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RESEARCH METHODOLOGY & STATISTICS LECTURE 6: THE NORMAL DISTRIBUTION AND CONFIDENCE INTERVALS Addictions Department MSc(Addictions) From sample to population… units population inference sample Background to statistical inference normal distribution sampling distribution standard normal distribution area under the curve percentage points confidence intervals p-values (significance) RESEARCH METHODS AND STATISTICS The normal distribution The normal distribution • Reasonable description of most continuous variables – given large enough sample size Standard deviation = 6.5cm Mean = 171.5cm 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 mean sample standard deviation SND 0.0951 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 Area under the curve of a normal distribution • Percentage between 165cm and 175cm? – Find proportions below and above this – Subtract from 1 (remember: total area under the curve is 1) -1 0.54 1 – 0.2946 – 0.1587 = 0.5467 0.1587 0.2946 Area under the curve of a normal distribution • 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 1 – 0.2946 – 0.1587 = 0.5467 0.1587 0.2946 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 point 2.5% 2.5% 95% COMPUTER EXERCISE The normal distribution Distributions and the area under the curve www.intmath.com/counting-probability/normal-distributiongraph-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 6 mean sample Sampling distributions population 6 5 mean sample Sampling distributions population 5 6 mean sample 6 Sampling distributions population 4 6 7 5 6 7 8 5 6 7 8 9 sampling distribution sampling distribution of the mean sample Relationship between distributions population the distribution of the mean is normal even if the distribution of the variable is not mean mean sample mean sampling distribution Relationship between distributions population how precisely the population mean is estimated by the sample mean standard error deviation standard deviation sampling distribution √sample size sample 95% confidence interval for a mean population mean mean mean mean -1.96 x s.e. sample mean +1.96 x s.e. 95% probability that sample mean is within 1.96 standard errors of the population mean 95% confidence interval for a mean population mean? mean mean mean -1.96 x s.e. sample mean +1.96 x s.e. 95% probability that population mean is within 1.96 standard errors of the sample mean 95% confidence interval for a mean population mean? mean mean mean -1.96 x s.d. √size sample mean +1.96 x s.d. √size 95% probability that population mean is within 1.96 standard errors of the sample mean Sampling and inference population mean? mean sample 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 samples 1. 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 tab 3. Click on the Edit Site List button 4. Click the Add button 5. Type http://wise.cgu.edu 6. Click the Add button again 7. 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?