Question

12: A researcher is interested in whether salaries for middle school teachers were less than salaries for nurses in Arkansas. A statewide salary survey is conducted using random sampling.

The Data Analysis output for the various tests used when comparing two group means are shown below. The significance level was .05. Identify the proper t-test from the scenario and the following outputs, and then answer the following questions using the correct t test output. (3.5 pts.)

F-Test Two-Sample for Variances
	Teachers	Nurses
Mean	45946.07	53365.13
Variance	68256753	86820446
Observations	300	300
df	299	299
F	0.7862
P(F<=f) one-tail	0.0190
F Critical one-tail	0.8265

t-Test: Paired Two Sample for Means
	Variable 1	Variable 2
Mean	45946.07	53365.13
Variance	68256753	86820446
Observations	300	300
Pearson Correlation	0.191488
Hypothesized Mean Difference	0
df	299
t Stat	-11.4663
P(T<=t) one-tail	9.01E-26
t Critical one-tail	1.649966
P(T<=t) two-tail	1.8E-25
t Critical two-tail	1.96793

t-Test: Two-Sample Assuming Equal Variances
	Teachers	Nurses
Mean	45946.07	53365.13
Variance	68256753	86820446
Observations	300	300
Pooled Variance	77538599
Hypothesized Mean Difference	0
df	598
t Stat	-10.319
P(T<=t) one-tail	2.19E-23
t Critical one-tail	1.647406
P(T<=t) two-tail	4.37E-23
t Critical two-tail	1.963939

t-Test: Two-Sample Assuming Unequal Variances
	Teachers	Nurses
Mean	45946.07	53365.13
Variance	68256753	86820446
Observations	300	300
Hypothesized Mean Difference	0
df	590
t Stat	-10.319
P(T<=t) one-tail	2.3E-23
t Critical one-tail	1.64744
P(T<=t) two-tail	4.61E-23
t Critical two-tail	1.963993

a) What is the appropriate two sample test to perform – the paired t test, the t test assuming equal variances, or the t test assuming unequal variances – for this research project?

b) State the H₀ and H_a.

c) Identify the decision rule using the critical value of t (round to three decimal places).

d) Identify the decision rule using the p value method.

e) State the test statistic (t _calc).

f) Do you reject or not reject Ho? Explain your decision.

Answer 1

12.

(a)

Here, two different groups (school teachers and nurses) are used to collect data about salary. Further we do not know population standard deviation (or variance). So, we have to perform two sample t-test. Now whether with equal variances or unequal variances is to be determined using F test for equality of variances.

We have to test for null hypothesis

against the alternative hypothesis

We reject our null hypothesis if

Here, we observe that

So, we cannot reject our null hypothesis.

Thus,based on the given data we can conclude that there is no significant evidence that variances are unequal.

Hence, we have to perform t test assuming equal variances.

(b)

Suppose, random variables X and Y denote salaries of school teachers and nurses respectively.

We have to test for null hypothesis

against the alternative hypothesis

(c)

Our alternative hypothesis is less than type. So, this is an one-tailed (more specifically left tailed) test.

We reject our null hypothesis if

(d)

Level of significance

We reject our null hypothesis if

(e)

Test statistic is

(f)

Here, we observe that

• 
• 

So, we reject our null hypothesis (any one of the above is sufficient to draw calculation, in fact both always draw same conclusion).

Hence, based on the given data we can conclude that there is significant evidence that salaries for middle school teachers were less than salaries for nurses in Arkansas.