Question

A sample of 21 randomly selected student cars have ages with a mean of 7.5 years and a standard deviation of 3.4 years, while a sample of 32 randomly selected faculty cars have ages with a mean of 5.5 years and a standard deviation of 2.8 years. First, define student cars as Population 1 and faculty cars as Population 2. Thus we have

n1=21, n2=32, x¯1=7.5, x¯2=5.5, s1=3.4, s2=2.8,

(a) Based on the study data, is there significant evidence that student cars are older than faculty cars, on average? Please select the competing hypotheses for this research questions in scientific notation.

A. H0:μ1=μ2 versus H1:μ1≠μ2
B. H0:x¯1=x¯2 versus H1:x¯1≠x¯2
C. H0:μ1=μ2 versus H1:μ1>μ2
D. H0:x¯1=x¯2 versus H1:x¯1>x¯2

(b) Can we trust the results from a two-sample t procedure for this study?

A. No, we need to know the population standard deviations σ1 and σ2 to use the t procedures.

B. Yes, we can rely on the Central Limit Theorem because the total sample size is reasonable large n1+n2>40.

C. No, we don't know if the data come from Normally distributed populations.

D. Yes, both populations are Normal and the combined sample size n1+n2 is 5 or larger.

(c) What is the test statistic for this hypothesis test?

A. -2.5142

B. 2.5142

C. 1.960

D. 2.2424

(d) The degrees of freedom for the corresponding t distribution is 37. What is the p-value for our hypothesis test? (Please obtain the p-value using the t table and use our 'conservative' approach for rounding the df)

A. Greater than 0.10

B. Between 0.05 and 0.10

C. Between 0.01 and 0.05

D. Less than 0.01

(e) Now suppose that we obtained a p-value of 0.011 (pretend this is the p-value that you calculated). What should we conclude?

A. Reject the null hypothesis at significance level α=0.01
B. Do not reject the null hypothesis at significance level α=0.01
C. Do not reject the null hypothesis at significance level α=0.05
D. Reject the null hypothesis at significance level α=0.001

Answer 1

(a)

The appropriate hypothesis are,

C. H0:μ1=μ2 versus H1:μ1>μ2


(b)

B. Yes, we can rely on the Central Limit Theorem because the total sample size is reasonable large n1+n2>40.

(c) What is the test statistic for this hypothesis test?

Standard error of mean , SE =

= 0.8918947

Test statistic, t = (x¯1 - x¯2)/SE = (7.5 - 5.5) / 0.8918947 =  2.2424

D. 2.2424

(d)

The degrees of freedom for the corresponding t distribution is 37.

Critical value of t at df = 37 and 0.01 significance level is 2.43

Critical value of t at df = 37 and 0.05 significance level is 1.69

Since the test statistic (2.2424) lie between 1.69 and 2.43, p-value lie bwteen 0.01 and 0.05

C. Between 0.01 and 0.05

(e)

p-value = 0.011

We reject the null hypothesis when p-value is less than significance level α

Since p-value > 0.01, we fail to reject the null hypothesis at α=0.01

B. Do not reject the null hypothesis at significance level α=0.01