Section 28-30 Standard Error, Confidence Interval

Question 1

Standard Error of the Means

Answer 1

SAMPLING DISTRIBUTION OF THE MEANS – Suppose that we drew not just one sample but many samples at random. That is, we drew a sample of 60, tested the subjects, and computed the mean … then drew another sample of 60, tested the subjects, and computed the mean … then drew a third sample of 60, tested the subjects, and computed the mean … and so on. We would have a very large number of means-known as the sampling distribution of means.

CENTRAL LIMIT THEOREM says that the distribution of these means is NORMAL in shape. The normal shape will emerge even if the underlying distribution is skewed, provided that the sample size is reasonably large (about 60 or more).

• The mean of an indefinitely large sampling distribution of means will equal the population mean.
• The standard deviation of the sampling distribution is known as the standard error of the means (SE_m).

STANDARD ERROR OF THE MEANS (SE_m) – is the STANDARD DEVIATION of the SAMPLING DISTRIBUTION OF THE MEANS

• Keep in mind that the means vary from each other only because of chance errors created by random sampling. That is, we are drawing random samples from the same population and administering the same test over and over, so all of the means should have the same value except for the effects of random errors. Therefore, there is variation among the means only because of sampling errors. For this reason, the standard deviation of the sampling distribution is known as the STANDARD ERROR OF THE MEANS.

To Determine the STANDARD ERROR OF THE MEANS without calculating multiple iterations of the sample mean, we have a formula (see below)

• ​Ex: For a randomly selected sample
  
  • m = 75.00
  • s = 16.00
  • n = 64.
  • If we divide 16.00 by the square root of 64 (i.e., 8), we estimate that the standard error of the mean equals 16 / 8 = 2.00. This is an estimate of a margin of error that we should keep in mind when interpreting the sample mean of 75.00.
  • Keep in mind, too, that the STANDARD ERROR OF THE MEAN is an ESTIMATE of the standard deviation of the sampling distribution of the means.
    
    • ​That means that The STANDARD ERROR OF THE MEANS should be treated like a STANDARD DEVIATION.
      
      • ​Meaning that the 2.00 calculated above is a STANDARD DEVIATION.
      • Since, in a NORMAL DISTRIBUTION, about 68% of the cases lie within one standard deviation unit of the mean. We would expect about 68% of all sample means to lie within +/- 2.00 points of the true (or population) mean.
      • Using the SAMPLE MEAN of 75.00 as an ESTIMATE of the POPULATION MEAN, we could estimate that odds are 68 out of 100 that the POPULATION MEAN lies between 73.00 (75.00- 2.00 = 73.00) and 77.00 (75.00 + 2.00 = 77.00).
      • The values 73.00 and 77.00 are known as the LIMITS of the 68% CONFIDENCE INTERVAL FOR THE MEAN.
        
        • That is, the TRUE MEAN lies between 73.00 and 77.00 for 68% of all possible samples.
• A small confidence interval is desirable because it indicates that the sample mean is probably close to the true population mean. Of the two variables that affect the size of the standard error of the means (sample size and the variability of the sample), we often have direct control of the sample size. By using reasonably large samples, we can minimize the standard error of the mean.
• The larger the sample, the smaller the standard error of the mean.
• This works for a reasonably large sample (60 or more)
• The less the variability in the population, the smaller the standard error of the means.
• We can use the standard deviation of the sample that we have drawn as an estimate of the amount of variability in the population. For instance, if we observed a very small standard deviation for a random sample, it would be reasonable to guess that the population has relatively little variation.