Question

A poll of 1068 adult Americans reveals that 48% of the voters surveyed prefer the Democratic candidate for the presidency. At a 0.05 level of significance, test the claim that atleast half of all voters prefer the Democrat.

Answer 1

Z = -1.333

P-value = 0.09176

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.

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 = (\hat p - p)/((√(p(1-p) ))/(n))}`

`Z = (0.48 - 0.5)/((√(0.5(1-0.5) ))/(1068))}`

`Z = \frac{-0.02}{\sqrt{\frac{{0.5(0.5) }}{1068}}}`

`Z = \frac{-0.02}{\sqrt{\frac{{0.25}}{1068}}}`

`Z = (-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.