Statistics 2

Question 1

What did Hermann Ebbinghaus pioneer?

Answer 1

Learning/experience curves.

Question 2

What shape does a learning curve of proficiency against effort expended take?

Answer 2

f

Question 3

What is the reasoning for the shape of a learning curve?

Answer 3

Learning how to do things more efficiently typically requires geometrically increasing effort and makes progress with positive, ever diminishing rewards.

Question 4

For power law:
y=ax^b

What returns are there when:
b > 1
b < 1

Answer 4

increasing returns

decreasing returns

Question 5

What does standard deviation measure?

Answer 5

The spread of values around their mean.

Question 6

What standard deviation was used in this PowerPoint?

Answer 6

Sigma = population.

Question 7

What is bootstrapping?

Answer 7

Resampling from a known sample to explore uncertainty about the sample mean.

Question 8

What is suggested when the bootstrapping method demonstrates uncertainty associated with the sample mean (getting very different results each time)?

Answer 8

It may not be a good representation of the true mean for the whole population which the sample was taken from.

Question 9

How do you show the range of uncertainty for a value?

Answer 9

Standard error bars.

The amount of uncertainty is represented by the size of the error bar.

Question 10

How do you calculate standard error?

Answer 10

The standard deviation (sigma) / the square root of the sample size.

Question 11

If a point appears anomalous and the sample mean lies within the standard error, what does this mean for the data point?

Answer 11

It is likely just a ‘blip’.

Question 12

How can you measure the linear trend of your data?

Answer 12

Using Pearson’s product-moment correlation coefficient (r).

Question 13

How do you work out r?

Answer 13

sum of ((values for x - mean of x)(values for y - mean of y)) / square root (sum of (values for x - mean of x)^2 x (values for y - mean of y)^2)

Question 14

What kind of data is finding r appropriate for?

Answer 14

Normally distributed data.

Question 15

What does r measure?

Answer 15

Linear correlation of x and y = the strength of a linear trend for paired data points from sample y plotted against sample x.

Question 16

What are the possible values of r?

Answer 16

-1 to +1

Question 17

What does it tell you if r is between -0.1 and +0.1?

Answer 17

Virtually no trend.

Question 18

What does it tell you if r is -0.3 or +0.3?

Answer 18

May be a trend.

Question 19

What shape is the line if r is -1 or + 1?

Answer 19

Perfectly straight line.

Question 20

Even after calculating r and obtaining a value of +0.1, why should you always do a plot of the data when exploring relationships?

Answer 20

Pearson’s correlation doesn’t always mean a lack of relationship.

Question 21

How can you check if a pattern occurred at random?

Answer 21

Randomise data values (like in bootstrapping) and calculate r.

Question 22

What can you conclude about a piece of data if 20 randomisations show no trend as strong as r = +0.68?

Answer 22

There is less than a 1 in 20 chance of seeing a trend this strong at random.

p < 0.05

Question 23

What is a p-value?

Answer 23

The likelihood of seeing a pattern at random.

Question 24

What p-values are we interested in?

Answer 24

p < 0.05 or p < 0.01 is even better.

Question 25

If p > 0.05, what do you do?

Answer 25

Report exact value and conclude non-significant.

Question 26

If 0.05 > p > 0.01, what do you do?

Answer 26

Report the exact value and conclude significant.

Question 27

If p < 0.01, what do you do?

Answer 27

Don’t report exact value and conclude (highly) significant.

Question 28

What does the significance of a correlation depend on?

Answer 28

It’s strength (r) and the size (n) of the sample.

Question 29

What combination of r and n gives a low p value?

Answer 29

high value of r and high value of n

Question 30

Correlation does not imply…

Answer 30

causation.

Question 31

When do you use a Mann-Whitney U test?

Answer 31

Aim = comparing samples.
Data = ranks/categories or measurements.

Data is not normally distributed (symmetrical about the mean).

Question 32

When do you use a t-test?

Answer 32

Aim = comparing samples.
Data = measurements.
Data is normally distributed.

Question 33

When do you use a chi-squared test?

Answer 33

Aim = comparing samples.
Data = frequencies.

Question 34

When do you use a regression test?

Answer 34

Aim = identify a trend/relation.
Variables = manipulated vs dependent.

Question 35

When do you use a correlation test?

Answer 35

Aim = identify a trend/relation.
Variables = Independent vs dependent.

Question 36

What is an alternative hypothesis?

Answer 36

A statement in hypothesis testing that proposes a difference, relationship, or effect exists in the population, challenging the null hypothesis.

Shown to be true/false by investigators.

e.g. the player’s scores in consecutive games are correlated.

Question 37

What is a null hypothesis?

Answer 37

A foundational concept in hypothesis testing that proposes there is no effect, no difference, or no relationship in a population or between variables.

It serves as the default or starting assumption that researchers aim to challenge or test against.

e.g. the player’s scores in consecutive games are not correlated.

Question 38

What do we do instead of proving an alternate hypothesis to be true?

Answer 38

We support a hypothesis by showing the null hypothesis is unlikely to be true.

Question 39

What does the p-value show in terms of the null and alternate hypotheses?

Answer 39

How unlikely our observed results would be if the null hypothesis were true.

Question 40

If our result is very unlikely e.g. p < 0.05 what do we do?

Answer 40

Reject the null hypothesis.

Question 41

What would 5 sigma indicate?

Answer 41

The observed statistic is more than 5 standard deviations from the mean.

For normal distribution, p < 0.0000003 (less than 1 in 3.5 million probability of seeing result by chance).