Example 4

EXAMPLE 4 Applying the strategy for constructing the hypotheses about $μ$

The mean annual rainfall in Arizona has been eight inches per year, according to the World Almanac. But weather researchers are interested in whether this already small amount of rain will decrease, leading to drought conditions in the state. Use the steps in the Strategy for Constructing the Hypotheses About $μ$ to write a null hypothesis and an alternative hypothesis for this situation.

Solution

Let's use our strategy to construct the hypotheses needed to test this claim.

Step 1 Search the word problem for certain key English words and select the appropriate symbol.

The problem uses the word “decrease,” which means “is less than.” Thus, we will write a hypothesis that contains the < symbol.

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Step 2 Determine the form of the hypotheses.

From Table 1, we see that the symbol < means that we use a left-tailed test:

$H_{0} : μ = μ_{0} versus H_{a} : μ < μ_{0}$
Step 3 Find the value for $μ_{0}$ and write your hypotheses.

The alternative hypothesis $H_{a}$ states that the mean annual rainfall in Arizona is less than some value $μ_{0}$ . Less than what? Eight inches per year. Write the two hypotheses with $μ_{0} = 8$ .

$H_{0} : μ = 8 versus H_{a} : μ < 8$

NOW YOU CAN DO

Exercises 13–18.