Lecture Nine - Twelve Tests of Significance

Tests of significance: The Z test, The t test, and The X² test

Tests of Significance What is a test of significance? A/ It is a formal procedure for comparing observed data with a hypothesis whose truth we want to assess. The results of tests are expressed in terms of a probability that measure how well the data and hypothesis agree

Stating Hypothesis A hypothesis is a statement about parameters in the population, ex: µ1= µ2 Hypotheses are only concerned with the population

Null hypothesis (Ho) A statistical test begins by supposing that the effect, we want, is not present. This assumption is called the null hypothesis Then we try to find evidence against this claim (hypothesis) Typically, Ho is a statement of “no difference” or “no effect” We also want to assess the strength against the null hypothesis

Alternative Hypothesis (Ha) It is the statement about the population parameter that we hope or suspect is true (i.e. what we are trying to prove or the effect we are hoping to see) Ha is a statement of difference or relationship It can be one tailed (< or >) (ex: Ha > Ho) or two tailed (< and >) (ex: Ha µ1≠ µ2)

Types of statistical tests: Parametric tests: assume that variables of interest are measured on interval scale or ratio scale, usually continuous quantitative variable. There is assumption that variables are normally distributed Non parametric tests: assumed that the variables are measured on a nominal or ordinal scale

Steps of hypothesis testing: State the null hypothesis State the alternative hypothesis State the level of significance Choose the correct test statistics Computed the test statistics Determine the critical value of a statistics (needed to reject the Ho) from a table of sampling distribution values Compare computed to critical value Accept or reject the Ho.

Significance level: Usually, it is represented as α It is the value of probability below which we start consider significant differences Typical levels used are 0.1, 0.05, 0.01 and 0.001 The usual alpha level considered in medicine is 0.05

The Z test

One sample Z – test That of one sample mean: Steps for testing one sample mean (with σ known), irrespective of sample size State the Ho (Ho: µ1= µ2) State the H1 (H1: M1≠ M2) State the level of significance (example 0.05) Calculate the test statistics:

Z =

5. Find the critical value a. for Z= 1.96 = 0.05 b. for Z= 2.58 = 0.01

6. Decision: Reject Ho if test statistics > critical value i.e. P value < the significance level

7. State your conclusion: If Ho is rejected, there is significant statistical evidence that the population mean is different than the sample mean If Ho is not rejected, there is no significant statistical evidence that the population mean is different from the sample mean

Z – test for sample proportion:

Z- Test for differences between 2 means: =

Testing the difference between 2 sample proportions: Z = Where Sp1-p2 = P (Pooled)=

T-test

One Sample T-test In small sample size, when σ is not known, the sample standard deviation is used to estimate σ and the Z-statistics is replaced by the T-statistics. t= x - µ S/ √n When the x is the mean of a random sample of size n from a normal distribution with mean µ, then t has a student t-distribution with n-1 degree of freedom (df)

The df is the number of scores in a sample that are free to vary The df is a function of the sample size determines how spread of the distribution is (compared to the normal distribution)

The T-distribution Example, using the normal curve, 1,96 is the cut-off for a two tailed test at the 0,05 level of significance On a t-distribution with 3 df (a sample size of 4), the cut-off is 3.18 for a 2-tailed test at the 0.05 level of significance If your estimate is based on a larger sample of 7, the cut-off is 2.45, a critical score closer to that for the normal curve

The t-distribution is a bell-shaped and symmetrical one that is used for testing small sample size (n < 30) The distribution of the values of t is not normal, but its use and the shape are some what analogous to those of the standard normal distribution of z. T spreads out more and more as the sample size gets small. The critical value of t is determined by its df equal to n-1

Finding tcrit using t-table T-table is very similar to the standard normal table The bigger the sample size (or df), the closer the t-distribution is to a normal distribution

T-test for two sample means t= lx1-x2l SE(x1-x2) SE(x1-x2) = Spooled * √ 1 + 1 n1 n2 2 2 Spooled = S1 (n1-1) + S2 (n2-1) n1 + n2 – 2 N.B df for 2 sample means in t-test = n1+ n2- 2

The X² Test

χ² = ∑ E= df= (r- 1) (c - 1)