One sample means Testing a sample mean against a population mean

Hypothesis testing for one population means We have measured a sample, and have a mean and SD, interval or ratio level data We have measured a sample, and have a mean and SD, interval or ratio level data Does the sample differ significantly from the population? Does the sample differ significantly from the population? Review of statistical significance: could this difference be chance, or is there a real difference? Review of statistical significance: could this difference be chance, or is there a real difference?

Statistical significance Usual rule of thumb: there must be less than a 5% probability (1 in 20) that this result could have been obtained by chance Usual rule of thumb: there must be less than a 5% probability (1 in 20) that this result could have been obtained by chance Null hypothesis: the mean of the sample is not different from that of the population Null hypothesis: the mean of the sample is not different from that of the population Non-directional alternative hypothesis: there is a difference—which one is higher is not known Non-directional alternative hypothesis: there is a difference—which one is higher is not known

Statistical significance Directional hypothesis: there is a difference and we specify whether the sample or the population will have the higher mean. Directional hypothesis: there is a difference and we specify whether the sample or the population will have the higher mean.

Testing a sample Assuming the sampling means form a normal distribution (central limit theorem) we can use z scores to determine if a sample differs significantly from a population Assuming the sampling means form a normal distribution (central limit theorem) we can use z scores to determine if a sample differs significantly from a population

Formula: Calculate Z statistic Z = (Mean of the sample-mean of population)/SD/square root of n (number of subjects) Z = (Mean of the sample-mean of population)/SD/square root of n (number of subjects) We can then use this z score to determine the likelihood of getting a result significantly different from the population We can then use this z score to determine the likelihood of getting a result significantly different from the population (the Z statistic should not be confused with z scores) (the Z statistic should not be confused with z scores)

Example The mean IQ of our training school population is 87 with a sample size (n) of 256 The mean IQ of our training school population is 87 with a sample size (n) of 256 The mean in the general population is 100 with a SD of 15 The mean in the general population is 100 with a SD of 15 Z = /(15/sqrt of n) Z = /(15/sqrt of n) -13/(15/16) = 13/.9375 = /(15/16) = 13/.9375 = 13.8 Statistically significant Statistically significant

Example See table on p. 284 for determining significance See table on p. 284 for determining significance Suppose that the mean is 102, sample size 100 Suppose that the mean is 102, sample size 100 Z = /(15)/10) = 1.33 Z = /(15)/10) = 1.33 Not statistically significant, the sample if from the same population Not statistically significant, the sample if from the same population