Download Hypothesis Testing: One-tail and Two

Hypothesis Testing: One-tail and Two-tail Tests
In order to understand the difference between one- and two-tail hypothesis tests, it is helpful
to examine the equation for the t-test.
While you do not have to use this equation (Excel plugs in the numbers and solves the equation
for you), if we look at the numerator of the fraction- we can see the bases of
one- and two-tail hypotheses.
A. Two-tail Hypothesis.
A two-tail hypothesis simply asserts that the means of the two samples are equal. In
math symbols, the null hypothesis is:
= 0. In words, this statement is:
there is no (0) difference between the means of the 2 samples; or, the mean of the first
sample equals the mean of the second sample.
With a null hypothesis in this form, we decide to reject the null hypothesis under these
conditions:
1) The sign of “t Stat” (in the Excel output) is either positive or negative; and
2) The p-value is equal to or less than the alpha (α) level we have chosen. [Recall that in the
social sciences this is typically 0.05 or 0.01.]
B. One-tail Hypothesis- Left-tail.
A left-tail test asserts that the mean of the first sample is LESS than the mean of the second
sample. In math symbols:
<0.
In words, this expression states that the mean of the first sample is less than the mean of the
second sample; if this hypothesis is valid, if we subtract the second mean from the first, the
result will be a number less than zero (a negative number).
This is a left-tail research hypothesis.
The corresponding null hypothesis is in the form:
≥ 0 . In words, this
hypothesis states that the mean of the first sample is greater than or equal to the mean of
the second sample. [Why is it called a left-tail test? Look at the direction of the inequality
sign in the research hypothesis- it points to the left.]
With a null hypothesis in this form, we decide to reject the null hypothesis under these
conditions:
1) The sign of “t Stat” (in the Excel output) is negative; and
2) The p-value [P(T<=t) one-tail] is equal to or less than the alpha (α) level we have chosen.
Note that both of these conditions must be met if we are to decide to reject the null
hypothesis. Note also that if the sign of t is positive, we cannot reject the null hypothesis.
B. One-tail Hypothesis- Right-tail.
A right-tail test asserts that the mean of the first sample is GREATER than the mean of the second
sample. In math symbols:
>0.
In words, this expression states that the mean of the first sample is greater than the mean of
the second sample; if this hypothesis is valid, if we subtract the second mean from the first, the
result will be a number greater than zero (a positive number).
This is a right-tail research hypothesis.
The corresponding null hypothesis is in the form:
< 0 . In words, this hypothesis
states that the mean of the first sample is less than or equal to the mean of the second sample.
[Why is it called a right-tail test? Look at the direction of the inequality sign in the research
hypothesis- it points to the right.]
With a null hypothesis in this form, we decide to reject the null hypothesis under these
conditions:
1) The sign of t (in the Excel output) is positive; and
2) The p-value [P(T<=t) one-tail] is equal to or less than the alpha (α) level we have chosen.
Note that both of these conditions must be met if we are to decide to reject the null
hypothesis. Note also that if the sign of t is negative, we cannot reject the null hypothesis.
Testing Hypotheses About Mean Incomes of Male and Female Respondents
A. Two-tail Hypothesis.
1. Null hypothesis:
2. Research hypothesis:
= 0; (Mean male income equal to mean female income;
no difference between mean incomes)
≠ 0; (Mean male income not equal to mean female
income)
3. Significance level (α): α = 0.05 ;
4. Data collection and descriptive statistics (data from “voter.xls” file);
5. Significant test: t-test – Excel output:
T Stat is negative (in 2-tail test, sign is not important).
P(T<=t) two-tail = 0.8343… (probability of Type I error).
6. Decision: Do not reject hull hypothesis. [P(T<=t) > 0.05]
Testing Hypotheses About Mean Incomes of Male and Female Respondents
A. Left-tail Hypothesis.
1. Null hypothesis:
≥ 0; (Mean male income greater than or equal to mean
female income)
2. Research hypothesis:
< 0; (Mean male income less than mean female
income)
3. Significance level (α): α = 0.05 ;
4. Data collection and descriptive statistics (data from “voter.xls” file);
5. Significant test: t-test – Excel output:
T Stat is negative *
P(T<=t) one-tail = 0.4171… (probability of Type I error).
