10 Testing a Single Population Mean 10.2 Drug Manufacturing Example 10.4 Z-Test on a Single Population Mean

10.3 Hypotheses Setups

In doing a hypothesis test on a single population mean $\mu,$ there are three possible setups for the competing hypotheses $H_{0}$ and $H_{1}.$ In the following, $\mu_{0}$ is a fixed real number, chosen by the researcher, to which the sample mean will be compared.

Case: Direction of Extreme to the Left Claims such as “the mean is smaller than

\mu_{0}

” or “the average is less than

\mu_{0}

” result in the following competing hypotheses:

\begin{array}[]{ll}H_{0}:&\mu\geq\mu_{0}\\ H_{1}:&\mu<\mu_{0}\\ \end{array}

In this case, sample means

\bar{x}

smaller than

\mu_{0}

give evidence against

H_{0},

and hence, the direction of extreme is to the left.

Case: Direction of Extreme to the Right Claims such as “the mean is greater than

\mu_{0}

” or “the average is larger than

\mu_{0}

” result in the following competing hypotheses:

\begin{array}[]{ll}H_{0}:&\mu\leq\mu_{0}\\ H_{1}:&\mu>\mu_{0}\\ \end{array}

In this case, sample means

\bar{x}

larger than

\mu_{0}

give evidence against

H_{0},

and so the direction of extreme is to the right.

Case: Two-Sided Direction of Extreme Claims such as “the mean is different from

\mu_{0}

” or “the average is not equal to

\mu_{0}

” result in the following competing hypotheses:

\begin{array}[]{ll}H_{0}:&\mu=\mu_{0}\\ H_{1}:&\mu\neq\mu_{0}\\ \end{array}

Now sample means

\bar{x}

that are either larger or smaller than

\mu_{0}

give evidence against

H_{0},

and so the direction of extreme is said to be two-sided.

Remarks: • Note the taken together,

H_{0}

and

H_{1}

cover all possibilities. So, if one is false, the other is true. • The case of

\mu=\mu_{0}

is always part of

H_{0},

i.e.,

H_{0}

is never a strict inequality. There are no exceptions to this.

Example 10.3.1.

A company advertises that their product will last at least 1 year before breaking, but a consumer protection agency suspects fraud. To test this, the agency took a random sample of size 25 of the product, and the average number of days until failure was 350.4 with a sample standard deviation of 54.1. State the competing hypotheses, and give the direction of extreme.

Solution: Let $\mu$ denote the mean time (in days) until the product breaks. The agency is concerned whether $\mu$ is smaller than 365 days, so the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&\mu\geq 365\\ H_{1}:&\mu<365\\ \end{array}

The direction of extreme is to the left. $\clubsuit$

Example 10.3.2.

A pet food manufacturer claims that each can of cat food they make contains 6 ounces of product. The manufacturer is concerned that the cans are being overfilled, so the company randomly selects 40 cans and measures the contents. The sample average and sample standard deviation for the sample are 6.20 and 0.51 ounces, respectively. State the competing hypotheses, and give the direction of extreme.

Solution: Let $\mu$ denote the mean weight of the cans. The manufacturer is concerned whether $\mu$ is greater than 6 ounces, so the competing hypotheses are:

\begin{array}[]{ll}H_{0}:&\mu\leq 6\\ H_{1}:&\mu>6\\ \end{array}

The direction of extreme is to the right. $\clubsuit$

Concepts Check: 1. Translate “the mean is smaller than 62” into the two competing hypotheses. Answer:

\begin{array}[]{ll}H_{0}:&\mu\geq 62\\ H_{1}:&\mu<62\\ \end{array}

2. What is the direction of extreme in the prior problem? Answer: To the left. 3. Translate “the mean is greater than 100” into the two competing hypotheses. Answer:

\begin{array}[]{ll}H_{0}:&\mu\leq 100\\ H_{1}:&\mu>100\\ \end{array}

4. What is the direction of extreme in the prior problem? Answer: To the right. 5. Translate “the mean is no larger than 10” into the two competing hypotheses. Answer:

\begin{array}[]{ll}H_{0}:&\mu\leq 10\\ H_{1}:&\mu>10\\ \end{array}

6. What is the direction of extreme in the prior problem? Answer: To the right. 7. Translate “the mean is 30” into the two competing hypotheses. Answer:

\begin{array}[]{ll}H_{0}:&\mu=30\\ H_{1}:&\mu\neq 30\\ \end{array}

8. What is the direction of extreme in the prior problem? Answer: Two-sided.

Exercises

1.

A provost believes that the average GPA for male students is different from 3.4. To test this, she took a random sample of 30 male students, and found a sample mean and sample standard deviation of 3.2 and 1.0, respectively. State the competing hypotheses, and give the direction of extreme.
2.

At a certain college, an athletic trainer wished to test whether a new method of weight training would reduce the average time students run a mile to under 7 minutes. To test the method, she took 6 subjects and subjected each to the weight training regimen for 3 weeks. Afterwards, each subject was timed in running a mile. The times are given below (times are in minutes):

$\begin{array}[]{cccccc}6.1&7.2&6.4&5.5&8.0&5.1\end{array}$

State the competing hypotheses, and give the direction of extreme.