Respuesta :
Answer:
Step-by-step explanation:
Step(i):-
Given first random sample size n₁ = 500
Given Roper survey reported that 65 out of 500 women ages 18-29 said that they had the most say when purchasing a computer.
First sample proportion
[tex]p^{-} _{1} = \frac{65}{500} = 0.13[/tex]
Given second sample size n₂ = 700
Given a sample of 700 men (unrelated to the women) ages 18-29 found that 133 men said that they had the most say when purchasing a computer.
second sample proportion
[tex]p^{-} _{2} = \frac{133}{700} = 0.19[/tex]
Level of significance = α = 0.05
critical value = 1.96
Step(ii):-
Null hypothesis : H₀: There is no significance difference between these proportions
Alternative Hypothesis :H₁: There is significance difference between these proportions
Test statistic
[tex]Z = \frac{p_{1} ^{-}-p^{-} _{2} }{\sqrt{PQ(\frac{1}{n_{1} } +\frac{1}{n_{2} } )} }[/tex]
where
[tex]P = \frac{n_{1} p^{-} _{1}+n_{2} p^{-} _{2} }{n_{1}+ n_{2} } = \frac{500 X 0.13+700 X0.19 }{500 + 700 } = 0.165[/tex]
Q = 1 - P = 1 - 0.165 = 0.835
[tex]Z = \frac{0.13-0.19 }{\sqrt{0.165 X0.835(\frac{1}{500 } +\frac{1}{700 } )} }[/tex]
Z = -2.76
|Z| = |-2.76| = 2.76 > 1.96 at 0.05 level of significance
Null hypothesis is rejected at 0.05 level of significance
Alternative hypothesis is accepted at 0.05 level of significance
Conclusion:-
There is there is a difference between these proportions at α = 0.05
The test statistics value will be "Z = - 2.76".
According to the question,
Women,
- [tex]\hat p_1 = \frac{65}{500}[/tex]
[tex]= 0.13[/tex]
Men,
- [tex]\hat p_2=\frac{300}{700}[/tex]
[tex]= 0.19[/tex]
Now,
The pooled estimate of the proportion will be:
→ [tex]\hat p=\frac{x_1+x_2}{n_1+n_2}[/tex]
By substituting the values,
[tex]= \frac{65+133}{500+700}[/tex]
[tex]= 0.165[/tex]
then,
→ [tex]\hat q = 1- \hat p[/tex]
[tex]= 1- 0.165[/tex]
[tex]= 0.835[/tex]
hence,
The test statistics:
→ [tex]Z = \frac{\hat p_1- \hat p_2}{\sqrt{\hat p\times \hat q\times (\frac{1}{n_1} +\frac{1}{n_2} )} }[/tex]
By putting the values, we get
[tex]= \frac{0.13-0.19}{\sqrt{0.165\times 0.835\times (\frac{1}{500} +\frac{1}{700} )} }[/tex]
[tex]= -2.76[/tex]
Thus the above response is right.
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