Comparing two populations

1
Comparing two populations

• Sometimes we want to compare two populations
  rather making decisions about a single
  population.
• For example, we might want to compare two
  population means or two population proportions to
  see if they are equal.
• Is the expected drying time for one type of paint
  lower than that of another type of paint?
• Is the proportion of republicans who favor
  withdrawing from Iraq higher than the proportion
  of democrats who favor withdrawal?

2
Comparing two population means

• Suppose we have two independent samples, X1,,Xm
  and Y1,,Yn, from two separate populations.
• A natural statistic for comparing the two
  population means, mX and mY, is .
• The distribution of is also Normal for m
  and n both large.

3
Large samples test for comparing population means

• To test H0 mX mY D0, use the test statistic

4
Home sales data

• A realtor in Albuquerque wants to argue that
  houses in the Northeast are more expensive on
  average than those in the rest of town. The
  data below contain sale prices (in 100s) for
  homes in the city. NE 1 indicates a home was
  in the Northeast. NE 0 indicates a home was
  not in the Northeast. Test the appropriate
  hypotheses with a 0.01.

5
Home sales data

• Interpret this p-value

6
Large samples confidence interval for the
difference between two population means

• A large sample (1-a)100 confidence interval for
  mX mY is
• For the home sales data, what is a 99 confidence
  interval for the difference between sale prices
  in the Northeast and the rest of town?
• Home sales data

7
Large sample confidence intervals

• Interpret the confidence interval

8
Equal population variances

• Suppose we assume that the two populations have a
  common variance s2.
• We can then estimate this common variance using
  the pooled sample variance

9
Small samples test for comparing population means
from Normal distributions with equal variances

• To test H0 mX mY D0, use the test statistic

10
THC example with equal variances

• The active component in marijuana is THC. An
  experiment was conducted to compare two slightly
  different configurations of this substance. The
  THC data set contains the time until the effect
  was perceived for 6 subjects exposed to each
  configuration. Is there any evidence that the
  mean time to perception is different between the
  two configurations using a 0.01?

11
Small samples confidence interval for the
difference between two population means

• Assuming equal variances, a small sample
  (1-a)100 confidence interval for mX mY is
• For the THC data, what is a 99 confidence
  interval for the mean difference between the
  detection times for the two configurations?
• THC data set

12
Unequal population variances

• The pooled procedures we have discussed
  previously are fairly robust to the assumption of
  equal variances.
• In other words if the two population variances
  are relatively close, the procedures perform
  well
• The level of significance for the hypothesis test
  is close to what it should be
• The coverage probability for the confidence
  interval is close to what it should be
• If the variances are quite different, then we
  need a different procedure.

13
Small samples test for comparing population means
from Normal distributions with unequal variances

• To test H0 mX mY D0, use the test statistic
• with degrees of freedom

14
THC example with unequal variances
15
Small samples confidence interval for the
difference between two population means

• Assuming unequal variances, a small sample
  (1-a)100 confidence interval for mX mY is
• For the THC data, what is a 99 confidence
  interval for the mean difference between the
  detection times for the two configurations?
• THC data set

16
Paired data

• Sometimes we have a third variable that connects
  elements from the X and Y samples.
• In this case, the assumption of independence
  between the two samples may be violated.
• Is there any evidence that the first twin and the
  second twin have different average weights among
  boy-boy twins?
• In this case, the twins are clearly connected by
  the mother.
• It might be better to base our test on the n
  pairwise differences, Di Xi Yi.

17
Paired test for comparing population means

• To test H0 mX mY D0, use the test statistic

18
Twins example

• Load the Twins data from StatCrunch sample data
  sets. Is there any evidence that Twin A and Twin
  B have different average weights among boy-boy
  twins with a 0.1?
• StatCrunch

19
Paired confidence interval for the difference
between two population means

• A small sample (1-a)100 confidence interval for
  mX mY is
• For the twins data, what is a 90 confidence
  interval for the mean difference between the twin
  A and twin B weights?
• StatCrunch

20
Comparing two population proportions

• A natural statistic for comparing the two
  population proportions, pX and pY, is .
• The distribution of is also Normal
  for m and n both large.

21
Large samples test for comparing population
proportions

• To test H0 pX pY 0, use the test statistic
• where

22
Polio example

• The following table summarizes a study of the
  efficacy of the Salk vaccine.
• Was the vaccine effective? Test at a 0.05.
• StatCrunch

23
Large samples confidence interval for the
difference between two population proportions

• A large sample (1-a)100 confidence interval for
  pX pY is
• For the Polio data, what is a 95 confidence
  interval for the difference between the
  proportion who contract the disease under each
  treatment?
• StatCrunch

24
Comparing two population variances

• Suppose two chemical companies can supply a raw
  material, but we suspect the variability in
  concentration may differ between the two.
• The standard deviation of concentration in a
  random sample of 15 batches from company 1 was
  found to be 4.7 g/l. A sample of 21 batches from
  company 2 yielded a standard deviation of 5.8
  g/l.
• Is there sufficient evidence to conclude that the
  variability in concentration differs for the two
  companies?

25
Test for comparing population variances from
Normal distributions

• To test H0 sX2 sY2, use the test statistic

F calculator
26
Chemical example

• Is there sufficient evidence to conclude that the
  variability in concentration differs for the two
  companies with a 0.05?
• F Calculator

27
Confidence interval for the ratio of two Normal
population variances

• A large sample (1-a)100 confidence interval for
  sX2/sY2 is
• For the THC example, what is a 95 confidence
  interval for the ratio of concentration
  variances?
• THC data set