Answer:
The p-value of the test is of 0.0287 < 0.05(standard significance level), which means that it can be concluded that less than half of HIV-positive smokers have used a nicotine patch.
Step-by-step explanation:
Test if less than half of HIV-positive smokers have used a nicotine patch:
At the null hypothesis, we test if the proportion is of at least half, that is:
At the alternative hypothesis, we test if the proportion is below 0.5, that is:
The test statistic is:
In which X is the sample mean, 
is the value tested at the null hypothesis, 
is the standard deviation and n is the size of the sample.
0.5 is tested at the null hypothesis:
This means that 
In a survey of 444 HIV-positive smokers, 202 reported that they had used a nicotine patch to try to quit smoking.
This means that 
Value of the test statistic:
P-value of the test and decision:
The p-value of the test is the probability of finding a sample proportion below 0.455, which is the p-value of z = -1.9.
Looking at the z-table, z = -1.9 has a p-value of 0.0287.
The p-value is of 0.0287 < 0.05(standard significance level), which means that it can be concluded that less than half of HIV-positive smokers have used a nicotine patch.
