Confidence Interval [Two-Sample Z-Procedure]

The Problem

Example #381: Let us estimate the difference in means between two populations using a confidence interval to indicate our uncertainty. To do this, we collect data from each of the two populations. The data from population 1 are:

29, 60, 53, 39, 38, 35, 29, 44, 37, 35, 38, 39, 65, 38, 42, 56, 37, 72, 18, 35, 41, 29, 38, 46, 54, 67, 45, 40, 65, 38, 52, 51, 24, 34, 20, 30, 29, 51, 39, 34, 30, 61, 44, 69, 38, 32, 54, 47, 38, 67, 44, 42, 29, 43

The data from population 2 are:

36, 58, 49, 39, 19, 42, 58, 32, 40, 47, 40, 39, 47, 41, 43, 49, 53, 47, 54, 29, 38, 35, 40, 30, 23, 43, 38, 50, 49, 59, 40, 40, 43, 33, 33, 44, 51, 50, 38, 46, 47, 40, 53, 36, 41, 19, 36, 60, 41, 36, 35, 46, 53, 34, 35, 33, 54, 46

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 = 144 and σ²₂ = 121.

With this information, calculate the endpoints of the symmetric 98% confidence interval for the difference in population means, μ₁ − μ₂.

Information given:

To summarize the above, the values of import are:

Summary statistics from the problem
\( \bar{x}_1 \)	=	42.6667
\( \bar{x}_2 \)	=	41.8966

\( \sigma_1 \)	=	12
\( \sigma_2 \)	=	11

\( n_1 \)	=	54
\( n_2 \)	=	58

\( \alpha \)	=	0.02

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 two endpoints of the 98% confidence interval for the population mean, then click on the “Check your answer!” button. Please round your answer to the ten-thousandths place.

98% Confidence Bounds: (, )

Make sure the lower limit is less than the upper limit.

Another Set of Data

Would you like to continue working on this topic? If so, click here for another data set.

Assistance

Show Formula

Show Solution

Show the R Code

Hide the R Code

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, and there is no z-test in the base R package.

The following code will perform the above steps, however. Copy and paste the following code into your R script window, then run it from there.

mud = 0

sigma1 = 12

sigma2 = 11

sample1 = c(29, 60, 53, 39, 38, 35, 29, 44, 37, 35, 38, 39, 65, 38, 42, 56, 37, 72, 18, 35, 41, 29, 38, 46, 54, 67, 45, 40, 65, 38, 52, 51, 24, 34, 20, 30, 29, 51, 39, 34, 30, 61, 44, 69, 38, 32, 54, 47, 38, 67, 44, 42, 29, 43)

sample2 = c(36, 58, 49, 39, 19, 42, 58, 32, 40, 47, 40, 39, 47, 41, 43, 49, 53, 47, 54, 29, 38, 35, 40, 30, 23, 43, 38, 50, 49, 59, 40, 40, 43, 33, 33, 44, 51, 50, 38, 46, 47, 40, 53, 36, 41, 19, 36, 60, 41, 36, 35, 46, 53, 34, 35, 33, 54, 46)

xbar1 = mean(sample1)

xbar2 = mean(sample2)

n1 = length(sample1)

n2 = length(sample2)

se = sqrt( sigma1^2/n1 + sigma2^2/n2 )

lcl = xbar1-xbar2 - qnorm(0.99)*se

ucl = xbar1-xbar2 + qnorm(0.99)*se

lcl; ucl

In the R output, the endpoints of the confidence interval are the numbers output by the script.

For practical reasons, the base R installation does not contain the z-procedures. However, I have created one for pedagogical purposes. If you are connected to the Internet, the following will also calculate the test statistic:

source("http://rfs.kvasaheim.com/Rfctns/z.test.R")

sample1 = c(29, 60, 53, 39, 38, 35, 29, 44, 37, 35, 38, 39, 65, 38, 42, 56, 37, 72, 18, 35, 41, 29, 38, 46, 54, 67, 45, 40, 65, 38, 52, 51, 24, 34, 20, 30, 29, 51, 39, 34, 30, 61, 44, 69, 38, 32, 54, 47, 38, 67, 44, 42, 29, 43)

sample2 = c(36, 58, 49, 39, 19, 42, 58, 32, 40, 47, 40, 39, 47, 41, 43, 49, 53, 47, 54, 29, 38, 35, 40, 30, 23, 43, 38, 50, 49, 59, 40, 40, 43, 33, 33, 44, 51, 50, 38, 46, 47, 40, 53, 36, 41, 19, 36, 60, 41, 36, 35, 46, 53, 34, 35, 33, 54, 46)

z.test(sample1, sample2, mu=0, sigmax=12, sigmay=11)

The test statistic follows “z =” near the top of the output.

Show the Excel Code

Hide the Excel Code

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
29	36	sigma1:	12
60	58	sigma2:	11
53	49	se:	=SQRT(D2^2/COUNT(A:A)+D3^2/COUNT(B:B))
39	39
38	19	lower:	=AVERAGE(A:A)-AVERAGE(B:B)-ABS(NORM.S.INV((1-0.98)/2))*D4
35	42	upper:	=AVERAGE(A:A)-AVERAGE(B:B)+ABS(NORM.S.INV((1-0.98)/2))*D4
29	58
44	32
37	40
35	47
38	40
39	39
65	47
38	41
42	43
56	49
37	53
72	47
18	54
35	29
41	38
29	35
38	40
46	30
54	23
67	43
45	38
40	50
65	49
38	59
52	40
51	40
24	43
34	33
20	33
30	44
29	51
51	50
39	38
34	46
30	47
61	40
44	53
69	36
38	41
32	19
54	36
47	60
38	41
67	36
44	35
42	46
29	53
43	34
	35
	33
	54
	46

The endpoints of the 98% confidence interval are given in cells D6 and D7. 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