Statistics 1

Question 1

Mathematical Model

Answer 1

A simplification of a real world situation

Question 2

Uses of a mathematical model (2)

Answer 2

1) Make predictions about a real world problem

2) To obtain an improved understanding of the situation through analysis

Question 3

Why mathematical models are useful (2)

Answer 3

1) They enable predictions to be made

2) They are quicker, easier and cheaper to use than the real situation

Question 4

Disadvantages of a mathematical model (3)

Answer 4

1) They only work for a restricted range of values.
2) They only give a partial description of the real situation (does not include all aspects)
3) The model may only work in certain situations

Question 5

Stages of designing a mathematical model (7)

Answer 5

1) A real world problem is observed
2) A mathematical model is devised
3) The mathematical model is used to make predictions about the expected behaviour of the real world problem
4) Experimental data is colelcted from teh real world
5) A comparison is made between the predicted and observed outcomes
6) Statistical tests are used to assess how well the model describes the real world
7) The mathematical model is refined, if necessary, to improve the match of predicted outcomes with (experimental) data

Question 6

What are statistical tests (2)

Answer 6

1) Used to assess how well a mathematical model matches a real world situation
2) An objective measure of deciding if the differences between the model’s predictions and the experimental data are within acceptable limits

Question 7

Quantitative variables

Answer 7

Variables associated with numerical observations

Question 8

Qualitative variables

Answer 8

Variables associated with non-numerical observations

Question 9

Continuous variable

Answer 9

A variable that can take any value in a given range

Question 10

Discrete variable

Answer 10

A variable that can take only specific values in a given range

Question 11

Median/IQR (what it shows, Adv, Disadv)

Answer 11

Use: To show how spread out the middle 50% of observations are
Adv: Not affected by extreme values; used when the data is skewed
Disadv: Does not use all of the data and therefore not a ‘true’ measure

Question 12

Why use mean/s.d.

Answer 12

These measures are a true measure of the data (use all the pieces of data) but are affected by extreme values

Question 13

Advantages of using a stem and leaf diagram (4) (Kangaroos eat quince tomorrow)

Answer 13

1) Keeps the shape of the data
2) Enables the shape of the distribution of the data to be revealed
3) Quartiles can easily be found from the diagram
4) Two sets of data can be compared by using back-to-back diagrams

Question 14

Disadvantage of using a stem and leaf diagram (1)

Answer 14

Can be time-consuming to do

Question 15

What is another name for the group encompassing mean and median

Answer 15

Measures of location

Question 16

What is another name for the group encompassing IQR and standard deviation

Answer 16

Measures of spread

Question 17

S.d./Mean (Use, Adv, Disadv)

Answer 17

When used: Used when the data are symmetrical/when the data size is not small
Adv: Uses all the pieces of data
Disadv: Affected by extreme values

Question 18

Give an example of all three of the measures of skewness (measures of location, quartiles, equation, boxplot)

Answer 18

Boxplot (self-evident)
Measures of location
- mode = median = mean –> symmetrical
- mode < median < mean –> positive skew
- mode > median > mean –> negative skew

Quartiles
Q2 - Q1 = Q3 - Q2 –> symmetrical
Q2 - Q1 < Q3 - Q2 –> positive skew
Q2 - Q1 > Q3 - Q2 –> negative skew

Equation
3(mean - median)/s.d. --> if value is large, skew = positive

Question 19

EQ: ‘Why should a histogram be used?’

Answer 19

A: The data are continuous

Question 20

Advantages of a histogram (2)

Answer 20

1) Gives a good pictures of how data are distributed
2) Enables you to see:
i) Rough location
ii) General shape of the data
iii) How spread out the data are

Question 21

What is the difference between a histogram and a bar chart? (2)

Answer 21

1) H: No gaps between the bars

2) H: Area of the bar is proportional to the frequency

Question 22

What are the three possible statuses for distribution

Answer 22

Symmetrical, positive skew, negative skew

Question 23

Define experiment

Answer 23

A repeatable process that gives rise to a number of outcomes

Question 24

Define event

Answer 24

A collection (or set) of one or more outcomes

Question 25

Define sample space

Answer 25

The set of all possible outcomes of an experiment

Question 26

Define discrete random variable

Answer 26

A value obtained by taking a measurement from an experiment in the real world

Question 27

Define random variable

Answer 27

A variable whose value is the outcome of an experimetn

Question 28

Equation for F(x)

Answer 28

P(X ≤ x)

Question 29

Equation for E(X)

Answer 29

Σ(xp(x))

Question 30

E(Xn)

Answer 30

ΣxnP(X = x)

Question 31

Var(X)

Answer 31

E(X2) – (E(X))^2

Question 32

E(aX + b)

Answer 32

aE(X) + b

Question 33

Var(aX + b)

Answer 33

a^2Var(X)

Question 34

s.d.(aX + b)

Answer 34

a(s.d.(X))