Confidence Intervals

1
Confidence Intervals

• WW, Chapter 8

2
Confidence Interval Review

• For the mean
• If ? is known
• ? M /- Z?/2(?/?N)
•  
• If ? is unknown and N is large (? 100)
• ? M /- Z?/2(s/?N)
•  
• If ? is unknown and N is small (lt 100)
• ? M /- t?/2(s/?N)

3
Confidence Interval for the Proportion

• If ? is known
• ? P ? Z?/2??(1-?)/n
•  
• If ? is unknown and n is large (at least 5
  successes and 5 failures turn up)
• ? P ? Z?/2?P(1-P)/n

4
Difference in Means

• Many times we are interested in comparing the
  means across groups, such as the mean GDP level
  for developed versus developing countries. To
  construct a confidence interval when the
  population variances are known
•  
• (?1 - ?2) (M1 - M2) /- Z?/2 ?(?12/n1 ?22/n2)

5
Difference of Means, Independent Samples

• It is usually the case that we are comparing
  means from two random samples and ?12 and ?22 are
  not known.
•  
• (?1 - ?2) (M1 - M2) /- t?/2 sp?(1/n1 1/n2)
•  
• where sp pooled variance (we assume that both
  populations have the same variance so that we can
  pool the samples together, or ?12 ?22)

6
Pooled Variance

• sp2 ?(X1 - M1)2 ?(X2 - M2)2
• (n1 - 1) (n2 - 1)
• df (n1 - 1) (n2 - 1)
• Example From a large class, a sample of 4 grades
  were drawn and from a second large class, an
  independent sample of 3 grades were drawn.
  Calculate the 95 confidence interval for the
  difference between the two class means, ?1 - ?2.

7
Example

• Class 1 Class 2
• X1 (X1 - M1)2 X2 (X2 - M2)2
• 64 (64-74)2 100 56 (56-60)2 16
• 66 (66-74)2 64 71 (71-60)2 121
• 89 (89-74)2 225 53 (53-60)2 49
• 77                (77-74)2 9
•  
• M1 296/4 74 M2 180/3 60
• ?(X1 - M1)2 398 ?(X2 - M2)2 186

8
Example

• sp2 398 186 584/5 117
• (4-1) (3-1)
• sp ?117 10.8
•  
• (?1 - ?2) (M1 - M2) /- t?/2 sp?(1/n1 1/n2)
• (74 - 60) /- 2.5710.8?(1/4 1/3)
•  
• df (n1 - 1) (n2 - 1) (4 - 1) (3 - 1) 5

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Interpretation

• (?1 - ?2) 14 /- 21 or -7 to 35
•  
• With 95 confidence, we can conclude that the
  average of the first class may be 7 marks below
  the average of the second class, or it may be 35
  marks above, or anywhere in between.

10
Difference of Means, Matched or Paired Samples

• In the previous example, the samples were
  independently drawn (two separate classes).
  Often we want to compare two variables that are
  not drawn independently. This is what we will
  refer to as matched, or paired samples.
• Suppose we want to compare the same students
  across 2 exams. We want to know how the
  students' grades changed or the difference
  between the two exam scores.
•  
• D X1 - X2
•  
• We can treat the differences in scores as a
  single sample.

11
Difference of Means, Matched or Paired Samples

• A confidence interval for matched samples is
  calculated as follows
• Dmean /- t?/2(sD/?N)
• where ? is the difference between the two scores
  in the populations, and Dmean is the average
  difference in our sample.
•  
• SD2 ?(D - Dmean)2
• N -1
•  which is a standard calculation for variance

12
Example

• Student X1 X2 DX1-X2 D - Dmean (D - Dmean)2
• Amy 64 57 7 -4 16
• Bill 66 57 9 -2 4
• Becky 89 73 16 5 25
• Mark 77 65 12 1 1
•  
• Dmean (791612)/4 44/4 11
•  
• SD2 (164251)/(4-1) 46/3 15.3
• SD ?15.3 3.9
•  
• Df N - 1 4 - 1 3

13
Example

• Dmean /- t?/2(sD/?N)
• 11 /- 3.18(3.9/?4) 11 /- 6 or 5 - 17
• Conclusion The difference in the two exams
  ranges from 5 to 17 points.
• Notice that the confidence interval shrinks when
  we are using paired or matched samples. This is
  because we are holding constant many extraneous
  variables (like year in college, IQ, hours spent
  studying, etc).

14
 Difference in two proportions for large samples

• (?1 - ?2) (P1 - P2) /- Z?/2? (P1(1-P1)/N1)
  (P2(1-P2)/N2)
• Example The Gallup poll periodically takes a
  random sample of about 1500 Americans. The
  percentage who favor the legalization of
  marijuana possession declined from 52 in 1980 to
  46 in 1985.

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Example

• Construct a 95 confidence interval for the
  population percentage in favor each year.
• 1980 ? P /- Z?/2(?(P(1-P)/N)
• .52 /- 1.96(?(.52)(.48)/1500) .52 /- .025
• .495 to .545 49.5 to 54.5 of people favored
  legalization
• 1985 ? P /- Z?/2(?(P(1-P)/N)
• .46 /- 1.96(?(.46)(.54)/1500) .46 /- .025
• .435 to .485 43.5 to 48.5 of people favored
  legalization

16
Example

• Find a 95 confidence interval for the change in
  this percentage.
• (?1 - ?2) (P1 - P2) /- Z?/2? (P1(1-P1)/N1)
  ((P2(1-P2)/N2)
• (.52 - .46) /- 1.96? ((.52)(.48)/1500)
  (.46)(.54)/1500))
• .06 /- .036
• We are 95 confident that the difference in
  legalization attitudes between 1985 and 1980 is
  between 2.4 and 9.6.