Elementary Statistics – Solution to Significance Testing Example 2

Example 2

By law, an industrial plant can discharge no more than 500 gallons of wastewater per hour, on the average, into a neighboring lake. An environmental activist thinks that a certain plant is breaking the law.

She takes a random sample of 50 hours and finds the sample mean discharge to be 504.6 gallons. Suppose that we know that the standard deviation of wastewater discharge is 180.

Carry out the significance test (state the hypotheses, calculate the test statistic, calculate the p-value, and interpret the results).

Hypotheses

Let be the mean wastewater discharge per hour for the plant. Then we want to test the hypotheses

[Note: The null hypothesis is always an equality statement. It may seem natural in this problem to write the null hypothesis as . Note, though, if you reject the null hypothesis in favor of the alternative, then you necessarily reject the hypothesis , too. Hence, we can stick with the simpler equality statement in .]

Sample-Size Check

Since the sample size is large (), we know the sample mean has an approximate normal distribution and we can carry out the test using the normal distribution.

Test Statistic

The test statistic is . (That is, our sample average wastewater discharge is only 0.18 standard deviations above the null-hypothesized value.)

P-value

The P-value is . Note: An appropriately labeled picture of a normal curve should be included here (it’s not on the web version simply because I can’t draw pictures in Word).

Definition of P-value and Conclusion

If the mean wastewater discharge for the plant is 500 gallons per hour, then there’s about a 43% chance of getting our sample mean discharge (504.6 gallons/hour) or a larger sample mean discharge. Because our data are not at all surprising, we do not have evidence against the mean discharge being 500 gallons/hour (i.e., we don’t have evidence the plant is breaking the law). These results are not statistically significant at any reasonable significance level.

[Note: Why do we say “we don’t have evidence against the null hypothesis,” rather than, “we think the null hypothesis is correct”? There are two types of possible error in significance testing. One type of error is when we reject the null hypothesis when it’s actually true. We control for this type of error by keeping the significance level small (e.g., 0.05). The other type of error is when we say the null hypothesis is true, when really it’s false. Unfortunately, we don’t typically control for this type of error. Hence, we don’t say “we think the null hypothesis is true,” since the chance of the latter error may be high. Instead we say, “we do not have enough evidence against the null hypothesis.”]