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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 $\hat p$ = 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:

$H_o : p \geq 0.50$

$H_1 : p < 0.50$

Using the Z test statistics which can be expressed by the formula:

$Z = \dfrac{\hat p - p}{\dfrac{\sqrt{p(1-p) }}{n}}}$

$Z = \dfrac{0.48 - 0.5}{\dfrac{\sqrt{0.5(1-0.5) }}{1068}}}$

$Z = \dfrac{-0.02}{\sqrt{\dfrac{{0.5(0.5) }}{1068}}}$

$Z = \dfrac{-0.02}{\sqrt{\dfrac{{0.25}}{1068}}}$

$Z = \dfrac{-0.02}{0.015}$

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 $H_o$ 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.

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3 years ago