Question

Dan has heard that assisting victims of natural disasters increases feelings of moral purpose in college students. He completes a diary study where students are asked to record thoughts about their moral worth each day, and he counts the number of statements each person makes about moral purpose. The data meet all of his assumptions and he uses α = 0.05.
Non-Assisting
Disaster Assisting
1
4
2
3
3
5
2
4
3
3
4
5
Mean=2.5
Mean=4
S2=1.1
S2=0.8

IF THESE DATA WERE COLLECTED FROM TWO DIFFERENT GROUPS OF PEOPLE,
Provide the null and alternative hypotheses for the appropriate test in symbols and in words.  
Set up the criteria for making a decision. That is, find the critical value(s).  
Summarize the data into the appropriate test statistic
State your conclusion. Did you reject or fail to reject the null hypothesis? What does this indicate in terms of your original research question?
What are some confounding variables in this study?

IF THESE DATA WERE COLLECTED FROM THE SAME PEOPLE TESTED TWICE,
Provide the null and alternative hypotheses for the appropriate test in symbols and in words.  
Set up the criteria for making the decision (that is, find the rejection region)
Test the new hypothesis and state your conclusions clearly.
What are some potential confounding variables in this study?

Answer 1

Non-Assisting 1 2 3 2 3 4

Disaster Assisting 4 3 5 4 3 5

Mean=2.5
Mean=4
S1=1.1
S2=0.8

1.COLLECTED FROM TWO DIFFERENT GROUPS:

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho:μ1​ = μ2 or natural disasters does not increase feelings of moral purpose in college students.

Ha: μ1​ ≠ μ2​ or natural disasters increases feelings of moral purpose in college students

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=10. In fact, the degrees of freedom are computed as follows, assuming that the population variances are equal:

Hence, it is found that the critical value for this two-tailed test is tc=2.228, for α=0.05 and df=10.

The rejection region for this two-tailed test is ={t:∣t∣>2.228}.

(3) Test Statistics

The provided sample means are shown below:

​

Also, the provided sample standard deviations are:

s1​=1.1 , s2=0.8

and the sample sizes are n1=6 and n2=6

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that ∣t∣=2.701>tc​=2.228, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0223, and since p=0.0223<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is different than μ2​, at the 0.05 significance level.

since, we rejected null hypothesis, hence, natural disasters increases feelings of moral purpose in college students

2.COLLECTED FROM TWO SAME GROUPS:

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μD​ = 0 or natural disasters does not increase feelings of moral purpose in college students.

Ha: μD​ ≠ 0 or  natural disasters increases feelings of moral purpose in college students

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and the degrees of freedom are df=5

Hence, it is found that the critical value for this two-tailed test istc​=2.571, for α=0.05and df=5

The rejection region for this two-tailed test is R={t:∣t∣>2.571}.

(3) Test Statistics

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	1	4	-3
	2	3	-1
	3	5	-2
	2	4	-2
	3	3	0
	4	5	-1
Average	2.5	4	-1.5
St. Dev.	1.049	0.894	1.049
n	6	6	6

From the sample data, it is found that the corresponding sample means are:

​,

Also, the provided sample standard deviations are:

s1=1.1, s2=0.8

and the sample size is n = 6. For the score differences we have

sD​=1.049

(3) Test Statistics

The t-statistic is computed as shown in the following formula:

(4) Decision about the null hypothesis

Since it is observed that ∣t∣=3.503>tc​=2.571, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0172, and since p=0.0172<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is different than μ2​, at the 0.05 significance level.

since, we rejected null hypothesis, hence, natural disasters increases feelings of moral purpose in college students.

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