Class 36: Topic 22: Comparing Two Means

Held: Monday, 28 April 2008

Summary: We consider tests that can be used to determine what samples of two populations reveal about the differences of means between those two populations.

Notes:

• Due Wednesday: 22-5, 22-6, 22-9, 22-14, 22-27.

• Handouts: Applet: Test of Significance Calculator; R Notes for Topic 22.
• Due: 20-8, 20-9, 20-18, 20-20.

Overview:

• Comparing two means.
• A complication: Degrees of freedom.
• Using R.

Comparing Two Means

• Much like comparing two proportions.
• We typically call this test a two-sample t-test.
• Null hypothesis is typically the two means are the same.
• Three possible alternative hypotheses (which affect how we assign p-values.
  • The first is greater than the second
  • The first is less than the second
  • The two differ
• We use a different formula for the standard error: sqrt(s1^2/n1+s2^2/n2).
• We look up the p-value in the t-table, rather than than the standard normal probabilities table.
  • Remember, the standard error for means seems to vary more from the standard deviation than does the standard error for proportions.
  • Interestingly, the t-distribution is not a precise representation of this sampling distribution. It is, however, close enough.
• We use t^* rather than z^* for computing confidence intervals.

Degrees of Freedom

• Our book notes that the authors seem to compute degrees of freedom differently than do many statistics packages.
  • This is an effect of the use of the t-table to approximate the distribution.
• When you're doing two-sample t-tests by hand, use the smaller of n₁-1 and n₂-1.
• Most computers use a different formula.

  df = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1))
  

• No, I don't know where that comes from.

Using R

• If you have the original samples, you can use a two-sample t-test in R with t.test.

  t.test(Sample1,Sample2)
  

• You can also use t.test to compute a confidence interval.

  t.test(Sample1,Sample2,conf.level=.##)
  

• If you simply know the test statistic and degrees of freedom, you can compute the p-value with pt.
• You can also compute t^* values using qt, but, once again, you need to supply the area of the curve to the left of that value. For a 95% confidence interval and 5 degrees of freedom, we would write

  qt(.975,df=5)