Rejecting the null hypothesis when true sample proportion equal

Solved Step by Step With Explanation-Test statistics

Questions

I. n = 40 and p = 0.11

II. n = 100 and p = 0.73

Question 29

Consider Ho: p = 0.70 versus H₁: p 0.70

(C) Cannot test since the sample size is small

(D) Reject Ho since observed proportion 0.68 is different from 0.70

Question 28

The Central Limit Theorem (CLT) states that for a large enough sample size (n), the sampling distribution of the sample proportion (p̂) will approximately follow a normal distribution, regardless of the shape of the population distribution. This means that we can use the normal distribution to make a test of hypothesis about the population proportion if the sample size is large enough.

In case III, n = 80 and p = 0.05, so np(1 - p) = 80(0.05)(0.95) ≈ 3.8. Since 3.8 < 10, the sample size in case III is not large enough to use the normal distribution.

In case IV, n = 50 and p = 0.14, so np(1 - p) = 50(0.14)(0.86) ≈ 6.12. Since 6.12 ≥ 10, the sample size in case IV is large enough to use the normal distribution.

The sample proportion (p̂) is equal to 0.68. This is different from the null hypothesis value of 0.70, but we need to determine whether this difference is statistically significant.

To do this, we can calculate the test statistic:

n is the sample size (600)

Plugging in the values, we get: