Are store-brand grocery items actually cheaper than their name-brand counterpart

Question

Are store-brand grocery items actually cheaper than their name-brand counterparts? A data set constructed by Michael McKay, a business statistics student at Fairfield University, will help you answer this question. Michael collected data on the prices of store-brand and name-brand items sold at Groceries Plus (not its real name). To select his sample, he made a preliminary visit to Groceries Plus to study the layout of the store. Then he used a random-number generator to select a series of four random numbers. The first random number provided an aisle number, the second random number provided a distance into the aisle, the third (0 or 1) indicated the shelf on the right or the left side of the aisle, and the fourth provided a shelf number. The store-brand item that was located nearest to the position dictated by the set of four random numbers was pulled into his sample. You will work with a sample of grocery items that was randomly selected from Michael's data set. The sample data are shown n the Data View tool that follows. Which of the following statements best describes how the data were collected and how they should be analyzed to make inferences about mu_s, the mean difference in price between name-brand and store-brand items sold at Groceries Plus? Collected a angle sample of 10 pairs of data values; analyze a single sample of 10 differences Collected two independent samples of size 20 each; analyze two samples of 10 differences each Collected a angle sample of 10 pairs of data values; analyze a angle sample of 20 differences Collected two independent samples of size 10 each; analyze a single sample of 10 differences Groceries Plus advertises that its store brand provides a savings to its customers of $1.25 per item on average. As a statistics consultant to the Better Business Bureau, you are asked to investigate the grocery store's claim using a hypothesis test. If there is sufficient statistical evidence to infer that the average savings per item is less than $1.25, the Better Business Bureau will take action to have Groceries Plus correct its advertising. The Better Business Bureau requests a .10 level of significance for your hypothesis test. the Better Business Bureau will take action to have Groceries Plus correct its advertising. The Better Business Bureau requests a .10 level of significance for your hypothesis test. You conduct ___ hypothesis test with the full and alternative hypotheses formulated as: H_0: mu_e greaterthanorequalto 1.25; H_s: mu_e notequalto 1.25 H_0: mu_e lessthanorequalto 1.25; H_0: mu_e > 1.25 H_0: mu_e greaterthanorequalto 1.25; H_0: mu_e

Explanation / Answer

Solution:-

Collected a single sample of 10 pairs of data values, analyse a single sample of 10 differences.

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: d = 0

Alternative hypothesis: d < 1.25

Note that these hypotheses constitute a one -tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a matched-pairs t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard deviation of the differences (s), the standard error (SE) of the mean difference, the degrees of freedom (DF), and the t statistic test statistic (t).

s = sqrt [ ((di - d)2 / (n - 1) ]

s =

SE = s / sqrt(n) =

S.E =

DF = n - 1 = 10 - 1

D.F = 9

t = [ (x1 - x2) - D ] / SE

t =

where di is the observed difference for pair i, d is mean difference between sample pairs, D is the hypothesized mean difference between population pairs, and n is the number of pairs.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 9 degrees of freedom is more extreme than

Interpret results. If the P-value is greater than the significance level (0.10), we cannot reject the null hypothesis.

We have insufficient evidence in the favor of the claim that the average saving per item is less than $1.25.

If the p value is less than the significance level (0.10), we can reject the null hypothesis.

We have sufficient evidence in the favor of the claim that the average saving per item is less than $1.25.

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