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A study by Consumer Reports showed that 64$\%$ of supermarket shoppers believe supermarket brands to be as good as national name brands. To investigate whether this result applies to its own product, the manufacturer of a national name-brand ketchup asked a sample of shoppers whether they believed that supermarket ketchup was as good as the national brand ketchup.a. Formulate the hypotheses that could be used to determine whether the percentage of supermarket shoppers who believe that the supermarket ketchup was as good as the national brand ketchup differed from 64$\% .$b. If a sample of 100 shoppers showed 52 stating that the supermarket brand was as good as the national brand, what is the $p$ -value?c. At $\alpha=.05,$ what is your conclusion?d. Should the national brand ketchup manufacturer be pleased with this conclusion? Explain.

a. $H_{0} : p=0.64, H_{a} : p \neq 0.64$b. $P=0.0124$c. Fail to reject the null hypothesis $H_{0}$d. $\mathrm{No}$

Intro Stats / AP Statistics

Chapter 9

Hypothesis Tests

Confidence Intervals

Hypothesis Testing with One Sample

Hypothesis Testing with Two Samples

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Lectures

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04:05

A study by Consumer Report…

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pLS ) study by Consumer R…

03:01

Supermarket Customer Satis…

So this problem. We're interested in seeing whether the supermarket brand of ketchup is as good as a national brand of ketchup. So the what were our alternative hypothesis is that mew is not equal to 0.64 the proportion of people that are interested in the national brand of ketchup and the no hypothesis is that our mu is equal to 0.64 So in order to compute a p value, we need a test statistic. And for our test statistic, we need to find a sample proportion. Our sample proportion is equal to 52 over 100. Because this is the proportion of shoppers from our sample that say the supermarket brand was as good as a national brand is equal to 0.52 and our end is equal to 100. So in order to find a test statistic, we will meet or P value. We will need to computer ze te statistic, and this is the formula. It is equal to the sample proportion of 0.52 minus the population proportion of 0.64 over the square root of the population proportion 0.64 times one minus the population proportionate. A 0.64 over the sample size of 100 all square rooted and we get a Z value of negative 2.5. So if we plot this on a normal distribution curve, you good Z equals zero Here, Z equals native 2.5. It's the left of Z equals zero and we're interested in finding the area under the curve into the left of Z equals negative 2.5. That area represents probability that Z is less than a Rico. Well, the negative 2.5. And in order to compute this value, we need to go to our ze table and I already marked wear. Our, uh, proportion value is its 0.62 But this is not our p value our p value, because we are looking at a two tailed T Z test. Um, and we know this is to tail because our mu differs from something from you, is not equal to something not greater than our less than it's just not equal to looking at both ends of the spectrum. We will have to multiply this value over here by two in order to find our P value. So R P value is equal to two times you're a 20.62 close on a new page our P value, which is the answer to Port B. Those two times 0.62 which is equal to 0.124 Karan sort of part. See, when we're comparing our P value against an Alfa of 0.5 we find that we should reject the no, because our P value 0.124 is less than our Alfa value of 0.5 So what does this mean? This means that we can we have sufficient evidence to conclude that the percentage of supermarket shoppers who believe the supermarket ketchup was as good as national brand Kush up differs from 64%. But is this good news for the national Super, our catch up manufacturer? Um so all this tells us all that the rejection of the no hypothesis tells us is the mu does not equal 0.64 in this supermarket, Theseus, for market catch up. The proportion of people that like the supermarket you think the supermarket ketchup is good as a national brand just differs in 0.64 but from the problem that we see that the proportion our sample proportion is 0.52 So because you're a 0.52 is less than 0.64 The supermarket are the national brand manufacturer should be pleased. So the national brand ketchup should be pleased because this mean this sample mean is less than the population me so a.

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