Question

Suppose we have collected times, in minutes, that it takes volunteers to complete a set of pencil and paper mazes. Volunteers are randomly assigned to one of two groups. Group 1 watch a 5 minute video explaining good strategies for completing the mazes. Group 2 watch a 5 minute video of other people successfully completing the mazes, but with no explanation given.
Researchers are interested in testing against the null hypothesis that there is no difference in population mean times that it takes people to complete mazes after watching either type of video.
There are the summary statistics after collecting data. Note that the sample sizes differ, because some subjects did not follow instructions correctly and so they were not included in the analysis.
x1 = 11.98, x2 = 9.15
s1 = 8.69, s2 = 7.75
n1 = 43, n2 = 52
a. Is this an experimental or observational study?
Observational
Experimental
b. What are the predictor and response variables, and are they categorical or quantitative? (You must get all correct)
Video type is categorical
Finishing time is quantitative
Video type is the response variable
Video type is the predictor variable
Video type is quantitative
Finishing time is categorical
Finishing time is the predictor variable
Finishing time is the response variable
How should the null hypothesis be written?
H0: x1 - x2 = 0
H0: μ1 - μ2 = 5
H0: μ = 0
H0: x = 5
H0: x1 - x2 = 5
H0: μ1 - μ2 = 0
(for the following calculations, round to 2 decimal places)
c. Calculate the standard error of x1 - x2:
d. Calculate the approximate 95% CI for μ1 - μ2: to
e. The result of the hypothesis test is:
We do not have evidence that there is a difference in population means, because the null value is inside the 95% CI.
We do not have evidence that there is a difference in population means, because the null value is outside the 95% CI.
We have evidence that there is a difference in population means, because the null value is inside the 95% CI.
We have evidence that there is a difference in population means, because the null value is outside the 95% CI.
f. What kind of error is it *possible* that we have made?
1. It is possible that we made a Type I error, because this is when you reject a false null hypothesis.
2. It is possible that we made a Type I error, because this is when you fail to reject a false null hypothesis.
3. It is possible that we made a Type I error, because this is when you reject a true null hypothesis.
4. It is possible that we made a Type I error, because this is when you fail to reject a true null hypothesis.
5. It is possible that we made a Type II error, because this is when you reject a false null hypothesis.
6. It is possible that we made a Type II error, because this is when you fail to reject a false null hypothesis.
7. It is possible that we made a Type II error, because this is when you reject a true null hypothesis.
8. It is possible that we made a Type II error, because this is when you fail to reject a true null hypothesis.

Answer 1

Answer:

Step-by-step explanation:

Hello!

A group of volunteers is randomly assigned to two groups:

Group 1: watch a 5-minute video explaining good strategies for completing the mazes.

X[bar]₁ = 11.98, S₁ = 8.69, n₁ = 43

Group 2: watch a 5-minute video of other people successfully completing the mazes, but with no explanation given.

X[bar]₂ = 9.15, S₂ = 7.75, n₂ = 52

After watching the videos the volunteers were given a set of pencil and paper mazes to resolve and the time, in minutes, it took them to resolve the mazes was measured.

a)

An observational study is one where the investigator has no control or intervenes on it. He just defines the variable of interest and merely collects and documents the information. These types of studies are usually made as precursors to a more formal experimental study, to have an idea of what's to be expected from the population.

An experimental study or experiment is one where the investigator intervenes by defining the variable of interest and artificially manipulates the study factor. It is also one of its characteristics the randomization of cases or subjects in groups (two or more, depending on what is the hypothesis of study).

⇒ Considering these definitions, and the description of the experiment, how the volunteers were treated exactly the same except for the videos and that the assignment of the groups was random, you can classify it as an experimental study.

b.

The response variable is the one that was measured by the researchers.

X: Time it takes the volunteer to complete the paper mazes.

This variable is quantitative continuous.

The predictor variable is the variable suspected to modify the response variable:

Y: Type of video assigned to the volunteer. Categorized: "Video on good strategies to solve mazes" and "Video showing people solving mazes"

This variable is a qualitative categorical.

If you want to compare the times it takes the volunteers of both groups the best is to do so trough the population means, so the parameter of interest is: μ₁ - μ₂

The claim is that there is no difference between the times that it takes people to complete mazes after watching either video so the hypotheses are:

H₀:  μ₁ - μ₂=0

H₁:  μ₁ - μ₂≠0

c.

Assuming X[bar]₁≈N and X[bar]₂≈N (since both samples n₁ and n₂ are large enough you can approximate the distribution of the sample means using the central limit theorem)

(X[bar]₁-X[bar]₂)≈N(μ₁-μ₂;σ₁²/n₁+σ₂²/n₂)

The estimation of the variance σ₁²/n₁+σ₂²/n₂ is:

V(X)= $\frac{S_1^2}{n_1} +\frac{S_2^2}{n_2} = \frac{(8.69)^2}{43} +\frac{(7.75)^2}{52}= 2.91$

Standard error= √V(X)= √2.91= 1.706

d.

(X[bar]₁-X[bar]₂)±$Z_{1-\alpha /2}$*$\sqrt{\frac{S_1^2}{n_1} +\frac{S_2^2}{n_2}}$

$Z_{1-\alpha /2}= Z_{0.975}= 1.96$

(11.98-9.15)±1.96*1.706

[-0.51; 6.17]minutes

e.

The 95% confidence interval contains the zero, so using this CI and at a complementary significance level of 5%, the test is not significant, which means that there is no evidence to reject the null hypothesis.

Correct option: We do not have evidence that there is a difference in population means, because the null value is inside the 95% CI.

f.

You can make two types of errors when deciding over a hypothesis test:

Type I error: Reject the null hypothesis when the hypothesis is true.

Type II error: Fail to reject the null hypothesis when the hypothesis is false.

Since the null hypothesis wasn't rejected, there is a chance that a type II error was committed.

The correct option is:

6. It is possible that we made a Type II error because this is when you fail to reject a false null hypothesis.

I hope this helps!