1 / 24

Overview

Overview. Central Limit Theorem The Normal Distribution The Standardised Normal Distribution Z Scores Estimation Confidence Intervals. Central Limit Theorem. The sampling distribution of the means of samples becomes normal as the sample size increases.

tiva
Download Presentation

Overview

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Overview • Central Limit Theorem • The Normal Distribution • The Standardised Normal Distribution • Z Scores • Estimation • Confidence Intervals

  2. Central Limit Theorem • The sampling distribution of the means of samples becomes normal as the sample size increases. • The sampling distribution of the mean for sufficiently large samples will be normal (N>30). • The sampling distribution of the mean will always be normal if the underlying population is also normal irrespective of sample sizes. • For small samples (N< 30) taken from a normally distribution population, we know the form of the sampling distribution of the mean.

  3. The Normal Distribution • The normal distribution is a specific distribution with a particular shape • It is mathematically defined by the expression • This defines the probability x with respect to two parameters, the mean and the variance of a population of scores

  4. Standardised Normal Distribution • We can define a normal distribution of standardised scores in the following way: • where • z is known as a standardised score

  5. Standardised Normal Distribution • Using calculus (specifically integration) we can calculate the area under a standardised normal distribution • This tells us the proportion of scores that fall under a particular area of the distribution. • We can use tables of standardised scores to estimate the probability of a particular score occurring in any set of data

  6. Using Z Scores • To use the standardised normal distribution we must adopt the basic assumption that the population of scores is normally distributed • What proportion of IQ scores are greater than 125? • IQ scores are normally distributed with a mean of 100 and a standard deviation of 15

  7. Using Z Scores • We have to calculate the z score associated with an IQ of 125. • To do this we calculate the difference between the mean and the score and divide by the standard deviation • We obtain:

  8. Using Z Scores • Now we look at the tables to find what proportion of scores are beyond the z score of 1.67. • Looking at the table entry for z = 1.67 we find that 0.04745 of the area of the curve lies beyond the z value of 1.67. • If we multiply this by 100 we obtain the percentage of scores that lie beyond this value. • In other words 100 x 0.04745 = 4.745% of the population have an IQ of greater than 125

  9. Using Z Scores • What proportion of IQ scores are less than 60? • We have to calculate the z score associated with an IQ of 60. • First calculate the difference between the mean and the score and divide by the standard deviation • We obtain:

  10. Using Z Scores • Now we look at the tables to find what proportion of scores are beyond the z score of -2.67 • Looking at the table entry for z = 2.67 we find that 0.00378 of the area of the curve lies beyond the z value of 2.67. • If we multiply this by 100 we obtain the percentage of scores that lie beyond this value. • In other words 100 x 0.00378 = 0.378% of the population have an IQ of less than 60

  11. Using Z Scores • What proportion of scores lie between 85 and 115 on the IQ scale • To do this we calculate the difference between the mean and the score and divide by the standard deviation for both points • We get:

  12. Using Z Scores • Now we look at the tables to find what proportion of scores are between the mean of the distribution and -1.00 and 1.00 respectively. • Looking at the table entry for z = -1.00 we find that 0.34134 of the area of the curve lies between the mean the z score -1.00. • Looking at the table entry for z = 1.00 we find that 0.34134 of the area of the curve lies between the mean the z score -1.00. • The total proportion is the addition of the two values, i.e. 0.34134+0.34134=0.68268 • In other words 100 x 0.68268 = 68.268% of the population have an IQ of between 85 and 115.

  13. Estimation • Most of the time we do not know about population parameters • We would like to be able to make a judgement about the population parameters • In parametric statistics we can make "best guess" judgements about the parameters of populations • These "best guesses" are known are estimates

  14. Point & Interval Estimation • There are two kinds of estimates, point and interval. • With point estimates we attempt to assign a particular value to a population parameter such as the mean or the variance • With interval estimates we try and construct a range in which the population parameter might fall and to which we can attach a probability

  15. Point Estimates • For an estimate to be considered a good estimate then it must be unbiased, sufficient and consistent. • Unbiased • The mean of the sampling distribution is equal to the population parameter being estimated. • Sufficient • The statistic on which the estimate is based uses all the information in the sample. • Consistent • Based on a statistics whose accuracy increases as sample size increases.

  16. Measures of Centre • All measures of centre i.e. mean, mode and median are unbiased measures of their respective parameters. • The mean, however, is the only one of the sample statistics which is both sufficient and consistent.

  17. Measures of Spread • Both the variance and the standard deviation are biased estimates of the population parameters s and s2 • The mean of the sampling distribution of the variance is too small as an estimate of the population variance by a factor of: • so that • is an unbiased estimate of s2

  18. Measures of spread • Too distinguish the sample variance and the sample based estimate of the population variance we will refer to the sample based estimate as: • Similarly the sample based estimate of the population standard deviation is referred to as:

  19. Standard Error of the Mean • The population standard error of the mean is defined as: • The sample based estimate of the population standard error of the mean is defined as:

  20. Interval Estimates • Interval estimates are calculated on the basis of three factors: • A point estimate for the parameter • A measure of spread in the population • A probability value

  21. Confidence Intervals • Suppose that we have tested the IQ of a number of subjects in an experiment. • We are going to calculate the 95% confidence interval for the population mean, µ • First we have to compute the population standard error of the mean: • The standard error of the mean of the population tells us how much we can expect the population mean to fluctuate.

  22. Confidence Intervals • Assuming the population is normally distributed means that the sampling distribution of the mean is also normally distributed (central limits theorem). • Now define an area of the sampling distribution that should contain the middle 95% of the possible sample means. • In order to do this we must use the standardised formula as applied to the sampling distribution:

  23. Confidence Intervals • If we look at the tables of z values we can find that the centre based z scores that include 95% of the distribution is equal to ±1.96. • The upper limit of our range of values for the population mean is: • The lower limit of our range of values for the population mean is:

  24. Confidence Intervals • In general terms, the formula for the 95% interval for µ is: • For any level of confidence, ranging from 0 to 99.999% we have:

More Related