Statistical Sciences 2244A/B Chapter Notes - Chapter 18: Sampling Distribution, Standard Deviation, Royal Society Of Biology
22 May 2018
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Stats 2244
Chapter 18
CHAPTER 18.1
Two-sample problems
- A two-sample problem can arise from a randomized comparative experiment that randomly
divides subjects into two groups and exposes each group to a different treatment
- Comparing random samples selected separately from two populations is also a two-sample
problem. Unlike the matched pairs designs studied earlier, there is no matching of
the individuals in the two samples, and the two samples can be of different sizes. Inference
procedures for two-sample data differ from those for matched pairs
- Here are some typical two-sample problems.
o Does regular physical therapy help with lower-back pain? A randomized experiment
assigned patients with lower-back pain to two groups: 142 received an examination and
advice from a physical therapist; another 144 received regular physical therapy for up
to five weeks. After a year, the change in their level of disability (0% to 100%) was
assessed by a doctor who did not know which treatment the patients had received.
o A field biologist observes gender-based behavior in wild chimpanzees. Twelve randomly
chosen young chimpanzees are tagged remotely with a dart and their behavior is
monitored. The amount of time each young chimpanzee spends in contact with its
mother is recorded. The biologist then compares the amount of time spent in contact
with the mother by young male and by young female chimpanzees.
o A physiologist compares the fermentation rates of yeast as it metabolizes glucose (a
simple carbohydrate) or starch (a complex carbohydrate). Live yeast suspensions are
placed in 20 fermentation flasks. Ten of the flasks contain a solution of water and
glucose, and the other 10 flasks contain a solution of water and starch. The volume of
CO2emitted per minute is measured for each flask.
CHAPTER 18.2
Comparing two population means
- Comparing two populations or the responses to two treatments starts with data analysis:
o Make boxplots, stemplots (for small samples), or histograms (for larger samples) and
compare the shapes, centers, and spreads of the two samples.
- The most common goal of inference is to compare the average or typical responses in the two
populations.
- When data analysis suggests that both population distributions are symmetric, and especially
when they are at least approximately Normal, we want to compare the population means.
- Here are the conditions for inference when comparing two means.
- Conditions for inference comparing two means:
o we have 2 SRSs, from 2 distinct populations
▪ the samples are independent →one sample has no influence on the other or
observations in one sample has no connection to observations in the other (ex:
matching violates independence)
▪ we measure the same variable for both samples
o both populations are normally distributed
▪ the means and standard deviations of the populations are unknown
- THIS IS WHAT WAUGH WROTE IN THE TEXTBOOK:
o The assumptions for the two-sample (independent samples) test of differences in means
are:
▪ We have 2 SRSs (SRS1 and SRS2)
▪ The sampling distribution of sample means for mu1 and the sampling distribution
of samplemeans for mu2 are at least approx. normally distributed
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• this will be the case if…
o population 1 and population 2 are both normal ORRR
o n1 and n2 are both large so that CLT holds for both sampling
distributions
- Call the variablewe measure x1 in the first population and x2 in the second, because the variable
may have different distributions in the two populations. Here is the notation we will use to
describe the two populations:
- There are four unknown parameters, the two means and the two standard deviations. The
subscripts remind us which population a parameter describes. We want to compare the two
population means, either by giving a confidence interval for their difference μ1−μ2 or by testing
the hypothesis of no difference, H0: μ1 = μ2, which is the same as H0: μ1 − μ2 = 0.
- We use the sample means and standard deviations to estimate the unknown
parameters. Again, subscripts remind us which sample a statistic comes from. Here is the
notation that describes the samples:
- To do inference about the difference μ1 − μ2 between the means of the two populations, we start
from the difference − between the means of the two samples.
CHAPTER 18.3
Two-sample t procedures
- Whether an observed difference in sample means is surprising depends on the spread of
the individual observations as well as on the two means
- Widely different means can arise just by chance if the individual observations vary a great deal
- How much the difference − can vary from sample to sample is given by its sampling
distribution
- When two random variables are Normally distributed, the new variable difference in sample
means also follows a Normal distribution, centered on the difference in the two variables’
means and with variance equal to the sum of the two variables’ variances.
- Recall: the sampling distributions of and have standard deviations and
, respectively
- Therefore, when we look at the difference − , the standard deviation of its sampling
distribution is
- This standard deviation gets larger as either populationgets more variable, that
is, as σ1 or σ2 increases. It gets smaller as the sample sizes n1 and n2 increase.
- Because we don’t know σ1 and σ2, we estimate them by the sample standard
deviations s1 and s2. The result is the standard error, or estimated standard deviation, of the
difference in sample means:
- When we standardize the estimate, we get
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Document Summary
A two-sample problem can arise from a randomized comparative experiment that randomly divides subjects into two groups and exposes each group to a different treatment. Comparing random samples selected separately from two populations is also a two-sample problem. Unlike the matched pairs designs studied earlier, there is no matching of the individuals in the two samples, and the two samples can be of different sizes. Inference procedures for two-sample data differ from those for matched pairs. A randomized experiment assigned patients with lower-back pain to two groups: 142 received an examination and advice from a physical therapist; another 144 received regular physical therapy for up to five weeks. After a year, the change in their level of disability (0% to 100%) was assessed by a doctor who did not know which treatment the patients had received: a field biologist observes gender-based behavior in wild chimpanzees.
