Question

A hospital director is told that 56% of the emergency room visitors are insured. The director wants to test the claim that the percentage of insured patients is below the expected percentage. A sample of 200 patients found that 100 were insured Make the decision to reject or fail to reject the null hypothesis at the 0.10 level. 2 A) Reject Null Hypothesis B) Fail to Reject Null Hypothesis

Answer 1

To Test :-

H0 :- P = 0.56

H1 :- P < 0.56

$p_{0} = 0.56$
$q_{0} = 1 - p_{0} = 0.44$
n = 200
P = X / n = 0.5

Test Statistic :-
$Z = ( P - P_{0}) / ( \sqrt{(P_{0} * q_{0})/n})$
$Z = ( 0.5 - 0.56 ) / ( \sqrt{( 0.56 * 0.44) /200})$
Z = -1.71

Test Criteria :-
Reject null hypothesis if $Z < -Z_{\alpha}$
$Z_{\alpha} = Z_{0.1} = 1.282$
$Z < -Z_{\alpha}$ = -1.71 < -1.282, hence we reject the null hypothesis
Conclusion :- We Reject H0

Decision based on P value
P value = P ( Z < -1.71 )
P value = 0.0436
Reject null hypothesis if P value < $\alpha = 0.10$  
Since P value = 0.0436 < 0.1, hence we reject the null hypothesis
Conclusion :- We Reject H0

There is sufficient evidence to support the claim that percentage of insured patients is below the expected percentage at 10% level of significance.