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Question

according to research study, 66% of preschool children living in povery have been exposed to cigarette smoke at home. In comparison, 44% of preschool children not in poverty have been exposed to cigarette smoke at home. if 200 preschool children living in poverty are selected at random, what is the probability that at least 125 have been exposed to cigarette smoke? what about not living in poverty?

Answer 1

That's a tricky question, so read carefully. You said "200 preschool children living in poverty are selected...", then you said "what about not living in poverty." That's two separate samples of 200 preschool students - 200 in poverty & 200 not in poverty.

 

I admit that I would prefer more information, and that I'm not particularly confident about my solution without further information. Nonetheless, here's my thinking & approach.

 

Preschool students living in poverty: 66% = 66/100 (66 out of every 100...)

The sample size is 2 times 100, so we should expect 2 times 66 to have been exposed to cigarette smoke at home - that's 132 students.

 

Preschool students not living in poverty: 44% = 44/100 (44 out of every 100...)

Again, a sample size of 2 times 100, so we expect 2 times 44 to have been exposed to cigarette smoke at home - that's 88 students.

 

These are rough expectations only, as the only sample so far comes from a single survey. We are speculating about the probability of a certain number of students in each case. It is definitely an issue that we are using results from a single study. Nonetheless, we can use these numbers to estimate a probability.

 

Since probability means we divide the number of successful outcomes by the total number of possibilities, we can determine how many successful outcomes are possible in each of these cases:

 

There are 76 numbers that are at least 125 & at most 200. Any of those 76 numbers would be successful outcomes in either case. The probability of getting one of those numbers is 76/200 - or 0.38.

 

But more subjects are expected to meet the criteria in the poverty group, so the probability will be higher for the poverty students. At first it might appear that 100% is a reasonable response for this group (the expected number of 132 is higher than the 125 threshold), but random sampling is fraught with all kinds of issues that affect your results. Frankly, some samples would yield more than 132 & other samples would yield less than 125. However, in the long run, we believe about 2/3 of the subjects would have been exposed to cigarette smoke at home.

 

For this reason, I would multiply 0.38 by 0.66 to get a probability of 0.2508 - almost exactly 25%. It may seem low, but it is the same as 1/4 - 1 out of every 4 samples will meet the criteria - not really low under the circumstances.

 

Now, apply this to the non-poverty case. Multiply 0.38 by 0.44. The 16.72% probability is a tiny bit over 1/6 - meaning that about 1 out of every 6 samples would meet the criteria. This reflects that, while it is possible for a sample to meet the 132 threshold, it will seldom occur.

 

I hope this helps!