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UWHC Scholarly Forum April 17, 2013 Ismor Fischer, Ph.D. UW Dept of Statistics,

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Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

April 17, 2013

IsmorFischer, Ph.D.

UW Dept of Statistics,

UW Dept of Biostatistics

and Medical Informatics

UWHC Scholarly Forum

April 17, 2013

IsmorFischer, Ph.D.

UW Dept of Statistics,

UW Dept of Biostatistics and Medical Informatics

All slides posted at http://www.stat.wisc.edu/~ifischer/UWHC

Click on image for full .pdf article

- Links in article to access datasets

POPULATION

Study Question:

Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4yrs old)?

Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

“population standard deviation”

“population mean”

- symmetric about its mean

Example: Body Temp (°F)

low

variability

98.6

- unimodal (i.e., one peak),
- with left and right “tails”

- models many (but not all)
- naturally-occurring systems

- useful mathematical
- properties…

“population standard deviation”

“population mean”

- symmetric about its mean

Example: Body Temp (°F)

low

variability

98.6

Example: IQ score

high

variability

100

- unimodal (i.e., one peak),
- with left and right “tails”

- models many (but not all)
- naturally-occurring systems

- useful mathematical
- properties…

“population standard deviation”

95%

2.5%

2.5%

≈ 2 σ

≈ 2 σ

“population mean”

- symmetric about its mean

Approximately 95% of the population values are contained between

– 2σ and + 2σ.

- unimodal (i.e., one peak),
- with left and right “tails”

- models many (but not all)
- naturally-occurring systems

95% is called the confidence level.

5% is called the significance level.

- useful mathematical
- properties…

POPULATION

via… “Hypothesis Testing”

Study Question:

Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

cannot be found with 100% certainty, but can be estimated with high confidence (e.g., 95%).

H0: pop mean age = 25.4

(i.e., no change since 2010)

“Null Hypothesis”

POPULATION

via… “Hypothesis Testing”

Study Question:

Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

Present Day: Assume “Mean Age at First Birth” follows a normal distribution (i.e., “bell curve”) in the population.

T-test

H0: pop mean age = 25.4

(i.e., no change since 2010)

Random Sample

size n = 400 ages

“Null Hypothesis”

x4

x1

x3

FORMULA

x2

x5

sample mean age

… etc…

?

x400

Do the data tend to support or refute the null hypothesis?

Is the difference STATISTICALLY SIGNIFICANT, at the 5% level?

95%

2.5%

2.5%

≈ 2 σ

≈ 2 σ

Approximately 95% of the population values are contained between

– 2σ and + 2σ.

Approximately 95% of the intervals from

to

contain , and approx 5% do not.

Approximately 95% of the sample mean values are contained between

and

POPULATION

via… “Hypothesis Testing”

Study Question:

Has “Mean (i.e., average) Age at First Birth” of women in the U.S. changed since 2010 (25.4 yrs old)?

“Null Hypothesis”

H0: pop mean age = 25.4

(i.e., no change since 2010)

FORMULA

SAMPLE

n = 400 ages

Approximately 95% of the intervals from

to

contain , and approx 5% do not.

x4

x1

x3

x2

x5

sample mean

… etc…

= 25.6

x400

PROBLEM!

σis unknown the vast majority of the time!

95% margin of error

POPULATION

via… “Hypothesis Testing”

Study Question:

“Null Hypothesis”

H0: pop mean age = 25.4

(i.e., no change since 2010)

FORMULA

SAMPLE

n = 400 ages

x4

x1

sample variance

x3

= modified average of the squared deviations from the mean

x2

x5

sample mean

… etc…

= 25.6

x400

sample standard deviation

95% margin of error

POPULATION

via… “Hypothesis Testing”

Study Question:

“Null Hypothesis”

H0: pop mean age = 25.4

(i.e., no change since 2010)

FORMULA

SAMPLE

n = 400 ages

x4

x1

sample variance

x3

x2

x5

sample mean

… etc…

= 25.6

x400

sample standard deviation

95% margin of error

= 1.6

= 0.16

= 0.16

= 0.16

25.44

25.76

BASED ON OUR SAMPLE DATA, the true value of μ today is between 25.44 and 25.76 years, with 95% “confidence” (…akin to “probability”).

