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Problem 30 Hard Difficulty

The population proportion is $30 .$ What is the probability that a sample proportion will be within $\pm .04$ of the population proportion for each of the following sample sizes?
a. $n=100$
b. $n=100$
c. $n=500$
d. $n=500$
e. What is the advantage of a larger sample size?

Answer

a. 0.6156
b. 0.7808
c. 0.9488
d. 0.9942
e. The estimates are more reliable.

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Video Transcript

All right. So this question asks us about sampling distributions with a point estimate of P hat equal 2.3. And it wants us to compute the probabilities for different sample sizes. So remember that fur sample proportions, the mean of the pee hats, is equal to the population proportion. And then the standard error of the Piaz is equal to squarely p times one minus p all over the sample size. So then, from there, it's asking, What's the probability that we're within 0.4 of the population mean, which is the same thing is asking what is the probability that were between 0.26 and 0.34? So now that's the same is asking normal CDF are lower bound is 0.26 Our upper bound is 0.34 our mean this 0.30 and our standard deviation is what changes, which is in this case, we're always having the same population proportion. It's 0.3 times 0.7 oven all over end. So this is the general form of our answer, where the only thing we're changing is that end in the denominator. So now let's go ahead and compete thes so the first question as an equal to 100. So based on this, that should be normal. CDF lower bound of 0.26 upper bound of 0.34 The mean of 0.3 and a standard deviation, or standard air rather of 0.3 times 0.7 over 100 all under a square root. And this works out to be point six 173 Now it wants us to do it for 200 so very similar. Normal CDF are lower bound. This 0.26 are upper bound is 0.34 Army and his 0.30 and our standard error this time is 0.3 times 0.7 over 200 on a square root, which this time works out to be 0.7830 Then it wants us to blow up the sample size to 500 So normal CDF of 0.26 0.34 0.30 than 0.3 times 0.7 while divided by 500 which this time works out to 0.9490 and then finally it wants us to do it for 1000. So finally it's normal. CDF same lower bound, same upper bound, same man. But this time our sample size is 1000 which gives us an answer off 0.9942 So what can we gather from This is asking What's the benefit of a larger sample size? So as we increase the sample size, our probability keeps getting closer and closer to one. So keep saying that were, if we take a sample, we're gonna be very, very close to the mean, most likely the main of the piazza at us. So a larger sample size we'll give a more reliable. We'll give a more, a more reliable estimate because of the lower variability, more variability, which is quantified by the standard air of the pee hats. Because, as we can see from our formula, it's related to the sample size. And so as we get more and more samples are standard air, a ghost