Question

Question 3: Hypothesis Testing about the population mean (μ)   
A researcher claims that the average weekly spending for lunch by commuting students in US colleges is usually higher in winter semester than the of average of $30 in the prior fall semester. To verify this claim a sample of 100 commuting students were randomly selected in spring semester and their average weekly spending was reported as $32.5. From historical experience the population standard deviation of weekly spending by commuting students nationally is known to be $12.6. Use this information to answer these questions.
Note: Standard error of  is          OR        

1. In the context of this problem, μ denotes ……………………..   

2. This is a case 1, 2 or 3 (circle one) hypothesis testing problem because ……………
     ………………… and ………………………….

3. Based on your answer in part 2,  can only be transformed into (choose one)
    Z = (    -  )  ( )       OR           t = (    -  )  ( S)   

4a. From the three exhibits below, which pair of hypotheses would you use to verify the researcher’s claim.                                                                                                                                        (2 points.)
                                          
Exhibit 1

H₀: µ ≤ 30  
H_a: µ < 30

Exhibit 2

H₀: µ ≤ 30        
H_a: µ > 30

Exhibit 3

      H₀: µ  30        
  H_a: µ < 30

b. From your choice in part ‘a’, which hypothesis (H₀ or H_a) reflects/represents the research question?
                                                                                                                                                
c. From your answer in part 3 above, the standard error of the sample mean is ..................

5a. At α = .05, the p-value (PV) is ............................................

b. At α = .05, state the decision rules for rejecting or not rejecting H₀.                        

c. At α = .05, what is your conclusion?

6. Repeat the hypothesis test using the critical value (CV) approach.

a. At α = .05, state the decision rules for rejecting or not rejecting H₀.

b. At α = .05, what is your conclusion?           

Answer 1

1. The population mean (μ) be  the average weekly spending for lunch by commuting students in US colleges.

2. The hypothesis is test is one-tailed Z-test because the population standard deviation known.

3.

4.

(a) Hypothesis: Vs  

(b) The alternative hypothesis Ha represents the research question.

(c) The standard erorr is ,

5.

(a) The test statistic is ,

The p-value is ,

p-value=

; From standard normal distribution table

(b) Decision rule : Reject Ho , if p-value <

Here , p-value=0.0239<=0.05

Therefore , reject Ho.

(c) Conclusion : At =0.05 , there is sufficinet evidence to  to support the claims that the average weekly spending for lunch by commuting students in US colleges is usually higher in winter semester than the of average of $30 in the prior fall semester.

6.

The critical value is ,

; From Z-table

(a) Decision rule : Reject Ho , if Z-stat>1.64

Here , Z-stat=1.98>1.64

Therefore , reject Ho.

(b) Conclusion : At =0.05 , there is sufficinet evidence to  to support the claims that the average weekly spending for lunch by commuting students in US colleges is usually higher in winter semester than the of average of $30 in the prior fall semester.