Chapter 9 - Hypothesis tests for one population mean

tests involve 2 hypothesis

1. null hypothesis

2. alternative hypothesis

null hypothesis

a hypothesis to be tested, use symbol H₀ to represent this hypothesis

alternative hypothesis

a hypothesis to be considered as an alternative to the null hypothesis, use symbol H_a to represent this hypothesis

hypothesis test

the problem in a hypothesis test us ti decide whether the null hypothesis should be rejected in favor of the alternative hypothesis

choosing hypothesis - null hypothesis

First step: deciding on the null and alternative hypothesis

• hypothesis concerning a population mean, null hypothesis always specifies a single value for that parameter

• null hypothesis as H₀:μ = μ₀ (μ₀ = some #)

choosing hypothesis - alternative hypothesis

3 choices are possible for alternative hypothesis

1. two tailed

2. left tailed

3. right tailed

two tailed alternative test

• primary concern is deciding whether a population mean, μ, is different from a specified value μ₀

• express the alternative hypothesis as H_a: μ does not equal μ₀

left tailed alternative hypothesis

• primary concern is deciding whether a population mean μ, is LESS than a specified value μ₀

• we express the alternative hypothesis as H_a: μ < μ₀

right tailed alternative hypothesis

• primary concern is deciding whether a population mean, μ, is GREATER than a specified value μ_0,

• we express the alternative hypothesis as H_a: μ > μ₀

one tailed test

form is called this if it is either left or right tailed

logic of hypothesis testing

2nd step - after choosing null and alternative hypothesis, we must decide whether to reject the null hypothesis in favor of alternative hypothesis

• take random sample from population

• if the sample data are consistent with null hypothesis do not reject null hypothesis

• if the sample data are inconsistent with the null hypothesis and supportive of alternative hypothesis, reject null for alternative

Type I error

REJECTING the null hypothesis when it is in fact TRUE

denoted a

Type II error

NOT REJECTING the null hypothesis when it is in fact FALSE

denoted B

Type I and II probabilites

both should have small probabilities

• smaller we specify the significance level, a, the larger the probability of B of not rejecting a false null hypothesis

critical value approach to hypothesis testing

in approach we choose a cutoff point (s) based on significance level of the hypothesis

• uses criterion (z-scores)

rejection region

the set values for the test statistic that leads to rejection of the null hypothesis

non rejection region

the set values for the test statistic that leads to non rejection of null hypothesis

critical value (s)

value or values of the test statistic that separate the rejection and non rejection regions (critical value is considered part of rejection region)

2 tailed tests

• null hypothesis is rejected when test statistic is either too small or too large

• rejection region for test consists of 2 parts: one on left and one on right

left tailed test

• null hypothesis is rejected only when the test statistic is too small

• rejection region for test has one part, on the left

right tailed test

• null hypothesis is rejected only when test statistic is too large

• rejection region for test has one part, on the right

obtaining critical values

• a hypothesis test is to be performed @ the significance level, a, (the % of saying test is wrong, rejecting the test even when its actually right)

• critical value(s) must be chosen so that if the null hypothesis is true, the probability is a that the statistic willfall in rejection region 

significance level

probability of rejecting a true null hypothesis

one mean z-test

procedure is used to perform a hypothesis test for one population mean when the population standard deviation is known and the variable under consideration is normally distributed

central limit theorem

test will work when sample size is large

Table 9.4 - common critical values of Z

Z_0.10 = 1.28

Z_0.05 =1.645

Z_0.025 = 1.96

Z_0.01 = 2.33

Z_0.005 = 2.575

Table 9.5 - critical value approach to hypothesis testing 

1. state null and alternative hypotheses

2. decide on the significance level, a

3. compute the value of the test statistic

4. determine the critical value (s)

5. if value of the test statistic falls in rejection region, reject H₀; otherwise do not reject null

6. interpret the result of hypothesis test

p-value

probability of getting a test statistic (z-score) as small or smaller than the one we observed, if the null hypothesis is true (P); also known as observed significance level

obtaining criterion

• if p-value is less than or equal to specified significance level, we reject null hypothesis

• if p-value is greater than specified significance level, we do not reject null hypothesis 

*smaller (closer to 0) the p-value is the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis

Table 9.7: p-value approach to hypothesis testing

1. state the null and alternative hypotheses

2. decide on the significance level, a

3. compute the value of the test statistic

4. determine the p-value, P

5. if P is less than significance level reject null, otherwise do not reject

6. interpret the result of hypothesis test