# 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 One Tailed H0= k H1 ≠ k 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 μ 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 μ 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:

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.