Test Statistic [Two-Sample Z-Procedure]

The Problem

Example #228: Let us test whether the mean height of population 1 is 1 less than that of population 2. In symbols, this is:

H₀ : μ₁ − μ₂ = -1
H_A : μ₁ − μ₂ ≠ -1

To test this hypothesis, we collect data. The data from population 1 are:

37, 36, 40, 50, 36, 47, 43, 36, 44, 38, 37, 25, 40, 39, 41, 24, 37, 39, 38, 25, 43, 40, 34, 32, 38, 43, 21, 43, 25, 34, 41, 30, 32, 32, 34, 31, 39, 42, 37, 33, 29

The data from population 2 are:

38, 30, 46, 37, 40, 41, 32, 37, 31, 31, 30, 35, 36, 27, 31, 32, 33, 49, 49, 35, 52, 32, 28, 30, 32, 41, 38, 36, 36, 44, 49, 15, 45, 28, 32, 23, 24, 32, 45, 37, 39, 48, 44, 31, 37, 31, 18, 46, 44, 29, 33, 25

In addition to these data, we also know two important things. First, the data are generated from independent Normal processes; that is, the data in each sample come from independent Normal distributions. Second, we know that the population variances are σ²₁ = 49 and σ²₂ = 64. With this information, calculate the test statistic corresponding to the null hypothesis.

Information given:

To summarize the above, the values of import are:

Summary statistics from the problem
\( \mu_d \)	=	-1

\( \bar{x}_1 \)	=	36.2195
\( \bar{x}_2 \)	=	35.4615

\( \sigma_1 \)	=	7
\( \sigma_2 \)	=	8

\( n_1 \)	=	41
\( n_2 \)	=	52

Calculate these values yourself then hover your mouse over the grey spaces to see if you calculated them correctly.

Your Answer

In the box below, please enter the value of the test statistic for the null hypothesis and the data given above. Once you have done so, click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

Your guess:

Another Set of Data

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Assistance

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The z-procedures are sensitive to knowing the population variance. Logic dictates that if we do not know the population mean, then we will not know the population variance. As such, the z-procedures are rarely used now. As such, there is no z-test in the base Excel program.

Copy and paste the following code into your Excel spreadsheet window, making sure the value sample1 ends up in A1 after pasting.

How to calculate the expected value in Excel.
sample1	sample2
37	38	mu d:	-1
36	30	sigma1:	7
40	46	sigma2:	8
50	37
36	40	ts:	=(AVERAGE(A:A)-AVERAGE(B:B)-D2)/SQRT(D3^2/COUNT(A:A)+D4^2/COUNT(B:B))
47	41
43	32
36	37
44	31
38	31
37	30
25	35
40	36
39	27
41	31
24	32
37	33
39	49
38	49
25	35
43	52
40	32
34	28
32	30
38	32
43	41
21	38
43	36
25	36
34	44
41	49
30	15
32	45
32	28
34	32
31	23
39	24
42	32
37	45
33	37
29	39
	48
	44
	31
	37
	31
	18
	46
	44
	29
	33
	25

The z test statistic is the number calculated in cell D6. Again, when you paste this code into Excel, make sure that you start the pasting in cell A1. To help with that, you may want to also copy this notice. It seems to help.

© Ole J. Forsberg, Ph.D. 2024. All rights reserved.		.

Calculating the Test Statistic

The Problem

Information given:

Your Answer

Another Set of Data

Assistance