Question

4) Last year we went to Zanvoort Circuit Park in North Holland to retest how many laps women and men drivers could do in half an hour. Historical data available gave us a basis for standard deviations of 3.5 and 2.63 laps, respectively Use men pilots as  x̄1. Below are the summaries of our new findings:

(INCLUDE SKETCH) (ROUND 5 DECIMAL PLACES)

Laps/30 minutes	Mean	SD	n
Women Pilots	17.91	3.39	50
Men Pilots	19.39	2.52	50

a) Find a 95% confidence interval for the difference here. Explain the meaning of the results.

b) Now perform the appropriate hypothesis test, and explain your results.

Answer 1

a)

Level of Significance ,    α =    0.05          
z-critical value =    Z α/2 =    1.960   [excel function =normsinv(α/2) ]      
                  
std error , SE =    √(σ1²/n1+σ2²/n2) =    0.5974          
margin of error, E = Z*SE =    1.960   *   0.597   =   1.170822
                  
difference of means = x̅1 - x̅2 =    17.91   -   19.39   =   -1.480
confidence interval is                   
Interval Lower Limit= (x̅1 - x̅2) - E =    -1.480   -   1.171   =   -2.6508
Interval Upper Limit= (x̅1 - x̅2) + E =    -1.480   +   1.171   =   -0.3092

b)

Ho :   µ1 - µ2 =   0          
Ha :   µ1-µ2 <   0          
                  
Level of Significance ,    α =    0.05          
                  
sample #1   ------->              
mean of sample 1,    x̅1=   17.9100          
population std dev of sample 1,   σ1 =    3.39          
size of sample 1,    n1=   50          
                  
sample #2   --------->              
mean of sample 2,    x̅2=   19.3900          
population std dev of sample 2,   σ2 =    2.52          
size of sample 2,    n2=   50          
                  
difference in sample means = x̅1 - x̅2 =    17.91   -   19.39   =   -1.48
                  
std error , SE =    √(σ1²/n1+σ2²/n2) =    0.5974          
                  
Z-statistic = ((x̅1 - x̅2)-µd)/SE =    -1.48   /   0.5974   =   -2.478
                  
Z-critical value , Z* =        -1.6449   [excel function =NORMSINV(α)]      
                  
p-value =        0.0066   [excel function =NORMSDIST(z)]      
Desison:   p-value <α , Reject null hypothesis              
                  
Conclusion:   There is enough evidence to conclude that men drivers could do more laps in half an hour