An HIV test is not entirely accurate. The test developer claims the test produce

Question

An HIV test is not entirely accurate. The test developer claims the test produces less than 5% false positives and less than 1% false negatives. In order to investigate the developer’s claim, the test was administered to 1000 individuals known to have HIV and 10,000 individuals known not to have HIV. The results are below.

      Truly has HIV             Truly does not have HIV               

Positive    993                             591                              

Negative 7                                  9409

(a) Construct a 95% confidence interval for the proportion of false positives produced by the test.  

(b) Is there sufficient evidence to conclude, at the 5% level of significance, that the false positive rate is less than 5%? Explain.  

(c) Construct a 95% confidence interval for the proportion of false negatives produced by the test.  

(d) Is there sufficient evidence to conclude, at the 5% level of significance, that the false negative rate is less than 1%? Explain.  

(e) Which of the two error types, false negative or false positive, is more crucial to public safety? Explain.

(f) Suppose now that the false positive rate is truly about what the data suggest: 591/(993 + 591); and that the false negative rate is truly about what the data suggest: 7/(7 + 9409). Suppose also that we know that about 2% of the population has HIV.  

            a. If a random person from the population tests positive for HIV, what is the probability that person truly has HIV?

            b. If a random person from the population tests negative for HIV, what is the probability that person truly does not have HIV?

Explanation / Answer

a) total number of person = n = 1000+10000= 11000

p = sample proportion = 591/11000 = 0.053727

Z value for 95 % confidence interval is 1.96

confidence interval = p +- z sqrt ( p (1-p)/n)

= 0.053727 +- 1.96 sqrt [ 0.053727 (1-0.053727)/11000]

confidence interval

(

b) Ho : p <0.05)

H1 : p >= 0.05

level of significance =0.05

p = 0.053727

p0 = 0.05

Z = ( p-p0) / sqrt [ p0( 1-p0) /n ]

=( 0.053727 -0.05 ) / sqrt [ 0.05*0.95 / 11000 ]

=

1.793662

p value = 0.0367

p value < 0.05

so we reject the null hypothesis.

there is no evidence to conclude, at the 5% level of significance, that the false positive rate is less than 5%

c)

total number of person = n = 1000+10000= 11000

p = sample proportion = 9409/11000=0.855364

Z value for 95 % confidence interval is 1.96

confidence interval = p +- z sqrt ( p (1-p)/n)

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