From a list of all passengers onboard at the time of the col

From a list of all passengers onboard at the time of the collision, groups were formed with ten names in each group. Then three of these groups were were selected using simple random sampling without replacement. The thirty passengers from the three selected groups are in the table below. Cells with an * indicate that this information is not known.

LASTNAME FIRSTNAME GENDER AGE CLASS BOARDED STATUS
Bucknell Emma Female 59 First Cherbourg Survived
Carter William Male 36 First Southampton Survived
Crosby Edward Male 70 First Southampton Died
Daniel Robert Male 27 First Southampton Survived
Douglas Mary Female 27 First Cherbourg Survived
Guggenheim Benjamin Male 46 First Cherbourg Died
Harris Henry Male 45 First Southampton Died
Pears Edith Female 22 First Southampton Survived
Smith Lucien Male 24 First Cherbourg Died
Stephenson Martha Female 52 First Cherbourg Survived
Cameron Clear * 35 Second Southampton Survived
Corbett Irene Female 30 Second Southampton Died
Hocking Samuel Male 36 Second Southampton Died
Knight Robert Male 39 Second Belfast Died
Norman Robert Male 28 Second Southampton Died
Padron-Manent Julian Male 26 * Cherbourg Survived
Toomey Ellen Female 48 Second Southampton Survived
Trout Jessie * 26 Second Southampton Survived
West Constance Female 4 Second Southampton Survived
Williams Charles Male 23 Second Southampton Survived
Asplund Johan Male 23 Third Southampton Survived
Bradley Bridget Female 22 Third Queenstown Survived
Daly Margaret Female 30 Third Queenstown Survived
Devaney Margaret Female 19 Third Queenstown Survived
Dowdell Elizabeth Female 31 * Southampton Survived
Jirjis Shaniini * 22 Third Cherbourg Survived
Madsen Fridtjof Male 24 Third Southampton Survived
Strandberg Ida Female 22 Third Southampton Died
Torfa Assad Male 20 Third Cherbourg Died
Touma Gerios Male 8 Third Cherbourg Survived

1.

At a 5% significance level, is there evidence that at most half of the passengers survived?
Hint: Please show all steps, including both the CV and p-value, for full credit.

Solution

As we see, 20 out of this sample of 30 has survived.

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.5
Ha:   p   <   0.5

As we see, the hypothesized po =   0.5      

Getting the point estimate of p, p^,          
          
p^ = x / n =    0.666666667      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.091287093      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    1.825741858

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Here, for a left tailed 0.05 level test,

zcrit = -1.645 [answer]

**********************************************      
          
As this is a    1   tailed test, then, getting the left tailed p value of our z,  
          
p =    0.966055423 [answer]

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significance level =    0.05      

As p > 0.05, then we FAIL TO REJECT THE NULL HYPOTHESIS.

There no sufficient evidence to say that at most half of the passengers survived. [CONCLUSION]

[Actually from the very start, we didn\'t have to compute. 20/30 survived. Thus, this cannot be evidence that at most half (15) has survived.]