CH. 7 The t test (Sat.) Jin-ju Yang Talk outline 1.Application of the t distribution 2.Confidence interval for the mean from a small sample 3.Difference of sample mean from population mean (one sample t test) 4.Difference between means of two samples 5.Difference between means of paired samples (paired t test)

The CLT is very powerful, but it has two limitations: 1) it depends on a large sample size, and 2) to use it, we need to know the standard deviation of the population(σ). In reality, we usually don’t know the standard deviation of the population(σ) so we use the standard deviation of our sample (denoted as ‘s’) as an estim ate. Since we are estimating the standard deviation using our sample, the sampl ing distribution will not be normal (even though it appears bell-shaped). It is a little shorter and wider than a normal distribution, and it’s called a t-distribution. The t-distribution is actually a family of distributions – there is a different distribution for ea ch sample value of n-1 (degrees of freedom). The shape of t depends on the size of t he sample…the larger the sample size, the more confident we can be that ‘s’ is near ‘ σ’, and the closer t gets to Z.

1. Application of the t distribution The application of the t distribution to the following four types of problem will now be considered. 1.The calculation of a confidence interval for a sample mean. 2.The mean and standard deviation of a sample are calculated and a value i s postulated for the mean of the population. How significantly does the sa mple mean differ from the postulated population mean? 3.The means and standard deviations of two samples are calculated. Could both samples have been taken from the same population? 4.Paired observations are made on two samples (or in succession on one s ample). What is the significance of the difference between the means of th e two sets of observations?

2. CI for the mean from a small sample To find the number by which we must multiply the standard error to g ive the 95% confidence interval we enter table B at 17 in the left han d column and read across to the column headed 0.05 to discover th e number table B The 95% confidence intervals of the mean are now set as follows: Mean SE to Mean SE Likewise from table B the 99% confidence interval of the mean is as follows:table B Mean SE to Mean SE

3. One Sample t test To test a sample of normal continuous data, we need: An expected value = the population or true mean (μ) An observed mean = the average of your sample A measure of spread: standard error Degrees of freedom (df) = n-1 (number of values used to calculate SD or SE ) Then, we can calculate a test statistic to be compared to a known di stribution. In the case of continuous, normal data, it’s the t-statistic a nd the t-distribution.

4. Two Samples t test We can use the t-test to compare two different groups of continuous data as the outc ome and compare test statistic to appropriate distribution to get p-value. Under the null hypothesis, we propose that this difference equals 0. We can calculate an estimate of the SE of this difference from our data. H 0 : σ ₁ = σ ₂ = σ (Equal standard deviations) Obtain the standard deviation in sample 1: S ₁ Obtain the standard deviation in sample 2: S ₂ Multiply the square of the standard deviation of sample 1 by the degrees of freedom, which is the number of subjects minus one: repeated for sample 2 Add the two together and divide by the total degrees of freedom

The standard error of the difference between the means is When the difference between the means is divided by this standard error th e result is t. Thus, The table of the t distribution Table B (appendix) which gives two sided P val ues is entered at degrees of freedom.Table B (appendix) A 95% confidence interval is given by

H ₁ : σ ₁ ≠ σ ₂ (Unequal standard deviations) Rather than use the pooled estimate of variance, compute This is analogous to calculating the standard error of the difference in two proportio ns under the alternative hypothesis as described in Chapter 6Chapter 6 We now compute We then test this using a t statistic, in which the degrees of freedom are: There is a slight modification to allow for unequal variances – this modification adjusts the d.f for the test, using slightly different SE computation.

5. Paired t test Sometimes data are paired. In this case, the “before” and “after” are not ind ependent – they are taken from the same person. What you are testing is the change in the same individual. When your data are paired, you basically create one set of data by calculating each person’s change, then doing a one-sample t-test. Find the mean of the differences, Find the standard deviation of the differences, SD. Calculate the standard error of the mean To calculate t, divide the mean of the differences by the standard error of th e mean A 95% confidence interval for the mean difference is given by

Exercises 7.1 In 22 patients with an unusual liver disease the plasma alkaline phosphatase was found by a certain laboratory to have a mean value of 39 King-Armstrong units, standard deviation 3.4 units. What is the 95% confidence interval within which the mean of the population of su ch cases whose specimens come to the same laboratory may be expected to lie? 7.2 In the 18 patients with Everley's syndrome the mean level of plasma phosphate was 1.7 mmol/l, standard deviation 0.8. If the mean level in the general population is taken as 1.2 m mol/l, what is the significance of the difference between that mean and the mean of these 18 patients?