and are skewed right.

1. True
2. False

7. True/False: Interchanging the degrees of freedom for the F distribution changes the distribution, so order of parameters is very important.

1. True
2. False

8. True/False: ANOVA is not too sensitive to violations of Normality if all samples have similar sizes and no sample is very small.

1. True
2. False

Answer 1

Based on the given data,

1.

The effect of the drug is studied on the same set of a random selected group of 20 patients by observing CK recorded - initially, after 3 weeks, and again after 6 weeks.Hence, a simple One way ANOVA would be inappropriate here.This violates the Independence of observations in the three groups compared.Hence, the correct option would be:

c. The assumption that the data are independent for the three time points is unreasonable because the same subjects were observed each time.

We find that this is a repeated sample data. Hence, for comparing the repeatedly measured observations across more than 2 groups (Here, period of time), Repeated Measures ANOVA would be the appropriate statistical test here.

2.

Here, the same sample of 18 young adults who were habitual marijuana smokers were made to smoke a different marijuana cigarette: one with 3.9% THC, one with 1.8% THC, and one with no THC each time the came to the lab. Hence, the violation here, would be:

The assumption of independent random samples is not met.

3.

In order to examine the effectiveness of a new weight-loss pill, a total of 200 obese adults randomly assigned to one of four conditions: weight-loss pill alone, weight-loss pill with a low-fat diet, placebo pill alone, or placebo pill with a low-fat diet are compared.

Let denote the mean weight loss after six months of treatment recorded in pounds for subjects assigned with weight-loss pill alone, weight-loss pill with a low-fat diet, placebo pill alone, or placebo pill with a low-fat diet respectively.

We have to test:

Vs    Not all means are equal.

Here, we are comparing the Means of a continuous dependent variable, here - 'Weight loss' across more than 2 independent groups, the appropriate statistical tool would be a One way ANOVA F test.

4.

Let denote the mean mean score on steering instability recorded for subjects assigned with sedating and nonsedating antihistamines respectively.

We have to test:

Vs   

Here, we are comparing the Means of a continuous dependent variable, here - 'Steering instability' score across 2 independent groups, the appropriate statistical tool would be:

A t test (if the population standard deviations are unknown) A Z test (if the population standard deviations are known)

5.

Let denote the mean flower length ofhe flower length (in millimeters) of three varieties of Heliconiato respectively.

We have to test:

Vs    Not all means are equal.

Here, we are comparing the Means of a continuous dependent variable, here - 'Flower length' across more than 2 independent groups, the appropriate statistical tool would be a One way ANOVA F test.

6.

True/False: The F distributions are a family of distributions that take on only positive values and are skewed right.

The F-distribution depends on the degrees of freedom and is usually defined as the ratio of variances of two populations normally distributed. If n1 and n2 are the degrees of freedom of the two populations and s1 and s2 and the sample standard deviation of the two groups under study:

Since the variances are the square of the deviations and hence cannot assume negative values, the value of the F-distribution is always positive, or zero. Its value lies between 0 and ∞. Also, the shape of the F-distribution depends on its parameters n1 and n2 degrees of freedom, hence, the F-distribution is positively skewed and with the increase in the degrees of freedom ν1 and ν2, its skewness decreases.

Hence, the given statement is true.

7. True/False: Interchanging the degrees of freedom for the F distribution changes the distribution, so order of parameters is very important.

By the reciprocal property of F given by

We find that the given statement is true.

8.

True/False: ANOVA is not too sensitive to violations of Normality if all samples have similar sizes and no sample is very small.

Anova is more sensitive to homogeneity of variance assumption than the assumption of Normality. Unequal sample sizes may contribute to violation of this assumption. As far as the assumption of Normality is concerned, if the samples are sufficiently large, it would, by cental limit theorm, be approximately normally deistributed.

Hence, we may conclude that the give statement is true.