Question

Since an instant replay system for tennis was introduced at a major​ tournament, men challenged

14371437

referee​ calls, with the result that

431431

of the calls were overturned. Women challenged

745745

referee​ calls, and

227227

of the calls were overturned. Use a

0.010.01

significance level to test the claim that men and women have equal success in challenging calls. Complete parts​ (a) through​ (c) below.

a. Test the claim using a hypothesis test.

Consider the first sample to be the sample of male tennis players who challenged referee calls and the second sample to be the sample of female tennis players who challenged referee calls. What are the null and alternative hypotheses for the hypothesis​ test?

A.

Upper H 0H0​:

p 1p1not equals≠p 2p2

Upper H 1H1​:

p 1p1equals=p 2p2

B.

Upper H 0H0​:

p 1p1equals=p 2p2

Upper H 1H1​:

p 1p1greater than>p 2p2

C.

Upper H 0H0​:

p 1p1less than or equals≤p 2p2

Upper H 1H1​:

p 1p1not equals≠p 2p2

D.

Upper H 0H0​:

p 1p1equals=p 2p2

Upper H 1H1​:

p 1p1less than<p 2p2

E.

Upper H 0H0​:

p 1p1equals=p 2p2

Upper H 1H1​:

p 1p1not equals≠p 2p2

F.

Upper H 0H0​:

p 1p1greater than or equals≥p 2p2

Upper H 1H1​:

p 1p1not equals≠p 2p2

Identify the test statistic.

zequals=negative . 23−.23

​(Round to two decimal places as​ needed.)

Identify the​ P-value.

​P-valueequals=. 818.818

​(Round to three decimal places as​ needed.)

What is the conclusion based on the hypothesis​ test?

The​ P-value is

greater than

the significance level of

alphaαequals=0.010.01​,

so

fail to reject

the null hypothesis. There

is not sufficient

evidence to warrant rejection of the claim that women and men have equal success in challenging calls.

b. Test the claim by constructing an appropriate confidence interval.

The

9999​%

confidence interval is

nothingless than<left parenthesis p 1 minus p 2 right parenthesisp1−p2less than<nothing.

​(Round to three decimal places as​ needed.)

What is the conclusion based on the confidence​ interval?

Because the confidence interval limits

▼

do not include

include

​0, there

▼

does

does not

appear to be a significant difference between the two proportions. There

▼

is not sufficient

is sufficient

evidence to warrant rejection of the claim that men and women have equal success in challenging calls.

c. Based on the​ results, does it appear that men and women may have equal success in challenging​ calls?

A.

The confidence interval suggests that there is a significant difference between the success of men and women in challenging calls. It is reasonable to speculate that women have more success.

B.

The confidence interval suggests that there is no significant difference between the success of men and women in challenging calls.

C.

The confidence interval suggests that there is a significant difference between the success of men and women in challenging calls. It is reasonable to speculate that men have more success.

D.

There is not enough information to reach a conclusion.

Answer 1

a)

Ho:   p1 = p2
Ha:   p1 ╪ p2  

sample #1   ----->              
first sample size,     n1=   1437          
number of successes, sample 1 =     x1=   431          
proportion success of sample 1 , p̂1=   x1/n1=   0.2999          
                  
sample #2   ----->              
second sample size,     n2 =    745          
number of successes, sample 2 =     x2 =    227          
proportion success of sample 1 , p̂ 2=   x2/n2 =    0.3047          
                  
difference in sample proportions, p̂1 - p̂2 =     0.2999   -   0.3047   =   -0.0048
                  
pooled proportion , p =   (x1+x2)/(n1+n2)=   0.3016          
                  
std error ,SE =    =SQRT(p*(1-p)*(1/n1+ 1/n2)=   0.0207          
Z-statistic = (p̂1 - p̂2)/SE = (   -0.005   /   0.0207   ) =   -0.23

-------

​P-value = 0.818

---------

The​ P-value is

greater than

the significance level of

alpha=0.01

so

fail to reject

the null hypothesis. There

is not sufficient

evidence to warrant rejection of the claim that women and men have equal success in challenging calls.

===============

b)

level of significance, α =   0.01              
Z critical value =   Z α/2 =    2.576   [excel function: =normsinv(α/2)      
                  
Std error , SE =    SQRT(p̂1 * (1 - p̂1)/n1 + p̂2 * (1-p̂2)/n2) =     0.0207          
margin of error , E = Z*SE =    2.576   *   0.0207   =   0.0534
                  
confidence interval is                   
lower limit = (p̂1 - p̂2) - E =    -0.005   -   0.0534   =   -0.0582
upper limit = (p̂1 - p̂2) + E =    -0.005   +   0.0534   =   0.0487
                  
so, confidence interval is (   -0.058   < p1 - p2 <   0.049   )  

Because the confidence interval limits

▼

include

​0, there

▼

does not

appear to be a significant difference between the two proportions. There

▼

is not sufficient

evidence to warrant rejection of the claim that men and women have equal success in challenging calls.

==============

c)The confidence interval suggests that there is no significant difference between the success of men and women in challenging calls.