Question

A recent poll of 809 randomly selected registered voters revealed that 426 plan to vote for your candidate in the coming election.

1. Is the observed percentage more than 50%?

1. Is the observed percentage significantly more than 50%? How do you know? Base your answer on a two-sided test.

Answer 1

A recent poll of 809 randomly selected registered voters revealed that 426 plan to vote for your candidate in the coming election.

Is the observed percentage more than 50%?

We are given

n = 809

x = 426

Observed proportion = x/n = 426/809 = 0.526576

Observed percentage = 52.66%

Yes, observed percentage is more than 50%.

Is the observed percentage significantly more than 50%? How do you know? Base your answer on a two-sided test.

Here, we have to use z test for population proportion.

H₀: p = 0.5 versus H_a: p ≠ 0.5

This is a two sided test.

Test statistic formula for this test is given as below:

Z = (p̂ - p)/sqrt(pq/n)

Where, p̂ = Sample proportion, p is population proportion, q = 1 - p, and n is sample size

x = number of items of interest = 426

n = sample size = 809

p̂ = x/n = 0.526576

p = 0.5

q = 1 - p = 0.5

Z = (p̂ - p)/sqrt(pq/n)

Z = (0.526576 – 0.5)/sqrt(0.5*0.5/809)

Z = 1.5118

P-value = 0.1306

(by using z-table)

P-value > α = 0.05

So, we do not reject the null hypothesis

There is insufficient evidence to conclude that the population percentage is significantly more than 50%.

The observed percentage is not more than 50%.