100 randomly selected items were tested. It was found that 50 of the items tested positive.
Test the hypothesis that exactly 40% of the items tested positive at α = 0.05
State the null and alternative hypothesis:
H0: p = 0.4HA: p ≠ 0.4
Compute
| p^ = | x |
| n |
| p^ = | 50 |
| 100 |
p^ = 0.5
Calculate our test statistic z:
| z = | p^ - p |
| √p(1 - p)/n |
| z = | 0.5 - 0.4 |
| √0.4(1 - 0.4)/100 |
| z = | 0.1 |
| √0.4(0.6)/100 |
| z = | 0.1 |
| √0.0024 |
| z = | 0.1 |
| 0.048989794855664 |
z = 2.0412414523193
Checking our table of z-scores for α = 0.05%, we get:
Z = 1.6449Our rejection region is Z > 1.6449