Lecture 18 Slides

Analogy

Consider an accused person in court.

H0: not guilty, Ha: guilty

Reject H0, accept Ha =) send to prison

Not reject H0 =) let him go

Usually, H0 means that some e ect does not exist, Ha means it exists

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A test is defined by a function

f(X1; : : : ; Xn)

of a sample X1; : : : ; Xn, where f takes values 1 or 0 (reject/not reject).

The function f may take di erent values for di erent realizations of a sample X1(!); : : : ; Xn(!) =) the conclusion is random.

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Testing hypothesis about the mean of a population

Our standard setting will be as follows.

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Example

Suppose the population is normal, N( ; 2). We want to test

H0 : = 0; Ha : 6= 0

Example 1. Suppose we have the sample of 10 elements

0:96; 1:67; 1:44; 1:45; 1:05; 0:16; 0:69; 0:84; 1:1; 1:08

Solution: construct a 95% confidence interval,

sb

= X t0:025(9)p

10

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We have

= 0:754 0:63 at the confidence level 95% Thus it is unlikely that = 0 so we reject H0.

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Example 2

Suppose in the same example we obtained the sample (from another population)

0:72; 0:49; 0:05; 0:19; 0:52; 0:80; 1:87; 1:39; 0:46; 2:05

The 95% confidence interval is

= 0:072 0:81

We cannot reject H0 : = 0.

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Type I and Type II errors

The conclusion (reject/not reject H0) may be wrong.

Type I error: reject H0 when H0 is true

Type II error: not reject H0 when H0 is false

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The probability of type I error of a test is denoted by :

= P(reject H0 j H0 is true)

The probability of type II error of a test is denoted by :

= P(not reject H0 j H0 is false)

is the significance of the test.

1 is the power of the test.

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Example

The proportion of defective items produced by a plant was 9%. Improvement has been made with the goal to reduce the rate to 5%.

Consider the following test:

Randomly select n = 200 details and if the number of defective details is not greater than m = 8 the improvement is considered as successful, otherwise unsuccessful.

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Let X Bernoulli(p) is the current rate of defective items. We want to test

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