Question

It is believed that the percentage of people who support candidate A is 56%. A sample of size 100 indicated that 40 people support candidate A. Is there evidence at an alpha level of 10% to conclude that the percentage of people who support candidate A has changed?

Answer 1

Given data:

Sample size $=100$

Sample proportion of people support candidate A:

$p=\frac{40}{100}=0.40$

Population proportion of people support Candidate A :

$P=\frac{56}{100}=0.56$

Check the condition: $nP>10$ and $n(1-P)>10.$

$nP=100*0.40=40>10$

$n(1-P)=100*0.60=60>10$

Null Hyposthesis: $H_0: P=0.56$

Alternative Hypothesis: $H_\alpha :P\neq 0.40$

The test statistics:

$z=\frac{p-P}{\sqrt{\frac{P(1-P)}{n}}}$

Plug in p=0.56, P=0.40 and n=100 in the above formula,

$z=\frac{0.56-0.40}{\sqrt{\frac{0.40(1-0.40)}{100}}}$

$z=\frac{0.56-0.40}{\sqrt{\frac{0.40(1-0.40)}{100}}}=3.2659$

The table value of z at 10% of significance level is 1.28

Table value is less than the calculated value.

Hence $H_0$ is rejected at 10% level.

Therefore there is the evidence at an alpha level of 10% to conclude that the percentage of people who support candidate A has changed.