Review - Sum of Normal Random Variables

1
Review - Sum of Normal Random Variables

• The weighted sum of two independent normally
  distributed random
• variables has a normal distribution.
• Example

2
Two Sample Problem

• Sometimes we will be interested in comparing
  means in two
• independent populations (e.g. mean income for
  male and females).
• We consider two basic scenarios populations are
  normal or
• populations are not normal but the sample sizes
  are large.

3
CI for Difference between Two Means Case I

• Suppose we have two independent large random
  samples from two
• different populations and we are interested in
  comparing the means
• of these two populations.
• The confidence interval for the difference
  between two means in this case is

4
Example
5
Tests for Difference between Two Means

• Suppose we have two independent large random
  samples from two
• different populations.
• We are interested in testing H0 µx - µy d
  versus a one sided or a two sided alternative
• Then.

6
Example
7
Minitab CI and Hypothesis Tests for Two Means

• We can use Minitab to construct confidence
  intervals, conduct
• hypothesis tests, for the difference between two
  means, two independent
• large iid samples.
• The command in Minitab is
• Stat ? basic statistic ? 2 sample t ? complete
  dialogue box ? o.k.
• Note that the data can be in two different
  columns or in one column
• with another column denoting the population of
  each observation.
• We can also use Minitab to calculate the power of
  the two sample
• hypothesis test.
• The command in Minitab is
• Stat ? power and sample size ? 2 sample t ?
  complete dialogue ? o.k