Two main ways to conduct a formal hypothesis test:

95% CONFIDENCE INTERVAL FOR µ

25.44

25.76

BASED ON OUR SAMPLE DATA, the true value of μ today is between 25.44 and 25.76 years, with 95% “confidence” (…akin to “probability”).

IF H0 is true, then we would expect a random sample mean that is at least 0.2 years away from = 25.4 (as ours was), to occur with probability 1.24%.

“P-VALUE” of our sample

Very informally, the p-value of a sample is the probability (hence a number between 0 and 1) that it “agrees” with the null hypothesis.

Hence a very small p-value indicates strong evidence against the null hypothesis. The smaller the p-value, the stronger the evidence, and the more “statistically significant” the finding.

Two main ways to conduct a formal hypothesis test:

- FORMAL CONCLUSIONS:
- The 95% confidence interval corresponding to our sample mean does not contain the “null value” of the population mean, μ = 25.4 years.
- The p-value of our sample, .0124, is less than the predetermined α = .05 significance level.
- Based on our sample data, we may (moderately) reject the null hypothesis H0: μ = 25.4 in favor of the two-sided alternative hypothesis HA: μ ≠ 25.4, at the α = .05 significance level.
- INTERPRETATION: According to the results of this study, there exists a statistically significantdifference between the mean ages at first birth in 2010 (25.4 years old) and today, at the 5% significance level. Moreover, the evidence from the sample data would suggest that the population mean age today is significantly older than in 2010, rather than significantly younger.

95% CONFIDENCE INTERVAL FOR µ

25.44

25.76

BASED ON OUR SAMPLE DATA, the true value of μ today is between 25.44 and 25.76 years, with 95% “confidence” (…akin to “probability”).

IF H0 is true, then we would expect a random sample mean that is at least 0.2 years away from = 25.4 (as ours was), to occur with probability 1.24%.

“P-VALUE” of our sample

Very informally, the p-value of a sample is the probability (hence a number between 0 and 1) that it “agrees” with the null hypothesis.

Hence a very small p-value indicates strong evidence against the null hypothesis. The smaller the p-value, the stronger the evidence, and the more “statistically significant” the finding.

POPULATION

via… “Hypothesis Testing”

Study Question:

T-test

Two loose ends

H0: pop mean age = 25.4

(i.e., no change since 2010)

Random Sample

size n = 400 ages

“Null Hypothesis”

x4

x1

x3

FORMULA

x2

x5

sample mean age

… etc…

x400

Do the data tend to support or refute the null hypothesis?

Is the difference STATISTICALLY SIGNIFICANT, at the 5% level?

POPULATION

via… “Hypothesis Testing”

Study Question:

T-test

Two loose ends

H0: pop mean age = 25.4

(i.e., no change since 2010)

“Null Hypothesis”

Check?

The reasonableness of the normality assumption is empirically verifiable, and in fact formally testable from the sample data. If violated (e.g., skewed) or inconclusive (e.g., small sample size), then “distribution-free” nonparametric tests can be used instead of the T-test.

Examples: Sign Test, Wilcoxon Signed Rank Test (= Mann-Whitney Test)

POPULATION

via… “Hypothesis Testing”

Study Question:

T-test

Two loose ends

H0: pop mean age = 25.4

(i.e., no change since 2010)

Random Sample

size n = 400 ages

“Null Hypothesis”

x4

x1

x3

x2

Sample size npartially depends on the power of the test, i.e., the desired probability of correctly rejecting a false null hypothesis.HOWEVER……

x5

… etc…

x400

Approximately 95% of the population values are contained between

– 2σ and + 2σ.

“population standard deviation”

95%

2.5%

2.5%

≈ 2 σ

≈ 2 σ

“population mean”

Samples, size n

Approximately 95% of the intervals from

to

contain , and approx 5% do not.

Approximately 95% of the sample mean values are contained between

and

… etc…

Approximately 95% of the population values are contained between

– 2sand + 2s.

“population standard deviation”

95%

2.5%

2.5%

≈ 2 σ

≈ 2 σ

“population mean”

Samples, size n

Approximately 95% of the intervals from

to

contain , and approx 5% do not.

Approximately 95% of the sample mean values are contained between

and

…IFn is large, 30 traditionally.

But if n is small…

… this “T-score" increases (from ≈ 2 to a max of 12.706 for a 95% confidence level) as n decreases larger margin of error less power to reject.

… etc…

If n is large, T-score ≈ 2.

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