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.

To find if the average customer spends more in the store than the national average, first we need to find the Z-score, then use the hypothesis test to find the conclusion.

Z-score formula

[tex]\boxed{Z=\frac{\overline{x}-\mu_0}{\frac{\sigma}{\sqrt{n} } } }[/tex]

where:

[tex]Z=\texttt{Z-score}[/tex]
[tex]\overline{x}=\texttt{observed value}[/tex]
[tex]\mu_0=\texttt{population mean}[/tex]
[tex]\sigma=\texttt{population standard deviation}[/tex]
[tex]n=\texttt{number of samples}[/tex]

Given:

[tex]\overline{x}=150[/tex]
[tex]\mu_0=160[/tex]
[tex]\sigma=30.2[/tex]
[tex]n=40[/tex]

Then:

[tex]\begin{aligned}Z&=\frac{\overline{x}-\mu_0}{\frac{\sigma}{\sqrt{n} } } \\\\&=\frac{160-150}{\frac{30.2}{\sqrt{40} } } \\\\&\approx2.094\end{aligned}[/tex]

By using the calculation or Z-score table, we can find that the probability that Z > 2.094 is 0.0181 or 1.81%.

Hypothesis test

First we establish the H₀ and H₁:

H₀ = the average customers spend more in his store than the national average
H₁ = the average customers do not spend more in his store than the national average

Given the significance level (α) is 2.5%:

If the probability ≤ significance level, then we accept H₀
If the probability > significance level, then we reject H₀ and accept H₁

Since the probability is 1.81%, which is smaller than the significance level 2.5%, then we accept H₀ and conclude that "the customers spend more than the national average in his store".