Statistics

Question 1

State three advantages of random sampling

Answer 1

• Avoids suspected sources of bias
• Only a random sample enables proper statistical inference about the population to be undertaken
• because the probability basis on which the sample has been selected is known

Question 2

If the variance of X is v. What is the variance of X repeated twice and the results are added together?

Answer 2

2v
(Var(X1 + X2) = 2Var(X). Not to be confused with Var(2X) = 4Var(X))

Question 3

Conditions for a Poisson distribution to be appropriate

Answer 3

Events occur randomly at a uniform average rate, and independently of each other

Question 4

What is an independent variable?

Answer 4

A variable that is not subject to random variation

Question 5

Why is random sampling needed for proper statistical inference?

Answer 5

Because then the probability basis on which the sample has been selected is known

Question 6

State the distribution of the score from a fair six-sided dice

Answer 6

Uniformly distributed over the values { 1, 2, … , 6 }
include brackets, its a set!

Question 7

How to comment on goodness of fit on a regression line

Answer 7

• comment on r^2 (square r if needed)
• comment on how close points lie to straight line
  *…so fit is not very/fairly/very good indeed!

Question 8

Conditions for a reliable estimate to be made from regression line

Answer 8

• Interpolation
• and strong linear correlation (seen by points lying close to regression line)

Question 9

Advantages of larger sample sizes for tests using correlation coefficients

Answer 9

1. as sample size increases, random variation in sample tends to decrease
2. so the (pmcc/spearman’s rank) coefficient tends to get closer to population correlation coefficient
3. so one can be more confident that the correlation is genuine, rather than simply the result of random variation

Question 10

What is the difference between association and correlation?

Answer 10

• Association refers to any relationship between two variables
• Correlation refers to a linear relationship between two variables

Question 11

When is it appropriate to use the PMCC?

Answer 11

• Data is random-on-random
• Parent population follows a bivariate normal distribution (seen by grouping of points on scatter graph having a roughly elliptical shape)

Question 12

Sum of residuals ε1 + ε2 + … =

Answer 12

0

Question 13

r^2

Answer 13

• The coefficient of determination
• The proportion of the variation is one variable that can be explained by the variation of another

Question 14

Why might effect sizes be used instead of conducting a hypothesis test?

Answer 14

• For a large set of random on random bivariate data a small non-zero value of the PMCC is likely to lead to a rejection of the null hypothesis of no correlation in the population; the test is uninformative
• So the size of correlation is considered, rather than whether the population correlation is non-zero

Question 15

What does a critical value of 5% mean?

Answer 15

5% of the time we reject the null hypothesis when it is in fact true

Question 16

Situations when you might use Spearman’s rank instead of PMCC

Answer 16

• If either variable is non-random
• If the relationship in non-linear
• Subjective data
• Grouping of points on scatter diagram not roughly elliptical

Question 16

Null hypothesis for Spearman’s rank hypothesis test

Answer 16

H0: There is no association between x and y (in context) in the population

Question 17

4 conditions for a situation be modelled by a binomial distribution

Answer 17

• Events are independent of each other
• Events occur randomly at a constant probability
• Only success/failure possible
• Fixed number of trials

Question 18

Indicator of whether a Poisson distribution may be able to model a data set is if…

Answer 18

sample variance is reasonably close to sample mean

Question 19

If test statistic falls in LHS of the critical region (e.g. chi-squared)…

Answer 19

• Perhaps the model was constructed to fit the data
• Or some data has been removed in order to produce a better fit
• Or some of the data is not genuine

Question 20

Four desirable features of a sample

Answer 20

• Random
• Unbiased
• Representative of whole population
• Items are chosen independently

Question 21

Why it would not be sensible to predict the distance for 5 year olds?

Answer 21

• This would be extrapolation
• As the least age is 50yo (put into context)
• And the relationship may be different for 5 year olds

Question 22

Why is it sometimes not useful to plot to find the x on y equation?

Answer 22

If the values of x are non-random, then it makes no sense to try and predict them

Question 23

Assumption for a chi-squared test

Answer 23

Sample must be random

Question 24

Cohen’s guideline for effect sizes

Answer 24

ignored 0.0-0.1
small 0.1-0.3
medium 0.3-0.5
large 0.5-1.0

Question 25

Comment on outcome of hypothesis test considering the effect size of 0.165

Answer 25

• The test shows that there is almost certainly some real correlation in the population
• However, the test is uninformative since the effect size is so small

Question 26

Suggest a reason for not using an outlier in any analysis

Answer 26

Because it is not representative

Question 27

Explain why a census would not be used

Answer 27

Because it would be very expensive / impracticable to carry out

Question 28

Explain why they have decided to carry out a test based on PMCC

Answer 28

• Grouping of points in scatter diagram is roughly elliptical
• So there is evidence to suggest bivariate Normality in
  the population which is required for test using pmcc to be valid

Question 29

Disadvantage of using 10% significance level over 5% significance level

Answer 29

Null hypothesis is more likely to be wrongly rejected

Question 30

When is spearman’s rank test not appropriate?

Answer 30

If the scatter diagram shows no evidence of a monotonic relationship

Question 31

Disadvantage of spearman’s rank

Answer 31

Ranking data loses information, which might affect the outcome of a test

Question 32

Disadvantage of spearman’s rank

Answer 32

Ranking data loses information, which might affect the outcome of a test

Question 33

3 conditions for a situation be modelled by a geometric distribution

Answer 33

• Events are independent of each other
• Events occur randomly at a constant probability
• Only success/failure possible