Hypothesis tests applied to means: Two independent samples

1
Hypothesis tests applied to means Two
independent samples

• Chapter 14

2
Independent Samples

• Independent samples are samples drawn from two
  different populations
• The t-tests we have done up to this point have
  used only one population
• When hypothesis testing using independent samples
  we are interested in differences between the 2
  groups
• H1 ?1 ? ?2
• For the rest of the course we are only using 2
  tailed tests
• Then, the null is that there is no difference
• H0 ?1 ?2

3
Example

• We are interested in attitudes held by Athens
  residents towards Wal-Mart coming to Athens. We
  want to know if there is a difference in
  attitudes between OU students and local residents
• Hypotheses
• H1 ?OU Students ? ?Local Residents
• H0 ?OU Students ?Local Residents

4
Distribution of Differences between Means

• The distribution of the differences between means
  over repeated sampling from the same
  population(s)
• Mean of this distribution is (?1 - ?2)
• Distribution of differences should be normal
• Variance Sum Law
• The rule giving the variance of the sum (or
  difference) of 2 or more variables
• The variance of the sum or difference of 2
  independent variables is equal to the sum of
  their variances

5
Variance Sum Law

• From the central limit theorem we know that the
  variance of a sampling distribution is ?2 / n
• Using the variance sum law, we can say the
  variance of the differences is
• ?2X1-X2 ?12 ?22
• n1 n2
• Being this is a t-test we do not know ?, so
• s2X1-X2 s12 s22
• n1 n2
• The square root gives us the standard error of
  the differences

6
T-test for independent samples

• t (X1 - X2) - (?1 - ?2)
• sX1- sX2
• Because ?1-?2 0, this can be simplified to
• t X1 - X2
• sX1- sX2
• Where sX1- sX2 is
• s21 s22
  df n-1
• n1 n2

7
Example 1

• Suppose I want to see if what t.v. show you watch
  influences your mood. I randomly assign people to
  2 groups one will watch Friends and one will
  watch Survivor. The mood of each group was then
  recorded after the show was finished (higher
  scores indicate a better mood). Set ?.05.
• Friends n20 X93 s236
• Survivor n20 X87 s236

8
Unequal Ns

• What if samples have different n?
• In this case we pool the variance
• An assumption for independent t-test is that the
  samples come from populations with equal
  variances
• This is called Homogeneity of Variance
• When we have unequal sample sizes, we need to get
  a weighted average of each sample variance
• We weight each by its degrees of freedom
• We call this average the pooled variance

9
Weighted Average

• Degrees of freedom for pooled variance
• df n1 n2 - 2
• Our pooled variance is
• sp2 (n1 - 1) s12 (n2 - 1) s22
• n1 n2 - 2
• Once we have our pooled variance, we can plug it
  into our t-test.
• t X1 - X2
• sp2 sp2
• n1 n2

10
Example 2

• A researcher wants to know if taking vitamins
  affects memory. We get 2 groups of people. The
  first group of 25 people is given a vitamin. The
  second group of 35 people is given a placebo
  pill. Both group are then given a list of 100
  words to memorize. We recorded the number of
  words they remembered. Set ?.01.
• The mean number of words the vitamin group
  remembered was 40.5 (with a SD of 8.5). The mean
  number of words for the placebo group was 29.5
  (with a SD of 13.5).

11
Unequal Variances

• Heterogeneity of variance
• A situation in which samples are drawn from
  populations having different variances
• General rule
• If 1 sample variance is no more than 4 times the
  other and sample sizes are pretty much the same,
  go ahead and do the t-test using pooled variance
• If one variance is more than 4 times larger
• Do not pool the variance, get separate estimates
• Use df from the smaller variance for critical
  values

12
Example 3

• A researcher want to know caffeine pills really
  work. We randomly assign 30 people to a caffeine
  pill group and 22 people to a placebo group. We
  measure how long they stay awake. Data are as
  follows
• Caffeine group X 23.3 s21
• Placebo group X 16.6 s25
• Set ?.05

13
Confidence Intervals for ?1 - ?2

• CI (X1 - X2) ? (tcritical sX1 - X2)
• Step 1 Plug in X1 - X2 and sX1 - X2
• Step 2 Look up the critical value
• Remember it is always 2-tailed
• Step 3 Compute the upper lower limit
• Step 4 Put in the following form
• CILower ? ?1 - ?2 ? CIUpper

14
Example 4

• Compute the 95 confidence interval for Example 1.