Lecture 03 - Inferential Statistics 1

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Advanced Mathematics for Business

Topic 3:

Inferential Statistics 1


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



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



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



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



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%



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



Populations and Samples - 1

Population Parameters

Sample Statistics?

?



Distribution of Sample Means

Population DistributionDistribution of Sample Means



Central Limit Theorem

n > 30

Population Distribution Distribution of Sample Means




Population Parameters

Sample Statistics

?



95%

z = -1.96 z = 1.96



95%



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



Interpreting 95% Confidence Interval

1 in 20 chance of not containing population

mean

19 in 20 chance of containing population

mean



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



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



Large Sample CI of Proportion



Example



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)



Student t-distribution

t(1)

t(30)t(4)



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



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?



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



Introducing Hypothesis Testing - 2




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




2.5%2.5%

17.20 18.38

95%



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



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.



Topic 3 – Inferential Statistics 1

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Lecture 03 - Inferential Statistics 1

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