Question

Sleep deprivation Does sleep deprivation linger for more than a day? Researchers designed a study using $71$volunteer subjects between the ages of $18$and $25$. All $21$participants took a computer-based visual discrimination test at the start of the study. Then the subjects were randomly assigned into two groups. The$11$subjects in one group, D, were deprived of sleep for an entire night in a laboratory setting. The$10$subjects in the other group, A, were allowed unrestricted sleep for the night. Both groups were allowed as much sleep as they wanted for the next two nights. On Day $4$, all the subjects took the same visual discrimination test on the computer. Researchers recorded the improvement in time (measured in milliseconds) from Day $1$to Day $4$on the test for each subject. We used Fathom software to randomly reassign the $21$subjects to the two groups$1000$times, assuming the treatment received doesn’t affect each individual’s time improvement on the test. The dot plot shows the approximate randomization distribution of ${\overline{x}}_A-{\overline{x}}_D$

(a) Explain why the researchers didn’t let the subjects choose whether to be in the sleep deprivation group or the unrestricted sleep group.

(b) In the actual experiment, $x_A-x_D=15.92$. This value is marked with a blue line in the figure. What conclusion would you draw? Justify your answer with appropriate evidence.

(c) Based on your conclusion in part (b), could you have made a Type I error or a Type II error? Justify your answer

Short answer

(a) To ensure that the two groups are as similar as feasible and to exclude the effect of any variables that were not examined, the individuals are randomly assigned to the two groups.

(b) The volunteers who were allowed to sleep had a higher mean than the participants who were not allowed to sleep.

(c) Based on our conclusion in part (b), WE have made a Type I error.

Step-by-step solution

Part (a) Step 1: Given information

Subjects in group D$=11$

Subjects in group A $=10$

Part (a) step 2: Explanation

To ensure that the two groups are as similar as feasible and to exclude the effect of any variables that were not examined, the individuals are randomly assigned to the two groups.

Part (b) Step 1: Given information

Subjects in group D$=11$

Subjects in group A$=10$

Part (b) Step 2: Explanation

We can see that there are nearly no dots above $15.92$and to its right on the dot plot.

This suggests that a sample mean difference of $15.92$is extremely implausible, and we conclude that the mean appears to be much greater for those who were allowed to sleep than for those who were not.

The volunteers who were allowed to sleep had a higher mean than the participants who were not allowed to sleep.

Part (c) Step 1: Given information

Number of subjects in group D$=11$

Number of subjects in group A$=10$

Part (c) Step 2: Explanation

Find the hypothesis

$H_0:{\mu }_1={\mu }_2$

$H_a:{\mu }_1>{\mu }_2$

In part(b), we concluded that the means were significantly different and thus the null hypothesis $H_0$was rejected.

Type 1 error: Reject$H_0$, when$H_0$is true

Type 2 error: Fail to reject $H_0$, when $H_0$is false

Since we rejected the null hypothesis $H_0$, we could have only made a Type I error.