A cereal company advertises that 50%50\% of households purchase its brand every month. A market research team wonders if the true proportion is different, so they survey 200200 households, and 114114 report that they purchase the brand at least once per month.(A) Use α=0.10α=0.10 to test the claim.H0:H_0:_________ Ha:H_a:_________x=x=________ n=n=________ PP-value:______Because PP-value [ < | > ] αα, we [ REJECT | FAIL TO REJECT ] H0H_0, there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

Because P P P -value < α α α , we REJECT H 0 H_0 H 0 , there is ENOUGH evidence to suggest that p ≠ 0.5.

A cereal company advertises that 50 % 50\% of households purchase its brand every month. A market research team wonders if the true proportion is different, so they survey 200 200 households, and 114 114 report that they purchase the brand at least once per month. (A) Use α = 0.10 α=0.10 to test the claim. H 0 : H_0: _________ H a : H_a: _________ x = x= ________ n = n= ________ P P -value:______ Because P P -value [ < | > ] α α , we [ REJECT | FAIL TO REJECT ] H 0 H_0 , there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

Welcome back. In this problem, we have a cereal company advertising that 50% of households purchase its brand every month. A market research team wonders if the true proportion is different. So they survey 200 households, and 114 report that they purchased the brand at least once per month. We'd like to use alpha equals 0.1 to test the claim in part a, so let's begin by writing our hypothesis, where the null hypothesis is that the true proportion of households buying the brand is equal to what we expect it to be. And looking at the problem, we can see that the expectation is 50%. So our null hypothesis is that p is equal to 0.5 or 50%. The alternative hypothesis is what we're trying to test or find evidence for. We can see that the market research team wonders if the true proportion is different, not necessarily that it's higher or lower, just different. So our alternative hypothesis is that p is not equal to 0.5 or 50%. Before we pull out our TI 80 fours, it is a good idea to get some of those key values from the problem, starting with x, the number of people in the sample that do buy from the brand, and we can see that that is 114. We also need n, our sample size, which is 200. Now that we have those values, let's pull out our t I 80 fours. For step one, we're gonna click on the stat button and then scroll two to the right to the test menu and scroll down to option five, the one dash proxy test function. We can click enter, and that is step one all set. And for step two, we just need to enter in some information from the problem, starting with p sub zero, which is the expected value for the population proportion. We can get that from the null hypothesis, and we can see that it is 0.5. So let's put in 0.5 for p sub zero. X we found to be 114, and n we found to be 200, so let's put those in. And next, we need to choose the correct alternative hypothesis. Looking at the problem, we said that the alternative hypothesis was that p was not equal to 0.5, and because p sub zero equals 0.5, we could also say that, as p is not equal to p sub zero, so let's make sure that not equal to p sub zero has been selected. Then we can scroll down to calculate and click enter. That is step two all set, and the results of the test appear in the viewing window. We end up getting a p value of about 0.048, and we wanna compare it to our alpha of 0.1. We can see that 0.048 is smaller than 0.1, meaning our p value is smaller than alpha, and we reject the null hypothesis, so there is enough evidence to suggest that the alternative hypothesis is true, that p is not equal to 0.5, or that the true proportion of households buying from this brand at least once a month is not equal to 50% or point five. Great work.

A cereal company advertises that 50 % 50\% of households purchase its brand every month. A market research team wonders if the true proportion is different, so they survey 200 200 households, and 114 114 report that they purchase the brand at least once per month. (A) Use α = 0.10 α=0.10 to test the claim. H 0 : H_0: _________ H a : H_a: _________ x = x= ________ n = n= ________ P P -value:______ Because P P -value [ < | > ] α α , we [ REJECT | FAIL TO REJECT ] H 0 H_0 , there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

Because P P P -value < α α α , we REJECT H 0 H_0 H 0 , there is ENOUGH evidence to suggest that p ≠ 0.5.

Because P P P -value > α α α , we FAIL TO REJECT H 0 H_0 H 0 , there is NOT ENOUGH evidence to suggest that p = 0.5.

Because P P P -value > α α α , we FAIL TO REJECT H 0 H_0 H 0 , there is NOT ENOUGH evidence to suggest that p ≠ 0.5.

Because P P P -value < α α α , we REJECT H 0 H_0 H 0 , there is ENOUGH evidence to suggest that p = 0.5.

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 cereal company advertises that $50 %$ of households purchase its brand every month. A market research team wonders if the true proportion is different, so they survey $200$ households, and $114$ report that they purchase the brand at least once per month.
(A) Use $α = 0.10$ to test the claim.
$H_{0} :$ _________ $H_{a} :$ ___
$x =$ $n =$ $P$ -value:
Because $P$ -value [ < | > ] $α$ , we [ REJECT | FAIL TO REJECT ] $H_{0}$ , there is [ ENOUGH | NOT ENOUGH ] evidence to suggest…

Because $P$ -value > $α$ , we FAIL TO REJECT $H_0$ , there is NOT ENOUGH evidence to suggest that p = 0.5.

Because $P$ -value > $α$ , we FAIL TO REJECT $H_0$ , there is NOT ENOUGH evidence to suggest that p ≠ 0.5.

Because $P$ -value < $α$ , we REJECT $H_0$ , there is ENOUGH evidence to suggest that p = 0.5.

Because $P$ -value < $α$ , we REJECT $H_0$ , there is ENOUGH evidence to suggest that p ≠ 0.5.

Verified step by step guidance

Step 1: Define the null and alternative hypotheses. Since the company claims that 50% of households purchase the brand, the null hypothesis is $H_0: p = 0.50$. The market research team suspects the true proportion is different, so the alternative hypothesis is two-tailed: $H_a: p \neq 0.50$.

Step 2: Identify the sample statistics. The sample size is $n = 200$, and the number of households reporting purchase is $x = 114$. Calculate the sample proportion as $\hat{p} = \frac{x}{n} = \frac{114}{200}$.

Step 3: Calculate the test statistic for a population proportion. Use the formula for the z-test: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \] where $p_0 = 0.50$ is the hypothesized population proportion.

Step 4: Determine the $P$-value associated with the calculated $z$-value. Since this is a two-tailed test, the $P$-value is twice the area in the tail beyond the absolute value of $z$ in the standard normal distribution.

Step 5: Compare the $P$-value to the significance level $\alpha = 0.10$. If $P$-value $<$ $\alpha$, reject $H_0$; otherwise, fail to reject $H_0$. Then, interpret the result in the context of the problem, stating whether there is enough evidence to suggest the true proportion differs from 50%.

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