Statistics

20th January 2020

The views expressed in this presentation are those of

the presenter(s) and not necessarily of the Society of

Actuaries in Ireland

© The Society of Actuaries in Ireland

Disclaimer

• Statistics

• Question Focused Approach

• 2 hours - 10mins break @7pm

• Any questions? – Just Ask!!

Welcome!

• www.themathstutor.ie

Useful Resource for Project Maths

• Slides, questions and solutions available from website:

https://web.actuaries.ie/students/maths-tutorials-higher-

level-leaving-certificate-20192020

Or google ‘actuaries maths tutorials’

Material from tonight

•Read the question carefully

•Label your diagrams

•Key formulae: Pages 33 and 34.

•Tables: Pages 36 and 37.

Exam Technique

•Mean is the average value

•Median is the middle value

•Mode is the most frequent value

•Variance and Standard Deviation measure how the data differ from

the mean

Statistical Measures

Presenting Data

Histograms• A graph used to display the data and where the data is grouped• Bars in a histogram must be equal width• Examples:

Frequency Distribution

Distribution• The distribution of a data-set is a function that shows all possible values

of the data and how often they occur.

Shape• Symmetric

• The left side mirrors the right side

• Skewed to the left (Negative skew)• The tail is longer on the left hand side

• Skewed to the right (Positive skew)• The tail is longer on the right hand side

Normal Distribution

The normal distribution is a continuous, symmetric, bell shaped probability distribution, not skewed to the left or to the right.

10

Key Points:- Symmetrical about mean- Mean = Median = Mode- Bell Curve

Empirical Rule

For a Normal Distribution with mean 𝜇:

Standard Normal Distribution

This is a special case of a normal distribution where the mean is 0 and the standard deviation is 1

Z-Scores

A z-score is the number of standard deviations that a value lies above or below the mean.

This is calculated as follows:

Means

Population

• 𝜇: Population mean• 𝜎: Standard Deviation of

population

Sample

• 𝑛 =Sample size• ҧ𝑥: Sample mean• 𝜎 ҧ𝑥: Standard Error of sample

mean

• 𝜎 ҧ𝑥 =𝜎

𝑛

Proportions

Population

• 𝑝: Population proportion

Sample

• 𝑛 =Sample size• Ƹ𝑝: Sample proportion (p hat)• Standard error for sample

proportion 𝜎 ො𝑝

• 𝜎 ො𝑝 =𝑝(1−𝑝)

𝑛

Confidence Intervals

A confidence interval for the mean gives us a plausible estimate for the population mean.

A 95% confidence interval for the population mean is given by:ҧ𝑥 ± 1.96𝜎 ҧ𝑥

A 95% confidence interval for a population proportion is given by:

Ƹ𝑝 ± 1.96𝑝(1 − 𝑝)

𝑛

Hypothesis Test

Hypothesis Test for Mean1. State the null hypothesis (𝐻0), state the alternative hypothesis (𝐻𝐴)2. Calculate Z-Value (Test statistic)3. If 𝑍 < −1.96 𝑜𝑟 𝑍 > 1.96 reject the null hypothesis4. If −1.96 ≤ 𝑍 ≤ 1.96 fail to reject the null hypothesis5. State conclusion

Hypothesis Test for Proportion1. State the null hypothesis (𝐻0), state the alternative hypothesis (𝐻𝐴)2. Calculate Ƹ𝑝

3. Calculate confidence intervals for Ƹ𝑝 using Margin of Error, E =1

𝑛

4. If p lies within the confidence interval fail to reject 𝐻0, if p lies outside confidence interval reject 𝐻0

5. State conclusion

Key Points:The null hypothesis is a statement.

• Monday 27th January

• Trigonometry 1

• 6 - 8 pm

• Same Location

• https://web.actuaries.ie/students/maths-tutorials-higher-level-leaving-certificate-20192020or google “actuaries maths tutorials”

Next tutorial

• Follow ”saimathstutorials”

• New Instagram Page created for reminders when solutions are made available and providing information about upcoming tutorials.

Instagram Page