Question

Recall the survey you took during the first week of class. One of the questions was “do you agree
that it is inappropriate to speak on a cellphone while at a restaurant?” Of the 1913 females that
responded to the survey, 1729 agreed with this statement. Of the 1276 males that responded to this
survey, 1111 agreed with this statement. Test to see if there is any difference between males and
females with respect to how they feel about this issue. Use a significance level of .05.
(a) State the appropriate null and alternative hypotheses.

(b) Calculate the test statistic and report the p-value.

(c) State your conclusion in context of the problem.

(d) Based only on the results of the hypothesis test, would you expect a 95% confidence interval to
include 0? Explain.

(e) Calculate and interpret a 95% confidence interval for the difference between males and females.

2. In a recent baseball World Series, the Houston Astros were ordered to keep the roof of their stadium
open. The Houston team claimed that this would make them lose a home-field advantage, because
the noise from fans would be less effective. During the regular season, Houston won 36 of 53 games
played with the roof closed, and they won 15 of 26 games played with the roof open. Treat these
results as a simple random sample of games. Use a significance level of 0.05 to test the claim that
the proportion of wins at home is higher with a closed roof than with an open roof. Does the closed
roof appear to be an advantage?
(a) State the null and alternative hypotheses to determine whether the closed roof is an advantage.

(b) The test statistic for the above test was z = 0.89 which gives us a p-value of 0.1858. State a
conclusion in context of the problem.

(c) Based on the results of the hypothesis test, would you expect a confidence interval to include
0. Why or why not?

3. In a study of the effects of marijuana use during pregnancy, measurements on babies of mothers who
used marijuana during pregnancy were compared to measurements on babies of mothers who did
not. A 95% confidence interval for the difference in mean head circumference μ1 − μ2 (μ1 is mean
for mothers who don’t use marijuana and μ2 is the mean for mothers who use marijuana) was 0.61
to 1.19 cm.
(a) Interpret this confidence interval in context of the problem.

(b) Based on the confidence interval, what can we say about the p-value of the following hypothesis
test: H0 : μ1 = μ2 versus H1 : μ1 > μ2 at the 0.05 significance level? State your conclusion for
the test.

Answer 1

1)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 ╪   0         
                  
sample #1   ----->   experimental          
first sample size,     n1=   1913          
number of successes, sample 1 =     x1=   1729          
proportion success of sample 1 , p̂1=   x1/n1=   0.9038          
                  
sample #2   ----->   standard          
second sample size,     n2 =    1276          
number of successes, sample 2 =     x2 =    1111          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.871          
                  
difference in sample proportions, p̂1 - p̂2 =     0.9038   -   0.8707   =   0.0331
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.8906          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.0113          
Z-statistic = (p̂1 - p̂2)/SE = (   0.033   /   0.0113   ) =   2.9357
                  

p-value =        0.0033   [excel formula =2*NORMSDIST(z)]      
decision :    p-value<α,Reject null hypothesis               
                  
Conclusion:   There is enough evidence to SAY THAT THERE IS DIFFERENCE

.................

CI will not contain 0

...............

level of significance, α =   0.05              
Z critical value =   Z α/2 =    1.960   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.0116          
margin of error , E = Z*SE =    1.960   *   0.0116   =   0.0227
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    0.033   -   0.0227   =   0.0105
upper limit = (p̂1 - p̂2) + E =    0.033   +   0.0227   =   0.0558
                  
so, confidence interval is (   0.010   < p1 - p2 <   0.056   )  

..................................

2)

Ho:   p1 - p2 =   0          
Ha:   p1 - p2 >   0          
                  
sample #1   ----->   experimental          
first sample size,     n1=   53          
number of successes, sample 1 =     x1=   36          
proportion success of sample 1 , p̂1=   x1/n1=   0.6792          
                  
sample #2   ----->   standard          
second sample size,     n2 =    26          
number of successes, sample 2 =     x2 =    15          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.577          
                  
difference in sample proportions, p̂1 - p̂2 =     0.6792   -   0.5769   =   0.1023
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.6456          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.1145          
Z-statistic = (p̂1 - p̂2)/SE = (   0.102   /   0.1145   ) =   0.8934
                     
p-value =        0.1858   [excel function =NORMSDIST(-z)]      
decision :    p-value>α,Don't reject null hypothesis               
                  
Conclusion:   There is not enough evidence to say that proportion of wins at home is higher with a closed roof than with an open roof

..................

3)

CI = (0.61 , 1.19)

WE are 95% confident to say that population mean difference in mean head circumference of babies will lie in this interval

..............

0 is not in the interval, so reject Ho

mean of mothers who don’t use marijuana is greater than other