Estimation

1
Estimation

• Overview of estimation
• Purpose to estimate population parameters such
  as the population mean using a sample of data.
  For example, you may want to describe a typical
  Canadian family on a number of attributes.
• Approach take a sample, determine its mean, and
  then use that estimate and other information from
  the sample to determine the population mean

2
Estimation

• Overview
• process is based on sampling, hence there is
  sampling error associated with it this means
  that the sample estimate will differ from the
  population estimate, and that any given sample is
  part of a sampling distribution
• Two different types of estimates point estimate
  and interval estimate

3
Estimation

• Overview
• point estimate single number used to estimate an
  unknown parameter
• interval estimate range of values used to
  estimate an unknown quantity interval estimate
  accompanied by a level of confidence, it is
  called a confidence interval

4
Estimation

• Overview
• Estimation and hypothesis testing are similar in
  that they both use similar data to address
  related questions
• Difference is in the question addressed.
  Hypothesis testing determines whether or not
  treatment has an effect. Estimation determines
  magnitude of the effect

5
Estimation

• Overview
• When to use estimation
• Use estimation procedures when you want to
  determine the size of an effect
• E.g., when null hypothesis is rejected, when you
  are interested in knowing the size of an effect
  because you want to be able to describe an
  unknown population, when you want to gain a sense
  of the practical or clinical significance of a
  treatment

6
Estimation

• Overview
• Logic of estimation
• Both hypothesis testing and estimation are
  inferential statistics
• Both use samples to draw conclusions about
  populations
• Both attempt to take into consideration the
  discrepancy or error between a statistic and a
  population parameter

7
Estimation

• Point estimation with a z score
• Recall a z score is used when the population mean
  is unknown but the population SD is known

8
Estimation

• Point estimation with a z score
• Recall whenever, a sample of observations are
  obtained, one can think of the sample as
  belonging to a sampling distribution
• In the case of a z score distribution, the mean
  is 0 and the SD 1
• For a given sample with M X, the most probable
  outcome is that M ?. Why

9
Estimation

• Point estimation with a z score
• Recall Z (M - µ)/sM
• When z 0, this is the highest probability
  outcome. Z 0 when M µ
• Example. Evaluate a program designed to improve
  reading. A test is used to determine whether
  reading improved. The test designed to evaluate
  reading performance of the (n 25) students in
  the reading program has ? 80, ? 10.
  Performance on the test after being in the
  training program was M 88.
• Goal. Estimate ? after being in the reading
  program.

10
Estimation

• Point estimation with a z score
• Z (M - µ)/sM 0 (M - µ)/sM Thus, M µ

11
Estimation

• Interval estimation with a z score
• Want to determine a range of values as an
  estimate of an unknown quantity
• E.g., the high temperature today is 20-22 degrees
  centigrade
• E.g., it will take me 45 60 minutes to drive to
  Mississauga

12
Estimation

• Confidence interval with a z score
• In a confidence interval, a range of scores is
  presented with a confidence interval
• The length of time it takes to drive to
  Mississauga is 45 60 minutes with a 95
  confidence interval
• This means that mean time it takes to drive to
  Mississauga will be in this range of scores 95
  of the time

13
Estimation

• Confidence interval with a z score
• Determining a confidence interval from a sample
  of scores in which the ? is unknown but ? is
  known
• This sample is part of a z sampling distribution
• Recall Z (M - µ)/sM
• if we want to construct a 95 confidence
  interval, we can determine the z score associated
  with the top and bottom 2.5. That value is z
  1.96

14
Estimation

• Confidence interval with a z score
• Z (M - µ)/sM
• Now need to determine upper and lower values of µ
• First step, isolate µ
• ZsM (M - µ)
• µ M - zsM

15
Estimation

• Confidence interval with a z score
• The test designed to evaluate reading performance
  of the (n 25) students in the reading program
  has ? 80, ? 10. Performance on the test
  after being in the training program was M 88.
• sM ?/vn 10/5 2
• One extreme z 1.96
• µ M - zsM
• µ 88 1.96 (2) 84.08

16
Estimation

• Confidence interval with a z score
• other extreme z -1.96
• µ M - zsM
• µ 88 1.96 (2) 91.92
• Hence we conclude that mean performance after the
  reading program falls between 84.08 and 91.92
  with 95 confidence

