Hypothesis Tests

DNA Pot (c) 2009 - Current

Test population Mean with Small sample size, σ known

Here, let’s see how to test statistical hypotheses about one population mean when sample size is small. The sample size is considered small when it is less than 30.

Suppose we have a random sample of    X1, X2, X3, ..., Xn

the sample size is clearly n, and in this case it is small (n < 30)

This sample is drawn from a population

The probability density function is let say f(x)

The population mean is μ

The population standard deviation is σ

The population variance is σ2

The sample mean is X bar

The sample variance is S2

Then we have three hypothesis we want to test about the population mean, each having one null hypothesis versus an alternative hypothesis.

Step 1

State the null hypothesis

Ho : μ = μ0

Step 2

State the alternative hypothesis

(1) Ho : μ < μ0

(2) Ho : μ > μ0

(3) Ho : μ ≠ μ0

Step 3

Assign an appropriate value of level of significance α

Common Values: 0.01 (1%), 0.05 (5%), and 0.10 (10%)

Pick One: Here 0.05 is picked in this case, or it is given in the question

Step 4

Determine a suitable test statistic

The parameter under investigation is used as test statistic

The test statistic is

z = (X bar - μ)/(σ/√n)  for  μ

This test statistic is used to test hypothesis (1), (2), (3) about μ

Step 5

Determine the probability distribution in the test statistic

Since, the sample is small (n< 30), we must verify that the data come from a population that is approximately normal with no outliers.

Use the normal probability plot and boxplot to determine these requirements. Sometimes, the question will assume the population is normal.

Step 6

Locate all rejection region and find the critical point

The location of rejection region depends on the alternative hypothesis.

1. (1)Ho : μ < μ

The rejection region locates on the left  tail because μ < μ0

1. (2)Ho : μ > μ

The rejection region locates on the right tail because μ > μ0

1. (3)Ho : μ ≠ μ

The rejection region locates on both tails because μ ≠ μ0

Step 7

Find the critical point

The critical point depends on the value assigned to size of the rejection region or the level of significance which is type I error probability α

the level of significance = P( type I error) = α = 0.05

the critical point values can be read out of the table for α = 0.05

in each case (1),(2), and (3)

Level of significance α      Left-tail     Right Tail        Both Tail

0.01                                       -2.33          +2.33               -2.575 and +2.575

0.05                                       -1.645        +1.645             -1.96 and +1.96

0.10                                        -1.28          +1.28             -1.645 and +1.645

location of rejection region depends on the alternative hypothesis.

1. (1)Ho : μ < μ

The rejection region locates on the left  tail because μ < μ0

1. (2)Ho : μ > μ

The rejection region locates on the right tail because μ > μ0

1. (3)Ho : μ ≠ μ

The rejection region locates on both tails because μ ≠ μ0

Step 8

Calculate the test statistic

Take a random sample from a population in question, calculate the value of the test statistic.

Take a random the sample X1, X2, X3, ..., Xn

Make sure n < 30, and the population is given normal

Calculate X bar = sample mean

σ and n are given or known

μ is the μ0 from the null hypothesis

use all values in calculating test statistic “ z =(X bar - μ)/(σ/√n)  for  μ”

Step 9

Make decision

Determine whether it falls in the rejection region. If it falls in the rejection region, we reject the null hypothesis. If it doesn’t fall in the rejection region, keep it.

Check the value from the calculation of test statistic

“ z =(X bar - μ)/(σ/√n)  for  μ”

If it falls in the rejection regions (1),(2) and (3) mentioned above, we reject the null hypothesis. If it doesn’t fall in the rejection region, keep it.