Hypothesis Testing – Examples and Case Studies
Steps in Hypothesis Testing
1.
Determine the null hypothesis and the alternative hypothesis.
2.
Collect and summarize the data into a test statistic.
3.
Use the test statistic to determine the p-value.
4.
The result is statistically significant if the p-value is less than or equal to the level of
significance.
5.
Report your conclusions in terms of the original hypothesis
Hypothesis Testing – Examples and Case Studies
Inferences about means
•Now that we know how to test hypotheses about proportions, it’d be nice to be able to do the
same for means.
•Just as we did before, we will base both our confidence interval and our hypothesis test on the
sampling distribution model.
The Central Limit Theorem told us that the sampling distribution model for means is Normal
with mean equal to the population mean and standard deviation (SD):
population standard deviation
sample size
•All we need is a random sample of quantitative data and the true population standard
deviation. That’s the problem.
•Proportions have a link between the proportion value and the standard deviation of the
sample proportion.
•This is not the case with means—knowing the sample mean tells us nothing about the SD of
the sample mean.
Hypothesis Testing – Examples and Case Studies
Inferences about means
We’ll do the best we can: estimate the population standard deviation with the sample standard
deviation .
Our resulting standard error of sample mean (SEM) is :
sample standard deviation
sample size
Reminder: Conditions for Rule for Sample Means
Population of measurements is bell-shaped, and a random sample of any size is measured.
OR
Population of measurements of interest is not bell-shaped, but a large random sample is
measured. Sample of size 40 is considered “large,” but if there are extreme outliers, better to
have a larger sample.
Hypothesis Testing – Examples and Case Studies
Example: Executives' blood pressures
The National Center for Health Statistics reports that the mean systolic blood pressure for males
35 to 44 years of age is 128.
The medical director of a large company looks at the medical records of 72 executives in this age
group and finds that the mean systolic blood pressure in this sample mean is equal to 126.1 and
that the standard deviation is = 15.2.
Is this evidence that the company's executives have a different mean blood pressure from the
general population?
The hypotheses: The null hypothesis is "no difference“ from the national mean. The alternative is
two sided. Why? So the hypotheses about the unknown population mean of the executive
population are
Null hypothesis: population mean equal to 128
Alternative hypothesis: population mean not equal to 128
The sampling distribution: If the null hypothesis is true, the sample mean comes from a Normal
distribution with mean equal to 128. The standard error of the sample mean (SEM) is:
sample standard deviation
sample size
= 15.2/square root of (72) = 1.79
Hypothesis Testing – Examples and Case Studies
Example: Executives' blood pressures
The data: The sample mean is = 126.1. The standardized score (z-score) for this outcome is
standardized score (z-score) = observation – null value/standard error
= 126.1 – 128/1.79
= -1.06
The P-value: The figure locates the sample outcome -1.06 (in
the standard scale) on the Normal curve that represents the
sampling distribution if the null hypothesis is true.
The two-sided P-value is the probability of an outcome at
least this far out in either direction.
In Table 8.1, the closest standardized score to -1.08. This is
the 13th percentile of a Normal distribution. So the area to
the left of-1.08 is 0.14. The area to the left of -1.08 and to
the right of 1.08 is double this, equal to 0.28.
This is our approximate P-value. (The exact P-value, from
software, is P = 0.289.)
The conclusion?
Comparing Two Means Hypothesis Testing – Examples and Case Studies
This time the parameter of interest is the difference between the two population means
For independent random quantities, variances add.
So, the standard error(SE) or measure of variability of the difference between two sample means is
SE(Sample 1 Mean - Sample 2 Mean ) = square root of [(SEM1)2 + (SEM2)2]
where SEM1 = sample 1 standard deviation
sample 1 size
and SEM2 = sample 2 standard deviation
sample 2 size
Note: Rules for Sample Means apply for comparing two means.
measure of variability known as Standard Error of Difference (SED) =
[(SEM1)2 + (SEM2)2]
Null hypothesis: Population mean difference is zero
Alternative hypothesis: Population mean difference is not zero.
The sampling distribution: If the null hypothesis is true, the sample mean comes from a Normal
distribution with mean equal to zero and standard error equal to above value.
Hypothesis Testing – Examples and Case Studies
Example: Weight Loss for Diet vs Exercise
Determine the null and alternative hypotheses.
Null hypothesis: No difference in average fat lost in population for two methods. Population
mean difference is zero.
Alternative hypothesis: There is a difference in average fat lost in population for two methods.
Population mean difference is not zero.
Collect and summarize data into a test statistic.
The sample mean difference (Diet – Exercise) = 5.9 – 4.1 = 1.8 kg and the standard error of the
difference is 0.83.
standardized score = observation – null value/standard deviation
= sample mean difference – null value/ standard error of the difference
= 1.8 – 0/0.83 = 2.17
Determine the p-value.
The alternative hypothesis was two-sided.
p-value = 2  [proportion of bell-shaped curve above 2.17]
Table 8.1 => proportion is about 2  0.015 = 0.03.
95% Confidence Interval : 1.6 kg to 4.8 kg
Hypothesis Testing – Examples and Case Studies
Example. Cholesterol. A randomized controlled double-blind experiment was performed to
demonstrate the efficacy of a drug called "cholestyramine“ in reducing blood cholesterol levels
and preventing heart attacks.
•There were 3,806 subjects, who were all middle-aged men at high risk of heart attack;
•1,906 were randomly assigned the treatment group and the remaining 1,900 to the control
group.
•The subjects were followed for 7 years. The drug did reduce the cholesterol level in the
treatment group (by about 8%).
•Furthermore, there were 155 heart attacks in the treatment group, and 187 in the control
group: 8.1% versus 9.8%, z = -1.8, P =0.035 (one-tailed).
•This was called "strong evidence" that cholestyramine helps prevent heart attacks
Hypothesis Testing – Examples and Case Studies
How Journals Present Tests
Mozart, Relaxation, and Performance on Spatial Tasks
Three listening conditions— Mozart, a relaxation tape, and silence—and all subjects participated in all
three conditions.
Null hypothesis: No differences in population mean spatial reasoning IQ scores after each of three
listening conditions.
Alternative hypothesis: Population mean spatial reasoning IQ scores do differ for at least one of
the conditions compared with the others.
“A one-factor (listening condition) repeated measures analysis of variance … revealed that subjects
performed better on the abstract/spatial reasoning tests after listening to Mozart than after listening
to either the relaxation tape or to nothing (F[2,35] = 7.08, p = 0.002).”
“The music condition differed significantly from both the relaxation and silence conditions (t = 3.41,
p = 0.002; t = 3.67, p = 0.0008, two-tailed, respectively). The relaxation and silence conditions did not
differ (t = 0.795, p = 0.432, two-tailed).”
Example: Weight Loss - Diet versus Exercise: z-test statistic given as 2.17 is actually a t-test statistic,
because the sample standard deviations were used in the computation. The degrees of freedom (df)
for the t-test is (42 + 47 – 2) = 87. Excel: TDIST(2.17,87,2) gives a p-value of 0.0327, very close to
0.0300 found using the standard normal curve (z-curve).
Text Questions
Study: Alternations in Brain and Immune
Function Produced by Mindfulness
Meditation
Q16. The participants were right-handed
volunteers in a biotechnology company in
Madison, Wisconsin in the Midwestern
United States. However, for the
relationship studied in this randomized
experiment, there should be nothing
unusual about them, so the results
probably apply to adults who would
volunteer for a study like this one.