Section 9.2 ~ Hypothesis Tests for Population Means
Objective: After this section you will understand and interpret one- and two-tailed
hypothesis tests for claims made about population means, and learn to recognize and
avoid common errors (type I and type II errors) in hypothesis tests.
Background information:
~ Recall that there are two possible outcomes of a hypothesis test; to either reject or not
reject the null hypothesis.
~ To determine whether to reject or not, a level of significance (0.05 or 0.01) needs
to be found.
~ To find the level of significance, a P-value needs to be calculated.
~ To calculate a P-value, you must first understand the concepts of a normal distribution
(introduced in ch.5):
~ Recall that if a distribution is normal, you can use z-scores along with a z-score
table to find probabilities of certain values occurring.
~ Also recall that a distribution begins to take the shape of a normal distribution
when the sample size is at least 30 and becomes more and more normal as the
sample size increases (Central Limit Theorem).
~ In essence, a P-value (probability value) is the probability that is found using zscores and the z-score table.
~ Be sure that you are using the sample standard deviation, S x , when calculating
the z-score since you are comparing a sample (group mean or group proportion)
to the entire population.
One-Tailed Hypothesis Tests:
Example 1: Left-Tailed Hypothesis Test
Columbia College advertises that the mean starting salary of its graduates is $39,000.
The Committee for Truth in Advertising suspects that this claim is exaggerated and that
the mean starting salary for graduates is actually lower. They decide to conduct a
hypothesis test to seek evidence to support this suspicion.
Example 2: Right-Tailed Hypothesis Test
In the United States, the average car is driven about 12,000 miles each year. The owner
of a large rental car company suspects that for his fleet, the mean distance is greater than
12,000 miles each year. He selects a random sample of n = 225 cars from his fleet and
finds that the mean annual mileage for this sample is x  12,375 miles. Suppose that the
standard deviation for that sample is 2,415 miles. Interpret this claim by conducting a
hypothesis test.
Since we can decide to reject the null hypothesis if the P-value is .05 or lower (or .01 or
lower), we can use critical values as a quick guideline to decide if we should reject the
null hypothesis or not.
Critical values for a one-tailed test at the .05 significance level:
~ For a left-tailed test, the z-score that corresponds to a probability of .05 is
__________, so any z-score that is _________________________will be
statistically significant at the .05 level.
~ For a right-tailed test, the z-score that corresponds to a probability of .05 is
__________, so any z-score that is _________________________will be
statistically significant at the .05 level.
Critical values for a one-tailed test at the .01 significance level:
~ For a left-tailed test, the z-score that corresponds to a probability of .01 is
__________, so any z-score that is _________________________will be
statistically significant at the .01 level.
~ For a right-tailed test, the z-score that corresponds to a probability of .01 is
__________, so any z-score that is _________________________will be
statistically significant at the .01 level.
Two-Tailed Hypothesis Tests:
Since a two tailed test tests both above and below the claimed value, a .05 significance
level would have to be split between the two extremes thus looking for a z-score that
corresponds to a probability of .025.
Critical values for a two-tailed test at the .05 significance level:
~ The z-scores that correspond to a probability of .025 are____________________,
so for a two-tailed test, it is significant at the .05 level if the z-score is
_______________________ or __________________________.
Critical values for a two-tailed test at the .01 significance level:
~ The z-scores that correspond to a probability of .005 are____________________,
so for a two-tailed test, it is significant at the .01 level if the z-score is
_______________________ or __________________________.
Summary of critical values for a one-tailed test:
Summary of critical values for a two-tailed test:
.05 significance level:
.01 significance level:
Example 3:
Consider the study in which University of Maryland researchers measured body
temperatures in a sample of n = 106 healthy adults, finding a sample mean body
temperature of x  98.2 F with a sample standard deviation of 0.62°F. We will assume
that the population standard deviation is the same as the standard deviation found from
the sample. Determine whether this sample provides evidence for rejecting the common
belief that the mean human body temperature is   98.6 F .
Common errors in hypothesis testing:
Type I error:
Ex. ~ Consider a drug company that seeks to be sure that its “500-milligram”
aspirin tablets really contain 500 milligrams. If the tablets contain less than 500
milligrams, the consumers are not getting the advertised dose. If the tablets contain
more than 500 milligrams, the consumers are getting more than the advertised dose
(which could be potentially dangerous).
Suppose that a hypothesis test gave evidence to reject the null hypothesis,
stating that the drug does not contain 500 milligrams, when in reality it does
contain 500 milligrams.
Type II error:
Ex. ~ Consider a drug company that seeks to be sure that its “500-milligram”
aspirin tablets really contain 500 milligrams. If the tablets contain less than 500
milligrams, the consumers are not getting the advertised dose. If the tablets contain
more than 500 milligrams, the consumers are getting more than the advertised dose
(which could be potentially dangerous).
Suppose that a hypothesis test gave evidence to not reject the null hypothesis,
stating that the drug does contain 500 milligrams, when in reality it does not
contain 500 milligrams.
Example 4:
The success of precious metal mines depends on the purity (or grade) of ore removed and
the market price for the metal. Suppose the purity of gold ore must be at least 0.5 ounce
of gold per ton of ore in order to keep the mine open. Samples of gold ore are used to
estimate the purity of the ore for the entire mine. Discuss the impact of type I and type II
errors on two of the possible alternative hypotheses:
H a : purity < 0.5 ounce per ton
H a : purity > 0.5 ounce per ton