sampling distributions: use density curve to find prob. that

I have several problems but I will send this one and hope that it will help me with the others.

An SRS of 400 Anmerican adults is asked, "What do you think is the most serious problem facing our schools?" Suppose in fact 30% of all adults would answer 'drugs' if asked this question. The proportion X of the sample who answer 'drugs' will vary in repeated sampling. In fact, we can assign probabilities to values of X using the normal density curve with mean 0.3 and standard deviation 0.023. Use this density curve to find the probabilities of the following events:

(a) At least half of the sample believes that drugs are the schools' most serious problem.

(b) Less than 25% of the sample believes that drugs are the most serious problem.

(c) The sample proportion is between 0.25 and 0.35.

So, for (a): mu = 0.3, sigma = 0.023, n = 400 sigma/square root of n approximately equals 0.00115

P(X>= 50%) ( .50 - .3)/0.0115

This does not give the correct answer of approximately 0.

I think (b) is P(X< 25%)

I think (c) is P(0.25 < X < .35)

but I am unsure how to procede for b and c.

Any help will be appreciated. I am actually trying to help someone else and have been thrust into this chapter without benefit of having been taught (except by myself) the preceding work. I am doing my best and have understood everything relatively well except for this section.

Re: sampling distributions

For part (a),

\(\displaystyle $z = \frac{x - \mu}{\sigma} \approx 8.7$\)

That's basically off the charts... Probability (Q = 1 - P(z)) of this occurring is pretty much 1 in infinity. In other words, on the distribution you have described, there is basically no area under the normal curve to the right of .5. That is, the tail area is infinitely small. When you get out more than about 4 or 5 standard deviations on either side of the mean in a normal distribution (which I think you have here in X), you're looking at very small probabilities.

Likewise, for (b) and (c), the question is asking how much area is under the normal curve (total area under the curve from -infinity to +infinity = 1), between those values. It relates to the distribution itself and not to the size of your sample. If you wanted to compute confidence intervals, you would then need to know the sample size, but all we need here is the mean and standard deviation. For (b), the value is only about two and a fraction standard deviations below the mean. Compute the z value and calculate (or look up in a table) the area under the normal curve at that value for z. For (c), you need both .25 and .35, and the area you want is what is in between those two values (subtract the table values).

Thanks for your reply.

I have another question - are you telling me that the 400 is not important for this problem?

I think the words "not important" are a little stronger than what I would actually tell you. If the sample size is too low, your confidence intervals are going to be all over the place. However, what the question (parts a, b, and c anyway) is asking deals with the area under a normal curve.

To answer the question "What is the probability that X will be greater than some value?" we don't need to know how big the sample is. That's because the mean of your SRS has nothing to do with the size of the SRS.

The confidence intervals depend on the size of the sample, but I think the question, as it was asked, was just asking about the probability that the mean of your SRS (X) would be above a certain value. To find that, all you need to do is get the z value, which depends only on mu and sigma, which you were given, and determine how much area is under the normal curve (since you were told that the distribution was normal) at or above that value of z.

For the other parts of the question, you are asked what is the probability that the mean of your SRS will be between two values or below a certain value. Again, you are not asked about the confidence intervals, just the probability of getting a certain result, somewhere on a normal curve. That can be found by determining the area under the normal curve to the left of one value and then subtracting the area to the left of the other value (for between) or simply to the left of the value (for less than).

Beware of the extraneous information, sometimes inserted into homework problems to lead you down the wrong path.

Thanks again. I'll be able to look at all the problems tomorrow night and expect them to be easier.