Question

Suppose our medical experts tell us it is safe to reopen the economy if the proportion of the state's population that is infected with a certain disease drops below 5%. A random sample of 800 residents is selected and 28 of them test positive for this disease. Do you believe it is safe to reopen the economy at a 5% significance level?

The alternative hypothesis for this test is :  

The distribution (choose from Z, T, X^2, or F) for this test is  

The P-value of this test is:

Based on this information, we (choose reject or fail to reject)   the null hypothesis

Based on this we (choose from do or do not)   believe it is safe to reopen the economy.

Answer 1

Step 1;

Ho: p ≥ 0.05

Ha: p < 0.05

Null hypothesis states that the proportion of the state's population that is infected with a certain disease is greater than or equal to 0.05

ALternative hypothesis states that the proportion of the state's population that is infected with a certain disease is less than 0.05

Step 2: Test statistics

As np≥ 10 and np((1-p) ≥ 10, we will assume normal distribution -  Z

n = 800

x= 28

z = -1.947

p value = P(z < -1.947) = 1- P( z < 1.947) = 1- 0.9744 = 0.0256

Step 3:

As the p value (0.0256) is less than , we reject the Null hypothesis.

Critical Value of Z (Left Tailed): -1.65

Also the z stat falls in the rejection region, hence we reject the Null hypothesis.

Hence we have sufficient evidence to believe that the proportion of the state's population that is infected with a certain disease is less than 0.05. SO we believe that it is safe to reopen the economy.