Answer:
Z = -1.333
P-value = 0.09176
Decision Rule: Reject [tex]H_o[/tex] if ∝ is greater than the P-value
Conclusion: Since P-value is > the level of significance ∝, we fail to reject the null hypothesis, therefore there is insufficient evidence to conclude that at least half of all voters prefer the Democrat.
Step-by-step explanation:
Given that:
The sample size of the poll = 1068
The proportion of voters that preferred Democratic candidate is [tex]\hat p[/tex] = 0.48
To test the claim that at least half of all voters prefer the Democrat, i.e 1/2 = 0.5
The null hypothesis and the alternative hypothesis can be computed as:
[tex]H_o : p \geq 0.50[/tex]
[tex]H_1 : p < 0.50[/tex]
Using the Z test statistics which can be expressed by the formula:
[tex]Z = \dfrac{\hat p - p}{\dfrac{\sqrt{p(1-p) }}{n}}}[/tex]
[tex]Z = \dfrac{0.48 - 0.5}{\dfrac{\sqrt{0.5(1-0.5) }}{1068}}}[/tex]
[tex]Z = \dfrac{-0.02}{\sqrt{\dfrac{{0.5(0.5) }}{1068}}}[/tex]
[tex]Z = \dfrac{-0.02}{\sqrt{\dfrac{{0.25}}{1068}}}[/tex]
[tex]Z = \dfrac{-0.02}{0.015}[/tex]
Z = -1.333
P-value = P(Z< -1.33)
From z tables,
P-value = 0.09176
The level of significance ∝ = 0.05
Decision Rule: Reject [tex]H_o[/tex] if ∝ is greater than the P-value
Conclusion: Since P-value is > the level of significance ∝, we fail to reject the null hypothesis, therefore there is insufficient evidence to conclude that at least half of all voters prefer the Democrat.
