Question

Sleep deprivation Does sleep deprivation linger for more than a day? Researchers designed a study using 21 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 were deprived of sleep for an entire night in a laboratory setting. The 10 subjects in the other group 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.31 The table shows summary statistics for the two groups.

We used software to randomly reassign the 21 subjects to the two groups 100 times, assuming the treatment received doesn’t affect each individual’s time improvement on the test. A dot plot of the simulated difference (Unrestricted − Sleep-deprived) in meantime improvement is shown.

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

b. Etimate and interpret the P-value.

c. What conclusion would you make?

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

Short answer

Part a) This is necessary to do because else we would not be sure that the differences after the treatments are due to the treatment themselves or due to the difference among the groups that were already existing prior to the treatments.

Part b) There is $2\%$2 probability of getting similar sample results or more extreme results when there is no difference in the true mean time improvement for the people who were allowed unrestricted sleep and the time improvement for the people who were sleep-deprived.

Part c) There is convincing evidence that sleep deprivation lingers for more than a day

Part d) Type I error

Step-by-step solution

Part a) Step 1: Explanation

Prior to the treatments, the subjects were assigned to treatment groups at random so that the groups were as similar as possible.

This is necessary because otherwise, we would not be able to tell whether the differences observed after the treatments are due to the treatments themselves or to differences between the groups that existed prior to the treatments.

Part b) Step 1: Given information

$$\begin{gathered} {\overline{x}}_1=19.32 \\ {\overline{x}}_2=3.90 \end{gathered}$$

The given claim is that: Higher means creativity rating for the intrinsic group.

Part b) Step 2: Explanation

Either a null hypothesis or an alternative hypothesis is asserted.

$$\begin{gathered} H_0:{\mu }_1={\mu }_2 \\ {H}_0:{\mu }_1>{\mu }_2 \end{gathered}$$

Where,

${\mu }_1=$true mean time improvement for the people who were allowed unrestricted sleep.

${\mu }_2=$true mean time improvement for the people who were sleep-deprived

.As a result, the difference in sample means is,

$$\begin{gathered} {\overline{x}}_1-{\overline{x}}_2=19.32-3.90 \\ =15.92 \end{gathered}$$

When the null hypothesis is true, the $P$-value is the probability of getting the sample results or more extreme results.

we note that 2 of the 100 dots lies to the right of $15.92$then, we have,

$$\begin{gathered} P-value=\frac{2}{100} \\ =0.02 \\ 2\% \end{gathered}$$

When there is no difference in the true mean time improvement for people who were allowed unrestricted sleep and the time improvement for people who were sleep-deprived, we conclude that there is a $2\%$ chance of getting similar sample results or more extreme results.

Part c) Step 1: Explanation

From part (b) we have,

$P-value=2\%$

And we know that the null hypothesis is rejected if the $P$-value is less than or equal to the significance level.

$P<0.05\Rightarrow \text{Reject}{H}_0$

Therefore, we conclude that sleep deprivation lasts for more than a day based on the evidence.

Part d) Step 1: Explanation

Part (c) of our conclusion states,

"There is compelling evidence to support the claim that intrinsic rewards promote creativity more than extrinsic rewards."

When we reject a null hypothesis when the null hypothesis is true, we make a type I error. When we fail to reject the null hypothesis when the null hypothesis is false, we make a Type II error.

As a result, if we reject the null hypothesis, we have made a Type I error.