lecture 03 - inferential statistics 1

©NCC Education LimitedV1.0

Advanced Mathematics for Business

Topic 3:

Inferential Statistics 1

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.2

Scope and Coverage

This topic will cover:• Sampling distributions• Point estimates and confidence intervals• Introduction to hypothesis testing

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.3

Learning Outcomes

By the end of this topic students will be able to:• Recognise the terms sample statistic and

population parameter• Use confidence intervals to indicate the reliability of

estimates• Know when approximate large sample or exact

confidence intervals are appropriate

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.4

Probabilities and the Standard Normal Distribution

P(Z < 1) = 0.8413

P( - 1 <Z < 1) = 0.6826

z = 1.0

P(Z > 1) = 0.1587

P( 0 <Z < 1) = 0.3413

+ x

0.0 1.0 -1.0 1.0

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.5

Probabilities and the Standard Normal Distribution

P(Z < 1) = 0.8413

z = 1.0+ x

P( - 1 <Z < 1) = 0.6826

P( 0 <Z < 1) = 0.3413

0.0 1.0 -1.0 1.0

z 0 1 2 3

0.8 0.7881 0.7910 0.7939 0.7967

0.9 0.8159 0.8186 0.8212 0.8238

1.0 0.8413 0.8438 0.8461 0.8485

1.1 0.8643 0.8665 0.8686 0.8708

z 0 1 2 3

0.8 0.2881 0.2910 0.2939 0.2967

0.9 0.3159 0.3186 0.3212 0.3238

1.0 0.3413 0.3438 0.3461 0.3485

1.1 0.3643 0.3665 0.3686 0.3708

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.6

Percentage Points of the Normal Distribution - 1

5%

z = 1.6449

z = 1.96

5%5%

2.5%2.5%2.5%

P(Z > 1.6449) = 0.05

1.6449-1.6449

-1.96 1.96

95%

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.7

Percentage Points of the Normal Distribution - 2

5%

z = 1.6449

z = 1.96

5%5%

2.5%2.5%2.5%

1.6449-1.6449

-1.96 1.96

95%

1 2 z

5.0% 10% 1.6449

2.5% 5% 95% 1.9600

1.0% 2% 2.3263

0.5% 1% 99% 2.5758

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.8

Populations and Samples - 1

Population Parameters

Sample Statistics?

?

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.9

Populations and Samples - 2

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.10

Distribution of Sample Means

Population DistributionDistribution of Sample Means

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.11

Central Limit Theorem

n > 30

Population Distribution Distribution of Sample Means

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.12

Populations and Samples - 3

Population Parameters

Sample Statistics

?

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.13

95%

z = -1.96 z = 1.96

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.14

95%

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.15

Confidence Interval

1 2 z

5.0% 10% 1.6449

2.5% 5% 95% 1.9600

1.0% 2% 2.3263

0.5% 1% 99% 2.5758

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.16

Interpreting 95% Confidence Interval

1 in 20 chance of not containing population

mean

19 in 20 chance of containing population

mean

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.17

Example - 1• A machine produces golf balls. The diameters of a each

of a sample of 30 balls is measured. Find the 95% and 99% confidence interval estimates of the mean diameter of balls produced by the machine.

42.83 43.82 43.51 42.64 43.8243.71 43.37 42.76 43.18 43.2244.00 42.77 43.00 42.99 42.8542.75 43.90 43.36 42.81 42.9243.33 42.78 42.75 43.09 43.7243.70 43.85 42.91 43.32 43.67

1 2 z

5.0% 10% 1.6449

2.5% 5% 95% 1.9600

1.0% 2% 2.3263

0.5% 1% 99% 2.5758

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.18

Example - 2• A machine produces golf balls. The diameters of a each

of a sample of 30 balls is measured. Find the 95% and 99% confidence interval estimates of the mean diameter of balls produced by the machine.

1 2 z

5.0% 10% 1.6449

2.5% 5% 95% 1.9600

1.0% 2% 2.3263

0.5% 1% 99% 2.5758

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.19

Large Sample CI of Proportion

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.20

Example

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.21

Exact Confidence Interval• If normal model for population then

- exact sample confidence interval can be calculated- do not use z percentage points but t percentage

points- t percentage points come from a family of

distributions called Student’s t-distribution- family because t-distribution depends on degrees of

freedom = (n – 1)

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.22

Student t-distribution

t(1)

t(30)t(4)

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.23

Exact Confidence Interval

1 5.00% 2.50% 1.00% 0.50%2 10.00% 5.00% 2.00% 1.00% 90.00% 95.00% 98.00% 99.00%

= n - 11 6.3138 12.7062 31.8205 63.65672 2.9200 4.3027 6.9646 9.92483 2.3534 3.1824 4.5407 5.84094 2.1318 2.7764 3.7469 4.6041

10 1.8125 2.2281 2.7638 3.1693100 1.6602 1.9840 2.3642 2.6259

1000 1.6464 1.9623 2.3301 2.5808

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.24

Example• A random sample of 11 apples is weighed and are

found to have a sample mean of 93.25 grams and a sample standard deviation of 15.60 grams. Assuming the apples are a random sample drawn from a normal distribution what is the 95% CI for the mean?

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.25

Introducing Hypothesis Testing - 1• A car manufacturer releases a new car and claims

that its urban cycle fuel efficiency is 18.5 km per litre. A car enthusiast magazine decides to test this claim.

- Null hypothesis

• H0: = 18.5

- Alternative hypothesis is

• H1: ≠ 18.5

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.26

Introducing Hypothesis Testing - 2

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.27

Introducing Hypothesis Testing - 3

1 5.00% 2.50% 1.00% 0.50%2 10.00% 5.00% 2.00% 1.00% 90.00% 95.00% 98.00% 99.00%

= n - 11 6.3138 12.7062 31.8205 63.65672 2.9200 4.3027 6.9646 9.92483 2.3534 3.1824 4.5407 5.84094 2.1318 2.7764 3.7469 4.6041

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.28

Introducing Hypothesis Testing - 4

2.5%2.5%

17.20 18.38

95%

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.29

Recap

By the end of this topic students will be able to:• Recognise the terms sample statistic and

population parameter• Use confidence intervals to indicate the reliability of

estimates• Know when approximate large sample and exact

confidence intervals are appropriate

©NCC Education LimitedV1.0

Inferential Statistics 1 Topic 3 - 3.30

Bibliography• Dewhurst, F. Quantitative Methods for Business and

Management. McGraw-Hill.• Hinton, PR. Statistics Explained Routledge• Oakshot, L. Essential Quantitative Methods for

Business, Management and Finance. Palgrave Macmillan.

Inferential Statistics 1 Topic 3 - 3.31

©NCC Education LimitedV1.0

Topic 3 – Inferential Statistics 1

Any Questions?