Download - Lecture 03 - Inferential Statistics 1
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Advanced Mathematics for Business
Topic 3:
Inferential Statistics 1
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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
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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
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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
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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
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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%
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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
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Inferential Statistics 1 Topic 3 - 3.8
Populations and Samples - 1
Population Parameters
Sample Statistics?
?
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Inferential Statistics 1 Topic 3 - 3.9
Populations and Samples - 2
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Inferential Statistics 1 Topic 3 - 3.10
Distribution of Sample Means
Population DistributionDistribution of Sample Means
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Inferential Statistics 1 Topic 3 - 3.11
Central Limit Theorem
n > 30
Population Distribution Distribution of Sample Means
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Inferential Statistics 1 Topic 3 - 3.12
Populations and Samples - 3
Population Parameters
Sample Statistics
?
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Inferential Statistics 1 Topic 3 - 3.13
95%
z = -1.96 z = 1.96
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Inferential Statistics 1 Topic 3 - 3.14
95%
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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
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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
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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
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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
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Inferential Statistics 1 Topic 3 - 3.19
Large Sample CI of Proportion
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Inferential Statistics 1 Topic 3 - 3.20
Example
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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)
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Inferential Statistics 1 Topic 3 - 3.22
Student t-distribution
t(1)
t(30)t(4)
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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
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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?
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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
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Inferential Statistics 1 Topic 3 - 3.26
Introducing Hypothesis Testing - 2
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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
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Inferential Statistics 1 Topic 3 - 3.28
Introducing Hypothesis Testing - 4
2.5%2.5%
17.20 18.38
95%
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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
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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
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Topic 3 – Inferential Statistics 1
Any Questions?