Lecture 18 Slides
.pdfIntroduction 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:
qi = |
i 0:5 |
; zi = 1(qi) |
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n |
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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)
Reject H0 |
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Do not reject H0 |
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