We finish our discussion of the hypothesis test for a population mean with a review of the meaning of the P-value, along with a review of Type I and Type II errors.

Review of the Meaning of the P-value

At this point, we assume you know how to use a P-value to make a decision in a hypothesis test. The logic is always the same. If we pick a level of significance ( $α$ ), then we compare the P-value to $α$ .

If the P-value $\leq α$ , reject the null hypothesis. The data supports the alternative hypothesis.
If the P-value $> α$ , do not reject the null hypothesis. The data is not strong enough to support the alternative hypothesis.

In fact, we find that we often treat these as “rules” and apply them without thinking about why these rules make sense. When we understand what the P-value means, these rules are logical. So let’s pause here and review the meaning of the P-value, since it is the connection between probability and decision-making in inference and one of the most important concepts in this course.

EXAMPLE Birth Weights in a Town

Let’s return to the familiar context of birth weights for babies in a town. Suppose that babies in the town had a mean birth weight of 3,500 grams in 2010. This year, a random sample of 50 babies has a mean weight of about 3,400 grams with a standard deviation of about 500 grams. Here is the distribution of birth weights in the sample.

Dot plot of birth weights, ranging from around 2,000 grams to 4,000 grams.

Obviously, this sample weighs less on average than the population of babies in the town in 2010. A decrease in the town’s mean birth weight could indicate a decline in overall health of the town. But does this sample give strong evidence that the town’s mean birth weight is less than 3,500 grams this year?

We now know how to answer this question with a hypothesis test. Let’s use a significance level of 5%.

Let $μ$ = mean birth weight in the town this year. The null hypothesis says there is “no change from 2010.”

$H_{0}$ : $μ = 3500$
$H_{a}$ : $μ < 3500$

Since the sample is large, we can conduct the T-test (without worrying about the shape of the distribution of birth weights for individual babies.)

$T = \frac{3400 - 3500}{\frac{500}{\sqrt{50}}} = - 1.41$

Statistical software tells us the P-value is 0.082 = 8.2%. Since the P-value is greater than 0.05, we fail to reject the null hypothesis.

Our conclusion: This sample does not suggest that the mean birth weight this year is less than 3,500 grams (P-value = 0.082). The sample from this year has a mean of 3,400 grams, which is 100 grams lower than the mean in 2010. But this difference is not statistically significant. It can be explained by the chance fluctuation we expect to see in random sampling. Statistical software tells us the P-value is 0.082 = 8.2%. Since the P-value is greater than 0.05, we fail to reject the null hypothesis. $End of example$

What Does the P-Value of 0.082 Tell Us?

A simulation can help us understand the P-value. In a simulation, we assume that the population mean is 3,500 grams. This is the null hypothesis. We assume the null hypothesis is true and select 1,000 random samples from a population with a mean of 3,500 grams. The mean of the sampling distribution is at 3,500 (as predicted by the null hypothesis.) We see this in the simulated sampling distribution.

If the mean = 3,500 then 86 out of the 1,000 random samples have a sample mean less than 3,400. This is 0.086 = 8.6%

In the simulation, we can see that 86 of the 1,000 samples (8.6%) have a mean less than 3,400. Since probability is the relative frequency of an event in the long run, we say there is an 8.6% chance that a random sample of 500 babies has a mean less than 3,400 if the population mean is 3,500. We can see that the corresponding area to the left of T = −1.41 in the T-model (with df = 49) also gives us a good estimate of the probability. This area is the P-value, about 8.2%.

If we generalize this statement, we say the P-value is the probability that random samples have results more extreme than the data if the null hypothesis is true. (By more extreme, we mean further from value of the parameter, in the direction of the alternative hypothesis.)

Another way to describe the P-value is in terms of error, or more precisely, the absolute value of the error. Our sample has a 1,000 gram absolute error because 3,400 (our sample mean) – 3,500 (null value) = -1,000. We can say that 8.2% of the time, a random sample of 500 babies with a mean weight below 3,500 grams will have a larger absolute error than observed in our sample.

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