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AP Statistics
Statistical Inference
Hypothesis Testing
What is Hypothesis Testing?
A statistical hypothesis is an assumption about a population parameter. This assumption may or may not be true.
Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses. The
best way to determine whether a statistical hypothesis is true would be to examine the entire population. Since that is
often impractical, researchers typically examine a random sample from the population. If sample data are not
consistent with the statistical hypothesis, the hypothesis is rejected.
Hypothesis Tests
Statisticians follow a formal process to determine whether to reject a null hypothesis, based on sample data. This
process, called hypothesis testing, consists of four steps…See Inference Toolbox (p. 705).
Key Vocabulary
(Text Chapters 11 – 13)
Null Hypothesis
(691)
Confidence Intervals and Two-Sided Tests
(710)
Alternate Hypothesis
(691)
Type I and Type II Error
(723)
One-Sided Alternative
(692)
Power
(729)
Two-Sided Alternative
(692)
Power and Type II Error
(729)
Test Statistic
(695)
One-Sample t Statistic and t Distribution
(744)
p-Value
(696)
One-Sample t Test
(746)
Significance Level
(699)
One-Proportion z Test
(766)
Statistical Significance
(699)
Two-Sample Problems
(781)
Reject Ho
(700)
Conditions for Comparing Two Means
(782)
Fail to reject Ho
(700)
Two-Sample z Statistic
(787)
z-Test for a Population Mean (705)
Two-Sample t Procedures
(788)
One-Sample z-Statistic
Confidence Interval for Two Proportions
(811)
Significance Test for Two Proportions
(815)
(705)
You should be familiar with all vocabulary related to hypothesis testing.
AP Statistics
Statistical Inference
Hypothesis Testing
Examples of Hypothesis Testing
z-Test for a Population Mean (µ)
At the bakery where you work, loaves of bread are supposed to weigh 1 pound. From experience, the weights of
loaves produced at the bakery follow a Normal distribution with standard deviation  = 0.13 pounds. You
believe that new personnel are producing loaves that are heavier than 1 pound. As supervisor of Quality
Control, you want to test your claim at the 5% significance level. You weigh 20 loaves and obtain a mean
weight of 1.05 pounds.
Hypotheses:
Calculate the Test Statistic:
⁄√
p-Value = 1 – 0.9573 = 0.0427 (since we are testing µ > 1)
Since σ is known, we will use z.
α = 0.05 (5% significance level)
(p < α)
Solution/Interpretation:
Since p < α, there is sufficient evidence against Ho. Therefore, reject Ho in favor of the alternate Ha.
t – Test for a Population Mean (µ)
An inventor has developed a new, energy-efficient lawn mower engine. He claims that the engine will run
continuously for 5 hours (300 minutes) on a single gallon of regular gasoline. Suppose a simple random sample
of 50 engines is tested. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes.
Test the null hypothesis that the mean run time is 300 minutes against the alternative hypothesis that the mean
run time is not 300 minutes. Use a 0.05 level of significance. (Assume that run times for the population of
engines are normally distributed.)
Hypotheses:
Calculate the Test Statistic:
⁄√
Since σ is unknown, we will use t with df = 50 – 1 = 49 (use 40 df from Table C).
For a One-Sample t, use 2P(T ≥ |t|) to calculate the value of p. The p-value will be between 2(0.025) and 2(.05)
which gives us 0.05 < p < 0.1. For α = 0.05 (5% significance level), p is not less than α.
Solution/Interpretation:
Since p > α, there is not sufficient evidence against Ho. Therefore, we cannot reject Ho in favor of the alternate Ha.
AP Statistics
Statistical Inference
z-Test for a Population Mean (µ)
Hypothesis Testing
(2-Sided Test and a Confidence Interval)
Your friend uses Minitab to generate 25 observations at random from a Normal distribution with known mean
and standard deviation. Unfortunately, he forgot to save the file. He remembers that the standard deviation was
4, and he thinks that the mean was 20. Before he closed the program, your friend did manage to print the
following output.
Descriptive Statistics: rand
Variable
N
Mean
Median
TrMean
StDev
SE Mean
25
18.792
18.663
18.791
3.241
0.648
Minimum
Maximum
Q1
Q3
11.192
26.422
17.223
21.128
rand
Variable
rand
Use an appropriate test to determine whether you believe your friend’s claim that the mean was 20.
Hypotheses:
Calculate a 95% Confidence Interval for µ:
(
√
Since σ is known, we will use z.
)
α = 0.05 (5% significance level)
Solution/Interpretation:
Since the hypothesized value
falls within a 95% confidence interval, there is not sufficient evidence against
Ho. Therefore, we cannot reject Ho in favor of the alternate Ha.
Alternate Solution
Hypotheses:
Calculate the Test Statistic:
⁄√
p-Value = 2(0.0655) = 0.131 (since we are testing 2-sided)
Since σ is known, we will use z.
α = 0.05 (5% significance level)
(p < α)
Solution/Interpretation:
Since p > α, there is not sufficient evidence against Ho. Therefore, we cannot reject Ho in favor of the alternate Ha.
AP Statistics
Statistical Inference
Hypothesis Testing
The amount of lead in a certain type of soil, when released by a standard extraction method, averages 86 parts per million
(ppm). A new extraction method is tried on 40 specimens of the soil, yielding a mean of 83 ppm lead and a standard
deviation of 10 ppm. Is there significant evidence at the 5% level that the new method frees less lead from the soil? Carry
out an appropriate test to help answer this question.
Hypotheses:
Solution/Interpretation:
Since p < α, there is sufficient evidence against Ho. Therefore, reject Ho in favor of the alternate Ha.