91Ó°ÊÓ

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.

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

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

Either a null hypothesis or an alternative hypothesis is asserted.

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

Where,

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

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

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

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

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.

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.

$P<0.05⇒\text{Reject}{H}_0$

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

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.