research methodology & statistics lecture 6: the normal distribution and confidence intervals...

RESEARCH METHODOLOGY & STATISTICSLECTURE 6: THE NORMAL DISTRIBUTION AND CONFIDENCE INTERVALS

MSc(Addictions)

Addictions Department

population

units

sample

From sample to population…

inference

Background to statistical inference

normaldistribution

samplingdistribution

standardnormal

distributionarea under the

curvepercentage

points

confidence intervals

p-values(significance)

RESEARCH METHODS AND STATISTICS

The normal distribution

The normal distribution

Mean = 171.5cm

Standard deviation = 6.5cm

• Reasonable description of most continuous variables – given large enough sample size

The normal distribution

• Reasonable description of most continuous variables

– given large enough sample size• Location determined by the mean• Shape determined by standard deviation• Total area under the curve sums to 1

The standard normal distribution• Has a mean of 0 and a standard deviation of 1

mean standard deviation

The standard normal distribution• Has a mean of 0 and a standard deviation of 1

• Relates to any normally-distributed variable by conversion:

standard normal deviate = observation – variable

mean

variable standard deviation

• Calculations using the standard normal distribution can be converted to those for a distribution with any mean and standard deviation

Area under the curve of the normal distribution• Percentage of men taller than 180cm?

– Area under the frequency distribution curve

above 180cm– Standard normal deviate: (180 - 171.5)/6.5 =

1.31

sample SND

0.0951

mean standard deviation

Area under the curve of the normal distribution• Percentage of men taller than 180cm?

– Area under the frequency distribution curve

above 180cm– Standard normal deviate: (180 - 171.5)/6.5 =

1.31• Percentage of men taller than 180cm is 9.51%

0.0951

• Percentage between 165cm and 175cm?– Find proportions below and above this – Subtract from 1 (remember: total area under

the curve is 1)

Area under the curve of a normal distribution

-1 0.54

0.1587 0.2946

1 – 0.2946 – 0.1587 = 0.5467

• Percentage between 165cm and 175cm?– Find proportions below and above this – Subtract from 1 (remember: total area under

the curve is 1)• 54.6% of men have a height between 165cm and

175cm

Area under the curve of a normal distribution

0.1587 0.2946

1 – 0.2946 – 0.1587 = 0.5467

Percentage points of the normal distribution• The SND expresses variable values as number of standard deviations away from the mean

• Exactly 95% of the distribution lies between -1.96 and 1.96

– The z-score of 1.96 is therefore 5%

percentage point2.5% 2.5%

95%

COMPUTER EXERCISE

The normal distribution

Distributions and the area under the curvewww.intmath.com/counting-probability/normal-distribution-graph-interactive.php

Exercises

1. Drag the mean and standard deviation left and right to see the effect on the bell curve

2. My variable has mean = 6 and standard deviation = 0.9• What proportion of observations are between 5

and 7?• How does this change when standard deviation =

2?

Hint: click on “Show probability calculation”

3. Verify that 95% of observations are within 2 standard deviations of the mean for any distribution

Hint: the red dashed lines are standard deviation units

RESEARCH METHODS AND STATISTICS

Sampling distributions and confidence intervals

Sampling distributions

population

sample

6mean

Sampling distributions

population

sample

5mean

6

Sampling distributions

population

sample

6mean

65

Sampling distributions

population

sample

6

65 7 8

7

4

6

5

9

8

7

sampling distribution of the mean

sampling distribution

Relationship between distributions

population

sample

sampling distribution

meanmean

mean

the distribution of the mean is normal even if

the distribution of the variable is not

Relationship between distributionspopulation

sample

sampling distribution

standarddeviation

standarddeviation

√samplesize

standarderror

how precisely the population mean is estimated by the sample mean

95% confidence interval for a meanpopulation

sample

meanmean

meanmean +1.96 x s.e.mean -1.96 x s.e.

95% probability that sample meanis within 1.96 standard errors of the population mean

95% confidence interval for a meanpopulation

sample

mean

meanmean +1.96 x s.e.mean -1.96 x s.e.

95% probability that population meanis within 1.96 standard errors of the sample mean

mean?

95% confidence interval for a meanpopulation

sample

mean

meanmean +1.96 x s.d. √size

mean -1.96 x s.d. √size

95% probability that population meanis within 1.96 standard errors of the sample mean

mean?

Sampling and inferencepopulation

sample

mean

mean

mean?

sampling distribution

Interpreting confidence intervals

• Don’t say:“There is a 95% probability that the population

mean lies within the confidence interval”

• The population mean is unknown but it is a fixed number

• The confidence interval varies between samples1. Take multiple random, independent samples

2. For each, calculate 95% confidence interval

3. On average, 19/20 (95%) of the confidence

intervals will overlap the true population mean

COMPUTER EXERCISE

Confidence intervals

Modify Java settings

1. Go to the Java Control Panel (On Windows Click Start and then type Configure Java)

2. Click on the Security tab3. Click on the Edit Site List button4. Click the Add button5. Type http://wise.cgu.edu6. Click the Add button again7. Click Continue and OK on the security window

dialogue box

Creating confidence intervals

http://wise.cgu.edu/ci_creation/ci_creation_applet/index.html

Exercises

1. How does altering the sample size affect the confidence intervals calculated?

2. When the population distribution is skewed, how does this affect the confidence intervals calculated?