教育统计学课件6

For 30 or more samples, it is normally distributed Its mean is equal to the mean of the population Its standard deviation, called standard error of means, is equal to the standard deviation of the population by the square root of the sample size.
Group n LISTENING SCORE Control 19 Exper. 19 TOTAL ESL SCORE Control 19 Exper. 20
Mean 11.7 10.5 66.9 63.4
S 4.0 4.7 7.7 8.4
t-value df
p
Between two groups – t-test
t xe xc s( xe xc )
Between two groups – t-test
Lazaraton (1985) planned to evaluate a set of authentic language materials used in a beginning-level ESL class. There were two sections of beginning ESL, one of which would form the experimental group and the other the control group. While students were placed in the class on the basis of a placement test, it was important to equate the listening skills of the two classes prior to giving the instruction. Here is a table showing the scores of the two groups: ESLPE Listening Scores & Total Scores Two Classes
sx

sx N
t
X sx
Degree of freedom
Between two groups – t-test

Sampling distribution of differences between two means



to find an individual score in a normal distribution, xx z we use z-score s to place a sample mean in a distribution of x means, we use a t-test t s x to compare the difference between two means
Assumptions underlying t-test




There are only two groups of one independent variable to compare. You can’t cross-compare groups. Each observation is assigned to one and only one group. The data are truly continuous (interval or strongly continuous ordinal scores). The mean and standard deviation are the most appropriate measures to describe the data. The distribution in the respective populations from which the samples were drawn is normal, and variances are equivalent.
2 2 t 4 . 65 2 2 .55 2 t df 4.65 18

The of .55 is a very strong association. It tells us that 55% of the variability in this sample can be accounted for by independent variable. (45% of the variability cannot be accounted for by the independent variable. This variance is yet to be explained.)
N1 ( N1 1) U 1 N1 N 2 T1 2 N 2 ( N 2 1) U 2 N1 N 2 T2 2

U is the smaller of U1 and U2
The first table gives the critical values for significance at the p≤ 0.05 level in a twotailed/non-directional test, and for the p ≤ 0.025 level in a one-tailed/ directional test. The second table gives the critical values for the p≤ 0.01 level in a two-tailed/nondirectional test, and for the p≤ 0.005 level in a onetailed/directional test. For significance, the calculated value of U must be smaller than or equal to the critical value. N1 and N2 are the number of observations in the smaller and larger group, respectively.
2T1 n1 ( N 1) z ( n1 )(n2 )( N 1) 3
Example

Nonparametric comparison of two groups
• If there are approximately 20 or more subjects in each group, the distribution of U is relatively normal and the following formulae is used where N=N1+N2
Nonparametric comparison of two groups

Mann Whitney U

Scores for the two groups are combined and ranked Sum the ranks for each group Use the following formulae to compute the test statistic U
The investigator predicts that the means will differ significantly. Carry out a test to assess significance at the 5 per cent level, giving reasons for selecting the test you decide to use.
2
Mind work
Key
The following scores are obtained by two groups of subjects on a language proficiency test: Gr o up A 41 58 62 51 48 34 64 50 53 60 44 Group B 38 40 64 47 51 49 32 44 61
Strength of association

2 2( eta )
By rejecting the H0, we can show that there is an effect of the groups of the independent variable on the dependent variable. The two groups differ in their performance on the dependent variable. When the sample statistic is significant, one rough way of determining how much of the overall variability in the data can be accounted for by the independent variable is to determine its strength of association.

Research hypothesis? There is no effect of group on listening



comprehension (i.e., there is no difference in the means of the experimental and control groups) Significance level? .05 1- or 2-tailed? 2-tailed Design Dependent variable(s)? Listening comprehension Measurement? Scores (interval) Independent variable(s)? Group Measurement? Nominal (experimental vs. control) Independent or repeated-measures? Independent Statistical procedure? t-test