Question

a manager of a grocery store wants to determine if consumers are spending more than the national average. the national average is $150.00 with a standard deviation of $30.20. the manager collects 40 random receipts and finds that the average is $160. complete a hypothesis test with a significance level of 2.5% to determine if the average customer spends more in his store than the national average. which of the following is a valid conclusion for the manager based on this test?

• the customers spend more than the national average in his store.
• the manager should decrease prices in his store.
• the customers do not spend more than the national average in his store.
• the customers in his store just come from a rich neighborhood.

Explanation:

Step1: Identify Hypotheses

Null hypothesis \( H_0: \mu \leq 150 \) (national average), Alternative hypothesis \( H_1: \mu > 150 \) (store average > national).

Step2: Calculate Test Statistic

For z - test (since population standard deviation \( \sigma = 30.20 \) is known, sample size \( n = 40 \)):
Test statistic \( z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}=\frac{160 - 150}{\frac{30.20}{\sqrt{40}}}\approx\frac{10}{4.78}\approx2.09 \).

Step3: Find Critical Value

Significance level \( \alpha = 0.025 \), one - tailed test. Critical value \( z_{\alpha}=1.96 \) (from standard normal table).

Step4: Compare Test Statistic and Critical Value

Since \( z = 2.09>z_{\alpha}=1.96 \), we reject \( H_0 \). So, there is sufficient evidence that the average customer spends more than the national average in his store.

Now, analyze the options:

• "The customers spend more than the national average in his store." is valid as we rejected \( H_0 \) supporting \( H_1 \).
• "The manager should decrease prices in his store." is not a conclusion from the hypothesis test (test is about spending, not pricing strategy).
• "The customers do not spend more than the national average in his store." is wrong as we rejected \( H_0 \).
• "The customers in his store just come from a rich neighborhood." is not a statistical conclusion from the hypothesis test.

Answer:

The customers spend more than the national average in his store.