Hypothesis Testing
To perform a hypothesis test, there are four steps that must be followed.
1. State the null H0 and the alternative H1 hypothesis.
| Two Tailed | H0= k | H1 ≠ k |
|---|---|---|
| One Tailed | H0 ≥ k | H1 < k |
| H0 ≤ k | H1 > k |
2.From a confidence level, (1 − α), or a significance level, α, determine:
The value of zα/2 (two tailed), or zα (one tailed)
The limit of acceptance of the sample parameter (x o p').
3.Calculate: the value of x or p' from the sample.
4.If the sample parameter value is within the limit of acceptance, accept the hypothesis with a significance level α.
If not, reject it.
Two Tailed Test
A two-tailed test occurs when the null hypothesis is of the type H0: μ = k (or H0: p = k) and the alternative hypothesis, therefore, is of the type H1: μ≠ k (or H1: p≠ k).
The significance level, α, is concentrated in two parts (or tails) symmetrical about the mean.
The limit of acceptance in this case is the corresponding confidence interval for x or p', that is to say:
or:
Example
It is known that the standard deviation of the scores in a math exam was 2.4 and a sample of 36 students scored an average of 5.6. With this data, can the hypothesis be confirmed that the average test score was 6 with a confidence level of 95%?
1. State the null and alternative hypotheses:
H0 : μ = 6 The average test score has not varied.
H1 : μ ≠ 6 The average test score has varied.
2. Calculate the limit of acceptance:
For a significance level of α = 0.05, the corresponding critcal value is: zα/2 = 1.96.
Calculate the confidence interval for the mean:
(6 − 1.96 · 0.4, 6 + 1.96 · 0.4) = (5.22, 6.78)
3. Verify:
The value of the mean of the sample is: 5,6 .
4. Decide:
The nule hypothesis, H0, should be accepted with a confidence level of 95%.
One Tailed Test
Case 1
The nule hypothesis is of the type H0: μ ≥ k (or H0: p ≥ k).
The alternative hypothesis, therefore, is of the type H1: μ < k (or H1: p < k).
Critical Values
| 1 - α | α | zα |
|---|---|---|
| 0.90 | 0.10 | 1.28 |
| 0.95 | 0.05 | 1.645 |
| 0.99 | 0.01 | 2.33 |
The significance level, α, is concentrated in one part or tail.
The limit of acceptance in this case is:
or:
Example
A sociologist has predicted that in a given city, the level of absenteeism in the upcoming elections will be a minimum of 40%. From a random sample of 200 individuals from the voting population, 75 state they will likely vote. Determine with a significance level of 1%, if the hypothesis can be accepted.
1. State the null and alternative hypotheses:
H0 : p ≥ 0.40 The absenteeism will be a minimum of 40%.
H1 : p < 0.40 The absenteeism will be a maximum of 40%.
2. Calculate the limit of acceptance:
For a significance level of α = 0.01, the corresponding critcal value is: zα = 2.33.
Determine the confidence interval for the mean:
3. Verify:
4. Decide:
The nule hypothesis, H0, should be accepted as it can be stated with a significance level of 1% that absenteeism will be at least 40% for the upcoming election.
Case 2
The nule hypothesis is of type H0: μ ≤ k (or H0: p ≤ k).
The alternative hypothesis is, therefore, of type H1: μ > k (or H1: p > k).
The significance level, α, is concentrated in one part or tail.
The limit of acceptance in this case is:
or:
Example
A report indicates that the maximum price of a plane ticket between New York and Chicago is $120 with a standard deviation of $40. A sample of 100 passengers shows that the average price of their tickets was $128.
Can the above statement be accepted with a significance level equal to 0.1?
1. State the null and alternative hypotheses:
H0 : μ ≤ 120
H1 : μ > 120
2. Calculate the limit of acceptance:
For a significance level of α = 0.1, the corresponding critical value is: zα = 1.28.
Calculate the confidence interval for the mean:
3. Verify:
The value of the mean of the sample is: $128.
4. Decide:
The nule hypothesis, H0, cannot be accepted with a significance level equal to 0.1.
