Lecture 18 Slides

Introduction to Probability Theory and Statistics

Lecture 18, 3.02.2015

Plan

1.Graphical methods to check normality

2.Hypothesis testing

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How to check a sample is obtained from a normal population?

Method 1. Histogram (left: normal, right: non-normal)

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Method 2. Histogram + density function (left: normal, right: non-normal)

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Method 3. Normal probability plot

If a sample is from a normal population, the plot will be approximately linear

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The algorithm to graph a normal probability plot

Suppose you have a sample x(1); : : : ; x(n), sorted from the lowest number to the highest.

Calculate the corresponding quantiles of the normal distribution:

Then plot the points (x(i); zi).

If the sample is obtained from N( ; 2), the points will lie along the line x = + z

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Examples of non-normal samples

Left: outliers, right – “skewed” npp

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An additional method: check the spread

If a sample is obtained from a normal population, then

approximately 68% of points are within b sb approximately 95% of points are within b 2sb

If these conditions are not satisfied, the population is unlikely to be normal.

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Statistical hypothesis testing

A statistical hypothesis is a statement about a population.

Examples of hypotheses:

1. The mean salary in the city is not less than 45 000 RUB

2.The distribution of students’ marks is normal

3.The die is fair

Not hypotheses:

1.The sample mean is 100

2.The sampling method produces a simple random sample

3.The number of heads when the coin is tossed 10 times will be 7

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The procedure of hypothesis testing

We formulate two hypotheses: the null hypothesis H0 and the alternative hypothesis Ha.

Then we try to reject H0 based on evidence from a sample. If H0 is rejected, we accept Ha.

Example

H0: The concentration of CO in the city is within the safe limit, i.e. H0 : 6 x

Ha: The concentration of CO is above the safe limit, i.e. Ha : > x

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Sample x1; : : : ; xn

Test

(process data)

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