Question

Before the last presidential debates, 50% of registered votes indicated they were planning to vote for the incumbent president. In a recent poll of 1200 registered voters after the debates, 636 indicated they are planning to vote for the incumbent president. Has there been a significant increase in the proportion of registered voters who are planning to vote for the incumbent president?
a. State the null and alternative hypotheses to be tested.


b. Compute the test statistic.


c. The null hypothesis is to be tested using α = .05. Compute the p-value.


d. What do you conclude?

Answer 1

Answer:

a) H₀ : p = 0.5

$H_{a} : p > 0.5$

b) z = 2.0785

c) p - value = 0.0188

d) Our conclusion is that there has been a significant increase in the proportion of registered voters who are planning to vote for the incumbent president

Explanation:

Probability of those planning to vote for the incumbent president before the debate, P₀ = 50/100 = 0.5

a) The null and the alternative hypothesis

Let the null hypothesis be represented by $H_{0}$

and the alternative hypothesis be represented by $H_{a}$

H₀ : p = 0.5

$H_{a} : p > 0.5$

b) compute the test statistics

sample size, n = 1200

number of voters planning to vote for the incumbent president, x = 636

Probability of those planning to vote for the incumbent president in the selected sample:

$\bar{p} = x/n\\\bar{p} = 636/1200\\\bar{p} = 0.53$

Formula for the test statistic:

$z = \frac{\bar{p} - p_{0} }{\sqrt{\frac{p_{0}(1-p_{0} ) }{n} } }$

$z = \frac{0.53 - 0.5 }{\sqrt{\frac{0.5(1-0.5 ) }{1200} } }$

z = 2.0785

c)

For $\alpha = 0.05$, the critical value as checked from the normal table will be:

$z_{\alpha} = 1.645$

The P value for z > 2.0785

$P-value = P(z > 2.0785) \\P - value = \phi (-2.0785)\\P - value = 0.0188$

d) What do you conclude?

since $( z = 2.0785) > (z_{\alpha} = 1.645)$, we will reject the null hypothesis.

Our conclusion is that there has been a significant increase in the proportion of registered voters who are planning to vote for the incumbent president