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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?
