Comparison of two Means

Question 1

What is unpaired data?

Answer 1

Occurs when individual observations in one sample are independent of individual observations in the other

Question 2

What is paired data?

Answer 2

Occurs when the individual observations in the first sample are matched to individual observations in the second sample. (i.e. not independent)

Question 3

What is a z-test?

Answer 3

If we can assume the distribution of mean differences is normal we can use a z-test to test the null hypothesis. The result tells us how many standard errors away from 0 is the sample mean difference.

Question 4

Give an example of paired data?

Answer 4

Measurements taken from the same individual. I.e. repeated blood pressure measurements.

Question 5

How is paired quantitative data measured?

Answer 5

Calculate the difference between the the events (di).
Calculate the mean of the index of the differences
Carry out a hypothesis test on mean of di

Question 6

How do you calculate a confidence interval for the sample mean difference and test the hypothesis that it is equal to zero (null hypothesis)

Answer 6

Must know the mean, standard deviation and standard error of the mean difference

Question 7

What is the null hypothesis?

Answer 7

The null hypothesis is the assertion that the mean population difference is zero

Question 8

How does a hypothesis test work?

Answer 8

If we can assume that the distribution of mean differences (sampling distribution) is normal. We can test the hypothesis using a z-test (when n> 30).

Question 9

How is a z-test performed?

Answer 9

z = (sample mean difference - hypothesised mean difference) / standard error of the mean difference

Question 10

What does the result of a z-test tell us?

Answer 10

How many standard errors away from zero the sample mean difference is/

Question 11

How does the analysis of two independent means differ from the analysis of paired data?

Answer 11

We look at the difference between two independent means rather than the mean difference of paired observations

Question 12

What can be said about the characteristics of the sampling distribution of the differences between the two independent means?

Answer 12

1. The mean of the sampling distribution is the difference between the two population means
2. The standard deviation of the sampling mean (standard error) depends on n1 and n2 (sample sizes)
3. As n1 and n2 increase, the distribution becomes increasingly normal

Question 13

How do we calculate the standard error of the mean of two independent sample distributions?

Answer 13

The SE of the difference between two means is a combination of the standard errors of the two independent sample distributions

Question 14

What is the square of the standard error of the mean?

Answer 14

The variance

Question 15

How do we calculate the confidence interval for the difference between two means?

Answer 15

The difference between the means of both samples +/- level of confidence * SE (where the standard deviation of the samples may be used instead of the population standard deviations as this is unknown in most scenarios)

Question 16

How does one obtain a P-value from a z-score?

Answer 16

Standard normal tables.

Question 17

In what scenario is it more prudent to use a t-test than a z-test?

Answer 17

When the sample size is small (conventionally n<30)

Question 18

What are degrees of freedom?

Answer 18

The number of values in the final calculation of a statistic that are free to vary. The number of independent ways by which a dynamic system can move, without violating any constraint imposed on it, is called number of degrees of freedom.

Question 19

How do you calculate the degrees of freedom?

Answer 19

n - (number of samples)

i.e. in a one sample test the degrees of freedom are equal to n-1. In two sample tests = (n1+n2)-2

Question 20

How do we calculate the variance in small samples?

Answer 20

One common variance is calculated using the data from two independent samples. It is a weighted average of the two sample variances.

Question 21

What assumptions are essential to make in using the z and t tests?

Answer 21

1. The population is normally distributed

2. The standard deviations of the populations are known

Question 22

How can we test the assumption that the populations in a hypothesis test are normally distributed?

Answer 22

Using histograms or normal plots

Question 23

What specific assumption does the two sample t test make in regards to measures of spread?

Answer 23

It assumes that the variances of the two samples are equal

Question 24

What do we do if the assumptions of normality are not possilbe?

Answer 24

1. Transformation (log or square root of data) to make it more approximately normal
2. Non parametric methods (these do not make any assumptions)

Question 25

What special type of t test can be used when the variance of a two sample scenario are not equal?

Answer 25

Welch’s test

Question 26

What is the Wilcoxon Rank Sum Test?

Answer 26

A non-parametric test that corresponds to the two sample t test (equivalent to the Mann-Whitney test)

Question 27

How does one perform the Wilcoxon Rank Sum Test?

Answer 27

1. Assume the two population distributions are the same
2. Define hypothesis (H0 = the two distributions overlap and H1 = they are shifted)
3. Rank all observations in ascending order (if the values are equal take an average)
4. Calculate T as the sum of their ranks in the smaller sample.
5. Compare T with tables of critical vaules for the test to obtain a P-vaule

Question 28

How do we interpret critical values for the Wilcoxon Rank Sum test?

Answer 28

If the T value is outside or at the limit of the critical value the value of P is less than the corresponding P-value of the aforementioned range. If it is inside the range then it is equivalent to P-value greater than that range P-value