Question

Paying high prices? A retailer entered into an exclusive agreement with a supplier who guaranteed to provide all products at competitive prices. To be sure the supplier honored the terms of the agreement, the retailer had an audit performed on a random sample of 25 invoices. The percent of purchases on each invoice for which an alternative supplier offered a lower price than the original supplier was recorded.17 For example, a data value

of 38 means that the price would be lower with a different supplier for 38% of the items on the invoice. A histogram and some numerical summaries of the data are shown here. The retailer would like to determine if there is convincing evidence that the mean percent of purchases for which an alternative supplier offered lower prices is greater than 50% in the population of this company’s invoices.

a. State appropriate hypotheses for the retailer’s test. Be sure to define your parameter.

b. Check if the conditions for performing the test in part (a) are met.

Short answer

Part a)$$\begin{gathered} H_0:\mu <50 \\ {H}_1:\mu >50 \end{gathered}$$

Part b) Large sample condition is not satisfied.

Step-by-step solution

Part a) Step 1: Given information 

The claim is that mean is bigger than$50\%$

Part a) Step 2: The objective is to explain the state appropriate hypothesis for the retailer's test 

The null hypothesis statement states that the population value is equal to the claim value:

$H_0:\mu <50$

The null hypothesis or the alternative hypothesis is the claim. The null hypothesis asserts that the population means equals the value specified in the claim. If the claim is the null hypothesis, then the alternative hypothesis statement is the inverse of the null hypothesis.

$H_1:\mu >50$

$\mu$denotes the average percentage of purchases for which an alternative supply provided lower prices.

Part b) Step 1: Given information

Given:

Part b) Step 2: The objective is to find the condition for performing the test in part (a) are met. 

Random, independent ($10\%$condition), and Normal/ Large samples are the three conditions.

Random: Satisfied because the sample was chosen at random.

Independent: Satisfied, because the sample of $25$invoices represents less than $10\%$of the total population of invoices.

Normal/large sample size: Not happy because the sample size of $25$invoices is small and the distribution is skewed (as the highest bar in the histogram is to the right in the histogram).

Because the Normal/Large sample condition is not met, a hypothesis test for the population mean is not appropriate.