Question

5. A study by a medical researcher of HIV infection rates among injection drug users reported that the rate of new infections steadily dropped over the last couple of decades. The latest report was that 2% of IV drug users are HIV positive. A drug counselor believes that the percentage of IV drug users in his city who are HIV positive is more than 2%. The counselor surveys a simple random sample of 415 drug users in the city and finds that 8 are HIV positive. Does this evidence support the counselor’s belief? Use a 0.01 level of significance.

Answer 1

We want to test the a drug counselor claim that the percentage of IV drug users in his city who are HIV positive is more than 2%.

To test this claim the counselor surveys a simple random sample of 415 drug users in the city and finds that 8 are HIV positive.

n= sample size= 415

X= number of HIV positive=8

p= proportion of IV drug users are HIV positive =x/n=8/415=0.01928

P = population proportion or percentage of  IV drug users are HIV positive

Hypothesis:

H_{​​​​​​0} :The percentage of IV drug users in his city who are HIV positive is 2%.

H₀: P=0.02

Against

H₁: The percentage of IV drug users in his city who are HIV positive is more than 2%

H₁: P > 0.02

Test statistic:

To test the above Hypothesis the test statistic is:

Follow N(0,1) Distribution.

Z=-0.1048

Critical value for right tailed at 0.01 level of Significance is:

= 2.326

Cal Z < tab

Therefore We fail to reject null hypothesis at 1% level of Significance.

Conclusion:

There is insufficient evidence to support the counselor’s belief that the percentage of IV drug users in his city who are HIV positive is more than 2%.