Revision: Descriptive Statistics and the t-test 2009 Methodology A - Lecture 2 1. Today’s Learning Objectives 2. Descriptive Statistics a. Central Tendency b. Variability 3. Hypothesis Testing 4. The t-test a. Degrees of freedom b. Effect Size 5. Review of Learning Objectives 6. Vocabulary Outline Today’s Learning Objectives Descriptive Statistics 1. What are the three most common measures of central tendency? 2. How do you calculate the mean? 3. How do you calculate the median? 4. How do you calculate the mode? 5. What are the two measures of variability? 6. How are these two measures related? Hypothesis Testing 7. What is the null hypothesis? 8. What is the difference between one-tailed and two-tailed alternative hypotheses? 9. How do p-values relate to the null and alternative hypotheses? 10. How do Type I and Type II errors differ? The t-test 11. What are the three types of t-test? 12. What do you need to know about your data to compute the t statistic? 13. What are the assumptions of the t- test? 14. How do you test for equal variance? 15. What do you do if variance is not equal? 16. What are degrees of freedom? 17. How do you calculate effect size? 18. How do you report the outcome of a t-test? 19. Given sample data, which type of t-test is most appropriate? Descriptive Statistics Ch. 3 Central Tendency Mean = (2+3+4+4+5+6+7+9)/8 = 5 Median = (4+5)/2 = 4.5 Mode = 4 2, 3, 4, 4, 5, 6, 7, 9 41-6 Variability Variance = !(x-x) 2 / (n-1) Standard Deviation = "variance x = 2, 3, 4, 4, 5, 6, 7, 9 74-7 x 5 6 7 8 9 4 3 2 1 Variability x = 2, 3, 4, 4, 5, 6, 7, 9 2 3 4 4 6 7 9 Variability x = 2, 3, 4, 4, 5, 6, 7, 9 = 36 / 7 5.14 = 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 !(x-x) 2 / (n-1) Variability x = 2, 3, 4, 4, 5, 6, 7, 9 5.14 SD = " 5.14 = 2.27 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
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1. What are the three most common measures of central tendency?
2. How do you calculate the mean?3. How do you calculate the median?4. How do you calculate the mode?5. What are the two measures of
variability?6. How are these two measures
related?
Hypothesis Testing7. What is the null hypothesis?8. What is the difference between
one-tailed and two-tailed alternative hypotheses?
9. How do p-values relate to the null and alternative hypotheses?
10. How do Type I and Type II errors differ?
The t-test11. What are the three types of t-test?12. What do you need to know about
your data to compute the t statistic?
13. What are the assumptions of the t-test?
14. How do you test for equal variance?
15. What do you do if variance is not equal?
16. What are degrees of freedom?17. How do you calculate effect size?18. How do you report the outcome of
a t-test?19. Given sample data, which type of
t-test is most appropriate?
Descriptive Statistics
Ch. 3
Central Tendency
Mean = (2+3+4+4+5+6+7+9)/8 = 5
Median = (4+5)/2 = 4.5
Mode = 4
2, 3, 4, 4, 5, 6, 7, 9
41-6
Variability
Variance = !(x-x)2 / (n-1)
Standard Deviation = "variance
x = 2, 3, 4, 4, 5, 6, 7, 9
74-7
x
5 6 7 8 94321
Variability
x = 2, 3, 4, 4, 5, 6, 7, 92 3 4 4 6 7 9
Variability
x = 2, 3, 4, 4, 5, 6, 7, 9
= 36 / 7 5.14=
363534333231302928
272625242322212019
181716151413121110
987654321
!(x-x)2 / (n-1)
Variability
x = 2, 3, 4, 4, 5, 6, 7, 9
5.14SD = " 5.14 = 2.27
363534333231302928
272625242322212019
181716151413121110
987654321
! Null Hypothesis (H0)
! Alternative Hypothesis (HA or Hi)
! Statistical significance (p-value)
! Type I Error
! Type II Error
Hypothesis Testing
Ch. 5
Null Hypothesis (H0)
! There is no difference between groups (paired-samples and independent-samples t-tests)
! The mean is not different from a specified value (one-sample t-test)
Alternative Hypothesis (HA)
! There is no difference between groups (paired-samples and independent-samples t-tests)
! The mean is not different from a specified value (one-sample t-test)
Statistical Significance! Gives guidelines for what allows us to
accept or reject the null hypothesis
! The p-value is the probability that you would get your test statistic (e.g. t) if the null hypothesis were true
! Alpha (#) is the critical p-value:
! accept H0 when p $ #! reject H0 when p < #
! In psychological research, usually # = .05
Tails! Two-tailed hypothesis: HA does not specify
in what direction the difference lies
! One-tailed hypothesis: the direction is specified
! One-tailed hypotheses require a convincing argument that the direction was predicted before the data were collected
! SPSS reports 2-tailed p-values for t-tests; since the t distribution is symmetric, divide p by 2 for the 1-tailed p-value
150-4
Types I and II ErrorsType I error:
! rejecting H0 when H0 is actually true
! a false positive
! p-value is ‘too small’
! e.g. a pregnancy test says you are pregnant when you are not
Type II error:
! accepting H0 when H0 is actually false
! a false negative
! p-value is ‘too big’
! e.g. a pregnancy test says you are not pregnant when you are
146-9
1. Independent-samples
2. Paired-samples
3. One-sample
The t-test
Ch. 7
Data are approximately normally distributed
For independent-samples t-tests:
Variance is equal or use a correction for
unequal variance
Assumptions
222-3
If Levene’s test is significant (p < .05), use ‘Equal variances not assumed’ (round df to the nearest whole number). If p $ .05, use ‘Equal variances assumed’.
One score from each subject and two groups of subjects (e.g. male and female)
Independent-samples t-test
Are the two groups of subjects doing something different?
223-30
Independent-samples t-test
tdifference between group means
variability of the groups=Two scores from each subject
Tests if the difference between scores is greater than 0
Paired-samples t-test
Are subjects doing something different in two conditions?
230-5
Paired-samples t-test
tdifference between paired scores
variability of the difference scores=
0
One score from each subject and a strong hypothesis about what the mean should be if H0 is true, usually 0 or chance (e.g. 50% in a forced-choice task)