z-Test

The z-test is the ideal hypothesis test to conduct in the presence of normal distribution of the random variable. In addition, the variance of the population must be known. The z-statistic refers to the test statistic computed for the purpose of hypothesis testing.

Testing H0: μ = μ0 using the z-Test

Given:

• A random sample of size n from a normally distributed population with mean μ and variance σ2,
• Sample mean X’

We can compute the z-statistic where

z-statistic = (X’ – μ0)/(σ/√n)

such that:

X’ is the sample mean

μ0 is the hypothesized mean of the population

σ is the standard deviation of the population

and n is the sample size.

Once computed, the z-statistic is compared to the critical value that corresponds to the level of significance of the test. For example, if the significance level is 5%, the z-statistic is screened against the upper/lower 95% point of the normal distribution (±1.96). The decision rule is to reject H0 if the z-statistic falls within the critical/rejection region.

Example 1

Academics carried out a study on 50 former United States presidents and found an average IQ of 135. You are required to carry out a 5% statistical test to determine whether the average IQ of presidents is greater than 130. (IQs are normally distributed and previous studies indicate σ = 25)

Solution

First, the candidate has to state the hypothesis:

H0: μ = 130

H1: μ > 130

Assuming H0 is true, (X’ – 130)/(σ/√n)  ~N(0,1)

The z-statistic is (135 – 130)/(25/√50) = 1.414

This is a right-tailed test. Therefore, we compare our test statistic to the upper 95% point of the standard normal distribution (1.6449). Since 1.414 is less than 1.6449, we do not have sufficient evidence to reject H0. As such, it would be reasonable to conclude that the average IQ of U.S. presidents is not more than 130.

The t-Test

The t-test is based on the t-distribution. The test is appropriate for testing the value of a population mean when:

• σ is unknown
• The sample size is large (n ≥ 30) and if n < 30, the distribution must either be normal or approx normal.

Testing H0: μ = μ0 Using the t-Test

We compute a t-statistic with n – 1 degrees of freedom as:

tn-1 = (X’ – μ0)/(s/√n)

Where

X’ is the sample mean

μ0 is the hypothesized mean of the population

s is the standard deviation of the sample

and n is the sample size

Example 2

The annual rate of rainfall (cm) in a certain equatorial country over the last 10 years is given below:

25   26   25    27   28   29   28   27   26   25

Financial analysts in the country wish to determine whether the average rate of rainfall has increased from its former value of 23. Carry out a statistical test at the 5% level.

Solution

As always, you should begin by stating the hypothesis:

H0: μ = 23

H1: μ > 23

If we assume that the annual rainfall quantities are normally distributed and recorded independently, then

(X’ – 23)/(S/√n)  ~tn- 1

Please confirm that X’ = 26.6 and S = 1.43

Therefore, our t-statistic = (26.6 – 23)/(1.43/√10) = 7.96

Our test statistic (7.96) is greater than the upper 95% point of the t9 distribution (1.833). Therefore, we have sufficient evidence to reject H0. As such, it’s reasonable to conclude that the average annual rainfall has increased from its former long-term average of 23.

Question

Determine the value of t9 in the example above if the level of significance is reduced from 5% to 0.5%. Does the reduction affect the decision rule?

A. 3.25

B. 2.821

C. 3.3

Solution

A quick glance at the t9 distribution when α = 0.5% gives a value of 3.25. However, the evidence against H0 is too strong since our test statistic (7.96) is still greater than 3.25. As such, the conclusion would remain unchanged.

(Note to candidates: You might as well work out the solutions to the above examples and question using p-values in lieu of critical values. The decision rules would remain unchanged provided you work out the p-values correctly.)

Identify the appropriate test statistic and interpret the results for a hypothesis test concerning the population mean of both large and small samples when the population is normally or approximately distributed and the variance is 1) known or 2) unknown

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