week 5. T-tests

Question 1

Central Limit Theorem

Answer 1

Tells us what the mean and standard deviation of the sampling distribution will be for any given size. As n increases, the sampling distribution will approach the normal distribution (for assessments which have a normal distribution).

Question 2

One Sample t-test

Answer 2

This statistic is similar to the z-score, but used when we do NOT know the population standard deviation. We use the one-sample t-test to determine if the sample mean is significantly different. t=X^--µ/S÷square root of n.

The t-statistic is a ratio of the observed difference between the sample and population means, and, the standard error.

thus, if the difference between the observed sample mean and population mean is greater than the standard eror, then t will be >1.

Question 3

Evaluating significance of one-tailed t-test

Answer 3

degrees of freedom =n-1

If t statistic_obtained is > t_critical, then the null hypothesis is rejected.

Note that it is only the magnitude of t which is important in this decision, ie treat a negative t same as a positive t.

Question 4

Matched sample t-test

Answer 4

Used when have matched or related samples, eg, 1 group of subjects tested multiple times.Thus, each person eg would be tested twice, and therefore obtain a difference score for each person (difference between 1st and 2nd reading).

t=D^--0/[SD÷square root of n]

Note, we use the minus 0, because the null hypothesis states that the mean of the standard deviation will be zero.

D^-=mean of the difference scores

SD =standard deviation of the difference scores

degrees of freedom =N-1.

(in matched sample tests, if test 10 people eg twice, then N=10)

If t_obtained>t_critical then we reject the null hypothesis

Question 5

Independent samples t-test

Answer 5

Used when have 2 independent samples and wish to know if they differ on a variable. Use when typically have one independent variable measured on a nominal scale, and one dependent variable measured on a continuous scale.

eg Are smokers more anxious than non-smokers?

eg do extroverts drink more alcohol than introverts? etc

The standard error now becomes the standard error of differences between the means.

H₀ states that µ₁=µ₂

a) for Equal sample sizes;

t=(X^-₁ - X^-₂)/[square root of{(S₁²/n₁) +(S₂²/n₂)}]

where S²=sample variance

b) for Unequal sample sizes:

t= (X^-₁ - X^-₂)/[square root of {(S²_p/n₁) +(S²_p/n₂)}]

where S²_p=pooled variance

S²_p=[(n₁ -1)S²₁ + (n₂ -1)S²₂] / (n₁ + n₂ -2)

Degrees of freedom is (n₁ -1) + (n₂-1)

Question 6

Confidence limits around a mean difference

Answer 6

If we consider t_critical at the alpha level of 0.05, we can generate a 95% confidence limit around the mean thus;

mean difference+/-t_critical

This gives us the range, in which we are 95% confident, that the actual mean lies.

Question 7

effect size Cohen’s d

Answer 7

Cohen’s d is used to comment on effect size for t-tests.

d=Cohen’s d

d=(X^-₁ -X^-₂)/S_p

effect size;

d=0.20=small

d=0.50=medium

d=0.80=large