Answer to Question #107197 in Statistics and Probability for Tinashe Tsimba

Question

In a random sample of 500 observations, 290 successes and 210 failures were detected.
Calculate the point estimate of the population proportion of successes
Estimate with 95% confidence the population proportion of successes.
Use a hypothesis test with significance level 0:05 to test whether the population proportion of successes is less than 0.6.
State the null and alternative hypothesis.
State and calculate the appropriate test statistic. Determine the critical value of the test and state the rejection region.
State whether or not you reject the null hypothesis. Draw an appropriate conclusion.
Does the conclusion of part (b) also follow from (a)

Answer 1

1) Point estimate of population proportion of successes $\hat{p}=\frac{290}{500}=0.58.$

2) $n\hat{p}=500(0.58)\geq10,$ $n(1-\hat{p})=500(1-0.58)\geq10.$ So we can approximate the binomial distribution with a normal distribution.

$z_{value}=1.96\\ \hat{q}=1-\hat{p}=1-0.58=0.42.\\ EBP=z_{value}\sqrt{\frac{\hat{p}\hat{q}}{n}}=1.96\sqrt{\frac{(0.58)(0.42)}{500}}\approx 0.043.\\ (\hat{p}-EBP,\hat{p}+EBP)\\ (0.58-0.043,0.58+0.043)\\ (0.537,0.623)\text{ is our confidence interval}.$

3) $H_0: p=0.6, H_1:p<0.6$ (left-tailed test).

$pn=(0.6)500>5\\ (1-p)n=(0.4)500>5$

So we will use z-test.

$z_{stat}=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}=\frac{0.58-0.6}{\sqrt{\frac{(0.6)(0.4)}{500}}}\approx -0.9129.\\ \Phi(z_{cr})=\frac{1-2\alpha}{2}=0.45.\\ z_{cr}=-1.64.\\ (-\infty,-1.64)\text{ is our critical region}.\\ z_{stat}\text{ is not in the critical region. So we accept (do not reject) } H_0.$

We constructed the 95% confidence interval. It tells us that the population proportion falls in this interval with probability 95% and falls outside this interval with probability 5%. Then we used a hypothesis test to check whether the population proportion equals 0.6 or less than 0.6. We accepted $H_0: p=0.6$ at significance level 0.05; 0.6 is in our confidence interval. The result of the 3rd part does not directly follow from the result in the 2nd part.