Answer: a) z=1.43
b) 0.1528
Step-by-step explanation:
The given set of hypothesis :
[tex]H_0:p=0.45[/tex]
[tex]H_a:p\neq0.45[/tex]
Since the alternative hypothesis [tex]H_a[/tex] is two-tailed , so we perform two-tailed test.
Also, it is given that : A random sample of n=76 Americans found 28 with brown eyes.
Sample proportion: [tex]\hat{p}=\dfrac{28}{76}=0.3684[/tex]
a) The z-statistic would be :-
[tex]z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex]
[tex]\Rightarrow\ z=\dfrac{0.3684-0.45}{\sqrt{\dfrac{0.45(1-0.45)}{76}}}=-1.4299121397\approx-1.43[/tex]
b) P-value for two-tailed test = 2P(Z>|z|)= 2P(z>|-1.43|)
=2P(z>1.43)
=2(1-P(z≤1.43)
=2-2P(z≤1.43)
= 2-2(0.9236)[Using standard z-table]
= 2-1.8472=0.1528
Hence, the P-value of the test= 0.1528
