A hospital director is told that 54%of the treated patients are insured. The dir

Question

A hospital director is told that 54%of the treated patients are insured. The director wants to test the claim that the percentage of insured patients is below the expected percentage. A sample of 450 patients found that 225 were insured. Answer the following questions, at the 0.10 level. Answer: Ho: & Ha: Calculate the value of the test statistic
A hospital director is told that 54%of the treated patients are insured. The director wants to test the claim that the percentage of insured patients is below the expected percentage. A sample of 450 patients found that 225 were insured. Answer the following questions, at the 0.10 level. Answer: Ho: & Ha: Calculate the value of the test statistic
A hospital director is told that 54%of the treated patients are insured. The director wants to test the claim that the percentage of insured patients is below the expected percentage. A sample of 450 patients found that 225 were insured. Answer the following questions, at the 0.10 level. Answer: Ho: & Ha: Calculate the value of the test statistic

Explanation / Answer

Given :--

sample size is N = 450, the number of favorable cases is X =225, and

Sample proportion is p¯= (X/N) = (225 / 450) = 0.5, and the significance level is =0.10

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p = 0.54

Ha: p < 0.54

This corresponds to a left-tailed test, for which a z-test for one population proportion needs to be used.

(2) Rejection Region

Based on the information provided, the significance level is =0.10, and the critical value for a left-tailed test is zc=1.28.

The rejection region for this left-tailed test is R={z:z<1.28}

(3) Test Statistics

The z-statistic is computed as follows:

Z=[ (p¯p0) / (p0 *(1-p0)/n) ] = [ (0.5-0.54) / (0.54*(1-0.54)/450) ] = -1.703

Since it is observed that z=1.703 < zc=1.28, it is then concluded that the null hypothesis is rejected.

P-value :- The p-value is p=0.0443, and since p=0.0443<0.10, it is concluded that the null hypothesis is rejected.

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