Estimation

1
Chapter 17

• Estimation

2
UB Height Example
Is the mean height of UB students 68?
Step 5) Make Decision
Step 6) Interpret Decision

• The mean height of UB students does not equal the
  national average.

• Hypothesis
• H0 mUB 68
• H1 mUB ? 68

• Hypothesis
• H0 mUB 68
• H1 mUB ? 68

• Decision Rule
• Reject H0
• zobt ? -1.96
• zobt ? 1.96

Test Statistic zobt 2.5
Based upon a sampling distribution with m? 68
s? .3
.3
3
Estimation

• Hypothesis Test vs Estimation
• Hypothesis test
• Determine if mreal equals a value
• Estimation
• Give an approximate value for mreal
• Two types of estimates
• Point estimates
• Interval Estimates
• Confidence Intervals

4
Point Estimates

• Giving a single value.

5
UB Height Example
Is the mean height of UB students 68?
Step 5) Make Decision
Step 6) Interpret Decision

• The mean height of UB students does not equal the
  national average.

• Hypothesis
• H0 mUB 68
• H1 mUB ? 68

• Hypothesis
• H0 mUB 68
• H1 mUB ? 68

• Decision Rule
• Reject H0
• zobt ? -1.96
• zobt ? 1.96

Test Statistic zobt 2.5
Based upon a sampling distribution with m? 68
s? .3
.3
6
Estimation

• What are Point Estimates?
• Estimating mreal with a single point.
• mreal ?
• Problem
• Not all ? equal mreal
• Variability amongst ?
• Shouldnt this information be included in your
  estimate?

7
Interval Estimates

• Confidence Intervals

8
Confidence Intervals

• What are interval estimates?
• Estimates that place mreal between two likely
  values.
• What are confidence intervals?
• Interval estimates with a calc-ed prob. of being
  correct.

9
Confidence Intervals

• Logic of Confidence Intervals (based on z
  distributions).
• Assumptions
• Normality
• Representative sample
• s known
• Start with point estimate
• mreal ? (the best guess based upon info)

10
Confidence Intervals

• Logic of Confidence Intervals (based on z
  distributions).
• Determine variability amongst ?s
• s?
• Find cutoff points off confidence region (level)
• Middle 95 centered around ?
• Convert to real scores

11
Confidence Intervals
95 Confidence Interval for Mean Height of UB
Students
95 confident that mean height of UB students is
between 68.32 70.68 inches
12
Confidence Intervals

• General Formula for Confidence Intervals
• ? ? zconfs?

13
Why Confidence Intervals Work

• 95 confident m between Interpretation

14
Why Conf. Intervals Work
15
Why Conf. Intervals Work
Potential Sampling Distributions
16
Why Conf. Intervals Work
If these were the true Sampling Distributions,
what is the probability of selecting this and
getting a false confidence interval?
17
Why Confidence Intervals Work

• 95 of the confident intervals be would be true
  
• Interpretation

18
Changing the Confidence Levels

• 99 Confidence

19
99 Confidence Interval
99 Confidence Interval for Mean Height of UB
Students
99 confident that mean height of UB students is
between 67.95 71.05 inches
20
Changing Confidence Level

• Increasing confidence level increases confidence
  interval

100 Confidence Interval
100 confident that mean height of UB students is
between -? ? inches
1.00
21
Getting a more precise estimate

• Narrow the confidence interval!

22
More Precision

• 2 ways to narrow interval
• Choose lower level of confidence

23
99 Confidence Interval
99 Confidence Interval for Mean Height of UB
Students
99 confident that mean height of UB students is
between 67.95 71.05 inches
24
95 Confidence Intervals
95 Confidence Interval for Mean Height of UB
Students
95 confident that mean height of UB students is
between 68.32 70.68 inches
25
More Precision

• 2 ways to narrow interval
• Choose lower level of confidence
• Increase sample size

26
95 Confidence Intervals
95 Confidence Interval for Mean Height of UB
Students
95 confident that mean height of UB students is
between 68.32 70.68 inches
27
95 Confidence Intervals
95 Confidence Interval for Mean Height of UB
Students
95 confident that mean height of UB students is
between 68.91 70.09 inches
28
More Precision

• 2 ways to narrow interval
• Choose lower level of confidence
• Increase sample size
• n Up gt s? down gt Interval narrow
• n down gt s? Up gt Interval WIDE