17
Estimation

• Confidence interval with a z score (Summary)
• In order to calculate a 95 confidence interval
  in a situation in which the SD is known, take the
  sample mean, then add or subtract the sampling
  error
• µ M zsM
• Population mean sample mean error

18
Estimation

• Confidence interval with t statistics
• Same process as with z statistic
• Pop. Mean sample mean t (standard error)
• Mean diff mean diff t (standard error)
• Estimation procedure. Use sample data to compute
  sample mean (or mean difference). Use that
  estimate to compute the population mean. Why most
  likely value for t 0. Then as before determine
  for a given confidence level the t scores, and
  use the formulae shown in the next slide to
  calculate the range

19
Estimation

• Confidence interval with a z score (Summary)
• In order to calculate a 95 confidence interval
  in a situation in which the SD is known, take the
  sample mean, then add or subtract the sampling
  error
• µ M zsM
• Population mean sample mean error

20
Estimation
21
Estimation

• Confidence interval with single sample scores in
  which pop. SD is unknown
• e.g., Determine mean age of people who purchase
  jeans. Sample obtained of 30 jean purchasers.
  Mean age of sample M 30.5 SS 709.
  Calculate point estimate and 95 confidence
  interval

22
Estimation

• Confidence interval with single sample scores in
  which pop. SD is unknown
• ? M tsM
• M 30.5
• sM vs2 /n, where s2 SS/(n-1)
• S2 709/29 24.45
• sM vs2 /n v24.45/30 .90

23
Estimation

• Confidence interval with single sample scores in
  which pop. SD is unknown
• ? M tsM
• Point estimate of ? is M or 30.5. Why? Hint, what
  is the mean of t? see page 284
• 95 confidence interval for t (29) is 2.045
• ? M tsM 30.5 2.045 .90 30.5 1.84
• ? 28.66 to 32.34
• Conclusion population mean of jean consumers
  falls between 28.66 and 32.34 with 95 confidence

24
Estimation

• Confidence interval for independent measures
  studies
• ?1 - ?2 X1 X2 ts m1 m2
• the logic is the same, only here the goal is to
  find a point estimate of the difference of the
  population means and a confidence interval for
  the same difference

25
Estimation

• Confidence interval for independent measures
  studies
• Example. Comparison of two different diets. DV
  learning performance of 2 samples

26
Estimation

• Confidence interval for independent measures
  studies
• ?1 - ?2 X1 X2 ts m1 m2
• Point estimate X1 X2 33-25 8
• s m1 m2 v(sp2/n1 sp2/n2)
• Where sp2 (SS1 SS2)/(df1 df2)
• sp2 (250 140)/(9 4) 390/13 30

27
Estimation

• Confidence interval for independent measures
  studies
• s m1 m2 v(sp2/n1 sp2/n2)
• s m1 m2 v(30/10 30/5)
• v(36) 3
• ?1 - ?2 X1 X2 ts m1 m2
• ?1 - ?2 8 t 3

28
Estimation

• Confidence interval for independent measures
  studies
• ?1 - ?2 8 t 3
• Suppose you want to compute an 80 confidence
  interval, then look up in table for
• t with df 10 -1 5 -1 13
• t (13) 1.35
• ?1 - ?2 8 1.35 3 8 4.05
• Conclusion. 80 confidence interval for mean
  increase in performance is 8 4.05

29
Estimation

• Confidence interval for repeated measures studies
• Logic is the same. Want to determine confidence
  interval for D, the difference between two
  related (or repeated) measures
• e.g. Purpose to determine effectiveness of a
  reading program. Test n16 students on a test of
  reading comprehension administer reading
  program re-administer test of reading
  comprehension
• Results Dbar 21 SS for difference scores
  1215. Construct point estimate of ?D and a 90
  confidence interval
• ?D Dbar tsDbar

30
Estimation

• Confidence interval for repeated measures studies
• ?D Dbar tsDbar
• sDbar vs2/n, where s2 SS/ (n-1) 1215/15 81
• sDbar vs2/n v81/16 2.25
• ?D Dbar tsDbar 21 t (2.25)
• But t (15) 1.753
• ?D 21 1.75 (2.25)
• 21 3.94
• Conclusion 90 confidence interval for mean
  improvement in reading comprehension is 21 3.94

31
Estimation

• Confidence interval width
• Width is affected by the sample size larger the
  sample size the smaller the confidence interval
• Width is also affected by the magnitude of the
  confidence higher the confidence greater the
  width