**Set Up 6:** According to the Retail Milk Prices Report produced by the United States Department of Agriculture, the mean price for a gallon whole milk in Chicago last month was **$5.36**. A sample from **38** different grocery stores in Chicago this month found a sample mean price per gallon of **$5.16.** Whole milk prices have a population standard deviation of **$0.65** and are normally distributed.

Suppose an economist is asked if the price of whole milk per gallon has decreased in Chicago. What is the null hypothesis for this test?

In hypothesis testing, the null hypothesis ($H_0$) represents the status quo or a statement of no effect or no change. It is usually what the test seeks to disprove or reject.

For this problem, the economist is asked to determine whether the price of whole milk per gallon has decreased in Chicago. This means we are testing if the mean price has fallen below last month's mean of $5.36.

The null hypothesis ($H_0$) is the hypothesis that there is no decrease in the mean price, which means the mean price is greater than or equal to $5.36.

**Null Hypothesis ($H_0$)**:  
\[
H_0: \mu \geq 5.36
\]

Where $\mu$ is the population mean price of whole milk per gallon in Chicago this month.

**Alternative Hypothesis ($H_a$)**:  
\[
H_a: \mu < 5.36
\]

This alternative hypothesis states that the mean price of whole milk has decreased compared to last month's mean price.

Do you want more details or have any questions?

### Related Questions:
1. How do we calculate the test statistic for this hypothesis test?
2. What is the critical value for this test at a 5% significance level?
3. How do we interpret the p-value in this context?
4. What assumptions must be met to use the z-test in this scenario?
5. How would the result change if the sample size were smaller?

**Tip:** When conducting hypothesis tests, always define your null and alternative hypotheses clearly and make sure they reflect the question being asked.

Suppose an economist is asked if the price of whole milk per gallon has decreased in Chicago. What is the null hypothesis for this test?

In this problem, we conduct a hypothesis test to determine if the price of whole milk per gallon has decreased in Chicago. We set up the null hypothesis (H0: μ ≥ 5.36) and alternative hypothesis (Ha: μ < 5.36) based on a population standard deviation of $0.65 and a sample size of 38 stores. Learn how to apply the Z-test and interpret the results.

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