Question

The members of a population have been numbered 1-372. A sample of size 5 is to be taken from the population, using systematic random sampling.

(a) Apply Procedure 1.1 on page 17 to determine the sample ( i.e .. the numbers corresponding to the members of the population that are included in the sample ).

(b) Suppose that, in Step 2 of Procedure 1.1, the random number chosen is 10 ( i.e., k = 10 ). Determine the sample.

Short answer

Part (a) 65, 139, 213, 287, and 361.

Part (b) 10, 84, 158, 232, and 306.

Step-by-step solution

Part (a) Step 1. Given information.

The given statement is:

The members of a population have been numbered 1-372. A sample of size 5 is to be taken from the population, using systematic random sampling.

Part (a) Step 2. Obtain a sample of size 5 using systematic random sampling.

First, divide the population size by the sample size and then round off the result to the nearest whole number(m).

In this case, the population size is 372.

and the sample size is 5.

$$\begin{gathered} m=\frac{372}{5} \\ =74.4 \\ =74 \end{gathered}$$

Part (a) Step 3. Obtain a sample of size 5 using systematic random sampling.

Now obtain a number kbetween 1 to mby using MINITAB:

Step 1: Select Calc > Random Data Integer from the MINITAB menu.

Step 2: Enter twice your intended sample size in the Number of rows of data to create a section, just in case there are repeats. Enter 1 as the sample size.

Step 3: In the Store column, type Number in column C1

Step 4: Enter 01 as the minimum value.

Step 5: Enter 74 as the maximum value.

Step 6: Select OK.

OUTPUT FROM MINITAB:

NUMBER $k=65$.

Now select the numbers k, k + m,...., k + 4m to generate the sample.

$$\begin{gathered} k=65 \\ k+m=65+74 \\ =139 \\ k+2m=65+\left(2\right)\left(74\right) \\ =213 \\ k+3m=65+3\left(74\right) \\ =287 \\ k+4m=65+4\left(74\right) \\ =361 \end{gathered}$$

The sample size is 65, 139, 213, 287, and 361.

Part (b) Step 1. If k=10 obtain a sample of size 5 using systematic random sampling.

$$\begin{gathered} k=10 \\ k+m=10+74 \\ =84 \\ k+2m=10+2\left(74\right) \\ =158 \\ k+3m=10+3\left(74\right) \\ =232 \\ k+4m=10+4\left(74\right) \\ =306 \end{gathered}$$

The sample size is 10, 84, 158, 232, and 306.