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 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).


Two Tailed Test


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:

Confidence Interval

or:

Confidence Interval


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

One Tailed Test

One Tailed Test

One Tailed Test

One Tailed Test


The significance level, α, is concentrated in one part or tail.

The limit of acceptance in this case is:

Limit of Acceptance

or:

Limit of Acceptance


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:

Confidence Interval

3. Verify:

Proportion of the Sample

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).


One Tailed Test


The significance level, α, is concentrated in one part or tail.

The limit of acceptance in this case is:

Limit of Acceptance

or:

Limit of Acceptance


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:

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.





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