6. Decision: Do not reject hull hypothesis. [P(T<=t) > 0.05]
*In left-tail hypothesis, a negative T Stat might be the basis of a decision to reject the null
hypothesis. However, the P-value [P(T<=t) one-tail] must be lower than the chosen level
of significance (α). In this illustration, the P-value = 0.4171 is greater than 0.05; hence,
our decision is NOT to reject the null hypothesis.
Testing Hypotheses About Mean Incomes of Male and Female Respondents
A. Right-tail Hypothesis.
1. Null hypothesis:
≤ 0 ; (Mean male income less than or equal to mean
female income)
2. Research hypothesis:
> 0; (Mean male income greater than mean female
income)
3. Significance level (α): α = 0.05 ;
4. Data collection and descriptive statistics (data from “voter.xls” file);
5. Significant test: t-test – Excel output:
T Stat is negative
P(T<=t) one-tail = 0.4171… (probability of Type I error).
6. Decision: Do not reject hull hypothesis. [P(T<=t) > 0.05]
*In right-tail hypothesis, a negative T Stat is the basis for a decision to reject the null
Hypothesis. However, the P-value [P(T<=t) one-tail] must be lower than the chosen level
of significance (α). In this illustration, the P-value = 0.4171 is greater than 0.05; hence,
our decision is NOT to reject the null hypothesis.
What happens when we reverse the samples? That is, what will be the results of our
hypothesis testing if we treat the females as Sample 1 and the males as Sample 2? The
next 3 slides demonstrate the outcome(s) when we reverse the samples.
Testing Hypotheses About Mean Incomes of Male and Female Respondents
A. Two-tail Hypothesis.
1. Null hypothesis:
2. Research hypothesis:
= 0; (Mean female income equal to mean male income;
no difference between mean incomes)
≠ 0; (Mean female income not equal to mean male
income)
3. Significance level (α): α = 0.05 ;
4. Data collection and descriptive statistics (data from “voter.xls” file);
5. Significant test: t-test – Excel output:
T Stat is positive (in 2-tail test, sign is not important).
P(T<=t) two-tail = 0.8343… (probability of Type I error).
6. Decision: Do not reject hull hypothesis. [P(T<=t) > 0.05]
Testing Hypotheses About Mean Incomes of Male and Female Respondents
A. Left-tail Hypothesis.
1. Null hypothesis:
≥ 0; (Mean female income greater than or equal to mean
male income)
2. Research hypothesis:
< 0; (Mean female income less than mean male
income)
3. Significance level (α): α = 0.05 ;
4. Data collection and descriptive statistics (data from “voter.xls” file);
5. Significant test: t-test – Excel output:
T Stat is positive *
P(T<=t) one-tail = 0.4171… (probability of Type I error).
6. Decision: Do not reject hull hypothesis. [P(T<=t) > 0.05]
*In left-tail hypothesis, a positive T Stat might be the basis of a decision to reject the null
hypothesis. However, the P-value [P(T<=t) one-tail] must be lower than the chosen level
of significance (α). In this illustration, the P-value = 0.4171 is greater than 0.05; hence,
our decision is NOT to reject the null hypothesis.
Testing Hypotheses About Mean Incomes of Male and Female Respondents
A. Right-tail Hypothesis.
1. Null hypothesis:
≤ 0 ; (Mean male income less than or equal to mean
female income)
2. Research hypothesis:
> 0; (Mean male income greater than mean female
income)
3. Significance level (α): α = 0.05 ;
4. Data collection and descriptive statistics (data from “voter.xls” file);
5. Significant test: t-test – Excel output:
T Stat is positive
P(T<=t) one-tail = 0.4171… (probability of Type I error).
6. Decision: Do not reject hull hypothesis. [P(T<=t) > 0.05]
*In right-tail hypothesis, a positive T Stat is the basis for a decision to reject the null
hypothesis. However, the P-value [P(T<=t) one-tail] must be lower than the chosen level
of significance (α). In this illustration, the P-value = 0.4171 is greater than 0.05; hence,
our decision is NOT to reject the null hypothesis.