What is the probability that at least 42 people have satellite dishes?
In one Alberta town, 74% of the homes have satellite dishes. A pollster randomly sampled 50 homeoweners from the town. What is the probability that at least 42 of the people polled have satellite dishes?
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To answer this question, you must use the binomial distribution, which is for discrete probabilities, like that shown here. The distribution works in the following way: given a probability `p` that someone has a satellite dish and a sample group of `n` people, the probability that `k` people have a satellite dish is
`P = ((n), (k)) (1-p)^(n-k) p^k`
So, the probability that 42 people have a satellite dish out of 50 with a probability of 74% is the following:
`P = ((50),(42))(1-0.74)^(8)0.74^42 = 0.036`
Unfortunately, you have a different question, which asks you to see what is the probability of having at least42 people with a satellite dish, giving us the following formula:
`P = sum_(k=42)^50 ((50), (k)) (0.26)^(50-k)(0.74)^k`
You can evaluate this sum in your calculator with some paper, or you can use Excel or another computer program to do it much more quickly. You should get the following answer:
`P = 0.0684`
In other words, you have a 6.84% chance of finding at least 42 people that have satellite dishes.
For a quick common-sense check, you can see that 42 out of 50 people is 84% of the sample. So, finding that 84% of the sample has a satellite dish when only 74% of the whole popuation has one will be somewhat small!
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