This lesson explains how to conduct a hypothesis test for a population mean.

**Checking Conditions**

Before we can conduct a hypothesis test for a population mean, we first need to make sure the following conditions are met to ensure that our hypothesis test will be valid:

**Random:**A random sample or random experiment should be used to collect the data.**Normal:**The sampling distribution is normal or approximately normal.

If these conditions are met, we can then conduct the hypothesis test. The following two examples show how to conduct a hypothesis test for a population mean.

**Hypothesis Test for Population Mean (Two-Tailed)**

A factory claims that they produce tires that each weigh 200 pounds. An auditor picks 100 tires at random from the assembly line and weighs them. He finds that the mean weight is 198 pounds with a standard deviation of 10 pounds.

**Test the null hypothesis that the mean weight of a tire is 200 pounds against the alternative hypothesis that the mean weight of a tire is not 200 pounds. Use a 0.05 level of significance.**

**Step 1. State the hypotheses. **

The null hypothesis (H0): μ = 200

The alternative hypothesis: (Ha): μ ≠ 200

**Step 2. Determine a significance level to use.**

The problem tells us that we are to use a .05 level of significance.

**Step 3. Find the test statistic.**

test statistic t = (x-μ) / (s/√n) = (198-200) / (10/√100) = (-2) / (1) = **-2**

Use the T Score to P Value Calculator with a t score of -2, degrees of freedom of n-1 = 100-1 = 99, and a two-tailed test, to find that the p-value = **0.048**.

**Step 4. Reject or fail to reject the null hypothesis.**

Since the p-value is less than our significance level of .05, we reject the null hypothesis.

**Step 5. Interpret the results. **

Since we rejected the null hypothesis, we have sufficient evidence to say that the mean weight of a tire is not 200 pounds.

**Hypothesis Test for Population Mean (One-Tailed)**

A botanist believes that the mean height of a certain plant is at least 14 inches. To prove her point, she randomly selects 30 plants and measures them. She finds that the mean height is 13.5 inches with a standard deviation of 2 inches.

**Test the null hypothesis that the mean height of a plant is at least 14 inches against the alternative hypothesis that the mean height of a plant is less than 14 inches. Use a 0.01 level of significance.**

**Step 1. State the hypotheses. **

The null hypothesis (H0): μ ≥ 14

The alternative hypothesis: (Ha): μ < 14

**Step 2. Determine a significance level to use.**

The problem tells us that we are to use a .01 level of significance.

**Step 3. Find the test statistic.**

test statistic t = (x-μ) / (s/√n) = (13.5-14) / (2/√30) = -1.369

Use the T Score to P Value Calculator with a t score of -1.369 and degrees of freedom of n-1 = 30-1 = 29, to find that the p-value = **0.091**.

**Step 4. Reject or fail to reject the null hypothesis.**

Since the p-value is greater than our significance level of .01, we fail to reject the null hypothesis.

**Step 5. Interpret the results. **

Since we failed to reject the null hypothesis, we do not have sufficient evidence to say that the botanist is wrong about her belief that the mean height of a plant is at least 14 inches.