**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 H _{0}: μ = μ_{0} using the z-Test**

Given:

- A random sample of size
*n*from a normally distributed population with mean μ and variance σ^{2}, - Sample mean X’

We can compute the z-statistic where

z-statistic = (X’ – μ_{0})/(σ/√n)

such that:

X’ is the sample mean

μ_{0 }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 H_{0 }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:

H_{0}: μ = 130

H_{1}: μ > 130

Assuming H_{0} 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 H_{0}. 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 H _{0}: μ = μ_{0} Using the t-Test**

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

t_{n-1} = (X’ – μ_{0})/(s/√n)

Where

X’ is the sample mean

μ_{0 }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:

H_{0}: μ = 23

H_{1}: μ > 23

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

(X’ – 23)/(S/√n) ~t_{n- 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 t_{9} distribution (1.833). Therefore, we have **sufficient evidence** to reject H_{0}. As such, it’s **reasonable** to conclude that the average annual rainfall has increased from its former long-term average of 23.

QuestionDetermine the value of t

_{9}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

SolutionThe correct answer is A.

A quick glance at the t

_{9}distribution when α = 0.5% gives a value of 3.25. However, the evidence against H_{0}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.)*

*Reading 12LOS 12g:*

*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*