INTRODUCTORY STATISTICS
Chapter 10 HYPOTHESIS TESTING WITH TWO SAMPLES:
TWO MEANS, PAIRED DATA, TWO PROPORTIONS
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SEC. 10.2: TWO POPULATION MEANS WITH
UNKNOWN STANDARD DEVIATIONS
If you want to test a claim that involves two groups (the types of breakfasts eaten east
and west of the Mississippi River) you can use a slightly different technique when
conducting a hypothesis test. (credit: Chloe Lim)
OVERVIEW
The two independent samples are simple random samples from two
distinct populations.
For the two distinct populations:
a) if the sample sizes are small, the distributions are important
(should be normal)
b) if the sample sizes are large, the distributions are not important
(need not be normal)
In order to account for the variation of means and standard
deviation, we take the difference of the sample means,𝑋1 − 𝑋2 , and
divide by the standard error in order to standardize the difference.
The result is a t-score test statistic.
USING A CALCULATOR
When the population standard deviation is unknown, use the Student t test.
On your calculator, this is in the STAT menu under TESTS.
Choose 2-SampTTest and enter the mean, standard deviation and number of the
sample for each sample. Choose NO for the Pooled option. Choose CALCULATE to get
the test statistic and p-value, DRAW to get a sketch of the graph.
When making your decision, the same rules apply as for one sample:
If α > p-value, reject the null hypothesis
If α ≤ p-value, do not reject the null hypothesis
EXAMPLE
A statistics instructor believes that there is no significant difference
between the mean class scores of statistics day students on Exam 2
and statistics night students on Exam 2. She takes random samples
from each of the populations. The mean and standard deviation for 35
statistics day students were 75.86 and 16.91. The mean and standard
deviation for 37 statistics night students were 75.41 and 19.73. Is
there sufficient evidence to support her claim?
EXAMPLE
To test if increasing calcium intake lowers blood pressure, 21 healthy men
were recruited for an experiment. 10 of the men received a calcium
supplement for 12 weeks, while the control group of 11 men received a
placebo pill. The following data is the decrease in systolic blood pressure for
a subject after 12 weeks, in millimeters of mercury. An increase appears as a
negative response.
Calcium
7
-4
18
17
-3
-5
1
10
11
-2
Placebo
-1 12
-1
-3
3
-5
5
2
-11
-1
-3
Is there evidence to support the claim that calcium lowers blood pressure?
SEC. 10.3: TWO POPULATION MEANS WITH KNOWN
STANDARD DEVIATIONS
In the rare case that we are comparing two population means and
know the population standard deviations, we can use the normal
distribution.
On your calculator, use the 2-SampZTest in this case.
The test statistic will now be a z for z-score, but the p-value is
evaluated the same.
EXAMPLE
A hospital notices that the babies born in their community seem to
weigh less at birth than the state average. Of the babies born at the
hospital in the last five years, 433 babies were born with an average
weight of 6.8 lbs. and a standard deviation of 1.56 lbs. A sample of
550 babies in the state were surveyed that had an average weight of
7.1 lbs. The state health department states that the standard deviation
of baby weights is 0.87 lbs. Should the hospital be concerned about
low birth weights in their community?
SEC. 10.4: COMPARING TWO INDEPENDENT
POPULATION PROPORTIONS
When conducting a hypothesis test that compares two independent
population proportions, the following characteristics should be
present:
1. The two independent samples are simple random samples that
are independent.
2. The number of successes is at least five, and the number of
failures is at least five, for each of the samples.
3. Growing literature states that the population must be at least ten
or 20 times the size of the sample. This keeps each population
from being over-sampled and causing incorrect results.
On your calculator use the 2-PropZTest.
EXAMPLE
Suppose the Acme Drug Company develops a new drug, designed to
prevent colds. The company states that the drug is equally effective
for men and women. To test this claim, they choose a simple random
sample of 100 women and 200 men from a population of 100,000
volunteers.
At the end of the study, 41 of the women caught a cold; and 98 of the
men caught a cold. Based on these findings, can we reject the
company's claim that the drug is equally effective for men and
women? Use a 0.05 level of significance.
QUIZ REVIEW: CONFIDENCE INTERVALS
1. Suppose average pizza delivery times are normally distributed with an unknown
population mean and a population standard deviation of six minutes. A random
sample of 28 pizza delivery restaurants is taken and has a sample mean delivery
time of 36 minutes. Find a 90% confidence interval for the mean and the error
bound.
2. Suppose that 14 children, who were learning to ride two-wheel bikes, were
surveyed to determine how long they had to use training wheels. It was revealed
that they used them an average of six months with a sample standard deviation of
three months. Assume that the underlying population distribution is normal.
Construct a 95% confidence interval for the mean and find the error bound.
3. Suppose that a market research firm is hired to estimate the percent of adults
living in a large city who have cell phones. Five hundred randomly selected adult
residents in this city are surveyed to determine whether they have cell phones. Of
the 500 people surveyed, 421 responded yes they own cell phones. Using a 95%
confidence level, compute a confidence interval estimate for the true proportion of
adult residents of this city who have cell phones.
QUIZ REVIEW: HYPOTHESIS TESTS
1. A study is done to determine if Company A retains its workers longer
than Company B. Company A samples 15 workers, and their average
time with the company is five years with a standard deviation of 1.2.
Company B samples 20 workers, and their average time with the
company is 4.5 years with a standard deviation of 0.8. The populations
are normally distributed.
a. Are the population standard deviations known?
b. Conduct an appropriate hypothesis test. At the 5% significance level,
what is your conclusion?
2. Suppose a consumer group suspects that the proportion of households
that have three cell phones is 30%. A cell phone company has reason to
believe that the proportion is not 30%. Before they start a big advertising
campaign, they conduct a hypothesis test. Their marketing people survey
150 households with the result that 43 of the households have three cell
phones.
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