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Dr Teresa Amabile conducted a study involving $47$college students, who were randomly assigned to two treatment groups. The $23$students in one group were given a list of statements about external reasons (E) for writing, such as public recognition, making money, or pleasing their parents. The $24$students in the other group were given a list of statements about internal reasons (I) for writing, such as expressing yourself and enjoying playing with words. Both groups were then instructed to write a poem about laughter. Each student鈥檚 poem was rated separately by $12$different poets using a creativity scale. The $12$poets鈥� ratings of each student鈥檚 poem were averaged to obtain an overall creativity score. We used Fathom software to randomly reassign the $47$subjects to the two groups $1000$times, assuming the treatment received doesn鈥檛 affect each individual鈥檚 average creativity rating. The dot plot shows the approximate randomization distribution of ${\overline{x}}_1-{\overline{x}}_E$

(a) Why did researchers randomly assign the subjects to the two treatment groups?

(b) In the actual experiment, ${\overline{x}}_1-{\overline{x}}_E=4.15$. 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.

Step by step solution

01

Part (a) Step 1: Given information

Number of students in group$1=23$

Number of students in grouplocalid="1650357231039" $2=24$

02

Part (a) Step 2: Explanation

To ensure that the two treatment 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 treatment groups.

03

Part (b) Step 1: Given information

Number of students in group $1=23$

Number of students in group role="math" localid="1650357187414" $2=24$

04

Part (b) Step 2: Explanation

We can see that there are nearly no dots above $4.15$and to its right on the dot plot.

This suggests that a sample mean difference of $4.15$is extremely exceptional, leading us to the conclusion that the mean looks to be much larger for individuals with internal causes than for those with external reasons.

The mean for students who are studying for internal reasons is greater than the mean for students who are studying for external reasons.

05

Part (c) Step 1 : Given information

Number of students in group $1=23$

Number of students in group$2=24$

06

Part (c) Step 2: Explanation

Find the hypothesis

$H_0:{渭}_1={渭}_2$

$H_a:{渭}_1>{渭}_2$

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: Failed 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.

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