A snack company claims that at least 70% of people prefer its new low-sugar granola bar over the original version. To test this claim, a grocery chain surveys a random sample of 80 customers, & 50 say they prefer the new version. Use α = 0.10 α=0.10 to test whether more than 70% of customers prefer the new granola bar. Should the grocery chain stock more of the new product & reduce shelf space for the original version?

Welcome back. For this problem, we have a snack company claiming that at least 70% of people prefer its new low sugar granola bar over the original version. To test this claim, a grocery chain surveys a random sample of 80 customers, and 50 say they prefer the new version. We'd like to use alpha equaling 0.1 to test whether more than 70% of customers prefer the new granola bar. Also, we'd like to know if the grocery chain should stock more of the new product and reduce shelf space for the original version. So we definitely wanna do a hypothesis test here. But before we do, we will need to check the criteria holds. And before we do even that, we should take stock of the information we know. So it will be easy to plug in different values we need when checking our criteria and running the hypothesis test. Let's start with n, our sample size. We can see that they took a sample of 80 customers, so n is going to be 80. P is going to be the expected population proportion. We can see that the claim is that at least 70 of people prefer its new low sugar granola bar, and we wanna test whether more than 70% actually do prefer it, meaning the expected population proportion is going to be point seven or 70%. Also, we should get x the number of successes in our sample or, in this case, the number of people in the sample that prefer the new version of the granola bar, which in this case is going to be 50. Finally, we are gonna need our alpha value, which we can see in the problem is point one. Now that we have that information squared away, let's check our conditions. The first condition is that we have a random sample. And as we can see in the problem, we do actually have a random sample, so we can check that condition off. Next, I wanna check n times p and n times q, which are going to be the expected number of people preferring the granola bar and the expected number of people not preferring the new granola bar in our sample. Using the values we identified above, we know that n p is going to be 80 times point seven, which is 56, and that's definitely greater than or equal to five, so our second condition has been met. And q is going to be 80 times one minus p or one minus point seven, which is going to be 24. That's also greater than or equal to five, so our third condition has been met, and we can see that all of our criteria are satisfied. So we're ready to move on to the hypothesis test and start by building our hypotheses. As a reminder, the null hypothesis is going to be that the claim is true and that the population proportion is what we expect, which in this case is that 70% of people prefer the new low sugar granola bar, or in other words, that p is equal to point seven. Our alternative hypothesis is what we're testing. And in this example, we're testing whether that proportion is actually more than 70%. So our alternative hypothesis is going to be that p is greater than point seven. Our next step is going to be to find the z score, but before we do that, we should find that p hat sample proportion, which we're going to need to plug in. P hat is going to be equal to x divided by n. So in this case, our p hat is going to be 50 divided by 80, which is point six two five. Now we're ready to plug everything in and find our z score. It's going to be p hat point six two five minus p point seven divided by the square root of point seven times one minus point seven divided by 80. We're definitely gonna wanna plug that into a calculator, and when we do, we should end up getting approximately negative 1.46 as our z score. We're going to have to turn the z score into a p value. We can either use technology or we can use a z table. As a reminder, the alternative hypothesis is that p is greater than point seven, meaning that we're doing a right tailed test, and we wanna find the probability of getting a z score that's greater than negative 1.46. No matter which method you use, you should end up getting a p value of approximately point nine two eight, which we're going to compare to our alpha value, which is point one. Now, obviously, point nine two eight is greater than point one, meaning that our p value is greater than our alpha value. This means that we fail to reject our null hypothesis, and there is not enough evidence to suggest that the alternative hypothesis is true or to suggest that p is greater than point seven. In the case of this problem, we don't have enough evidence to suggest that more than 70% of customers prefer the new granola bar. So let's look at that last question. We wanna know if the grocery chain should stock more of the new product and reduce shelf space for the original version. The answer here is going to be no because we don't have any evidence to suggest that the proportion of customers preferring the new version of the granola bar is greater than 70%. So we don't have any reason or any evidence to suggest that it would be a good idea to stock more of that new version of the granola bar. Great work.

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Proportions

Statistics for Business

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Proportions

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Multiple Choice

A snack company claims that at least 70% of people prefer its new low-sugar granola bar over the original version. To test this claim, a grocery chain surveys a random sample of 80 customers, & 50 say they prefer the new version. Use $alpha equals 0.10$ to test whether more than 70% of customers prefer the new granola bar. Should the grocery chain stock more of the new product & reduce shelf space for the original version?

Yes

Verified step by step guidance

Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (Hₐ). The null hypothesis is H₀: p ≤ 0.70, which states that 70% or fewer customers prefer the new granola bar. The alternative hypothesis is Hₐ: p > 0.70, which states that more than 70% of customers prefer the new granola bar.

Step 2: Identify the significance level (α) and the sample statistics. The significance level is given as α = 0.10. The sample size (n) is 80, and the number of customers who prefer the new granola bar is 50. Calculate the sample proportion (p̂) using the formula p̂ = x / n, where x is the number of successes.

Step 3: Calculate the test statistic using the formula for a one-proportion z-test: z = (p̂ - p₀) / √(p₀(1 - p₀) / n), where p₀ is the hypothesized population proportion (0.70), p̂ is the sample proportion, and n is the sample size.

Step 4: Determine the critical value for a one-tailed test at α = 0.10. Use a z-table or statistical software to find the z-critical value corresponding to a right-tailed test with a significance level of 0.10.

Step 5: Compare the calculated z-test statistic to the critical value. If the z-test statistic is greater than the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Based on the conclusion, decide whether the grocery chain should stock more of the new product or maintain the current shelf space allocation.

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