Question

A coin is tested for fairness. To do this, a test is conducted as follows. The coin is flipped until our result is heads. The flip number, F on which this event happens is recorded (i.e. which flip they got heads on). This is repeated 100 times producing 100 results, F₁,F₂,...F₁₀₀. Our calculations reveal that [F with a horizontal line above it. Chegg doesn't let me type that character] = 1.8 with a sample standard deviation of 0.4

a) If the coin is fair, then what should we get for our mean of F?

b) Calculate with 99% confidence about the mean of F.

c) Calculate an estimate of p=P[heads] based on the experiment we performed.

d) Test that p>0.5 using a hypothesis test at the α = 0.05 level. Give both the conclusion and the location of the critical region.

Answer 1

a)

If coin is fair then mean should be 2.

b)

Level of Significance ,    α =    0.01          
degree of freedom=   DF=n-1=   99          
't value='   tα/2=   2.6264   [Excel formula =t.inv(α/2,df) ]      
                  
Standard Error , SE = s/√n =   0.4000   / √   100   =   0.040000
margin of error , E=t*SE =   2.6264   *   0.04000   =   0.105056
                  
confidence interval is                   
Interval Lower Limit = x̅ - E =    1.80   -   0.105056   =   1.694944
Interval Upper Limit = x̅ + E =    1.80   -   0.105056   =   1.905056
99%   confidence interval is (   1.69   < µ <   1.91   )

c)

P(Heads) =1/ 1.8

=0.5556

d)

Ho :   p =    0.5                  
H1 :   p >   0.5       (Right tail test)          
                          
Level of Significance,   α =    0.05                  
Number of Items of Interest,   x =   55.55555556                  
Sample Size,   n =    100                  
                          
Sample Proportion ,    p̂ = x/n =    0.5556                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0500                  
Z Test Statistic = ( p̂-p)/SE = (   0.5556   -   0.5   ) /   0.0500   =   1.1111
                          
  
                          
p-Value   =   0.1333 [Excel function =NORMSDIST(-z)              
Decision:   p value>α ,do not reject null hypothesis                       
There is not enough evidence that probability is greater than 0.5

THANKS

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