A researcher is interested in the proportion of children...

A researcher is interested in the proportion of children that prefer specific types of candy. It was originally believed that 34% of children prefer Reeses, 21% prefer Hersheys, 20% prefer Gummy Bears, and 25% SweetTarts. The researcher believes the proportions have changed, and thus the researcher takes a random sample of 500 children and finds that 36% of them prefer Reeses, 19% of them prefer Hersheys, 18% of them prefer Gummy Bears, and 27% of them prefer sweet tarts. At the 5% significance level, what conclusion can be reached? What is the rejection region for this test?

Transcript

00:01 So we are going to perform a hypothesis test concerning some pieces.

00:05 And we are testing the hypothesis that less than 45 % are orange.

00:10 So that would be p, population proportion, less than 0 .45, 45%.

00:15 And this is the alternative hypothesis.

00:19 The matching null hypothesis would be actually it's at least 45%.

00:24 I know which is which because the null hypothesis always has an equal sign of some kind, and the alternative doesn't.

00:31 So this must be the alternate hypothesis and matching up the null.

00:37 So that's the first two parts.

00:39 Now we're going to find the test statistic and the critical value, and we're going to compare them.

00:46 So all of this is based on the assumption that the null hypothesis is true.

00:50 We think p might be 0 .45, at least.

00:53 Now, the central limit theorem tells me that if i take every sample of a given size, here, our sample size, n is 400, take all of the sample proportions and plot them out, i get something approximately normal.

01:09 P hat, sample proportion, follows a normal distribution, its mean is p, its standard deviation is p, 1 minus p, over n.

01:16 If you're wondering where those come from, look at the mean and standard deviation of the binomial distribution, np, and root np on minus p.

01:24 We divide these by n, we get these parameters here.

01:27 In the same way that you can take the number of successes, which here's 166, divide by n to get p hat.

01:34 So 166 out of 400 is 0 .415.

01:42 So that is below 45 percent.

01:44 But maybe the population proportion is still 45 and we just happen to take a sample with fewer orange pieces than average.

01:53 That's possible.

01:55 And looking at this, we're going to work out if it's probable.

01:59 So the probability of getting a sample proportion exactly 0 .45? tiny.

02:04 Absolutely tiny.

02:05 But if i got something, say, here, it would be below 45%, but it would not be unlikely to happen if the null hypothesis is true...

Question

Please give Ace some feedback