Question

name: michael brooks mat 150.5 - october 4, 2023 directions: answer each question to the best of your ability. you can use your calculator and pen/pencil. scrap paper will be provided upon request - if you do work on scrap paper, label the work so you can receive partial credit. unless otherwise specified, all answers should be rounded to the nearest hundredth. statistics: 1.) in a psychology class consisting of 150 students, the professor wants to use the technique systematic sampling to select 10 students to participate in a survey. each student is given a number from 1 to 150. the professor selects the first student whose number is 8. a) the ratio of the population size. (k) (2 points) b) identify the position of all 10 students. (6 points)

Explanation:

Step1: Calculate the sampling - interval (k)

The formula for the sampling - interval $k$ in systematic sampling is $k=\frac{N}{n}$, where $N$ is the population size and $n$ is the sample size. Here, $N = 150$ and $n=10$. So, $k=\frac{150}{10}=15$.

Step2: Find the positions of the 10 students

The first student has a number $a = 8$. The positions of the students in systematic sampling follow the arithmetic - sequence formula $a_n=a+(n - 1)k$, where $a$ is the first term, $k$ is the common difference, and $n$ is the term number.
For $n = 1$, $a_1=8$
For $n = 2$, $a_2=8+(2 - 1)\times15=8 + 15=23$
For $n = 3$, $a_3=8+(3 - 1)\times15=8+30 = 38$
For $n = 4$, $a_4=8+(4 - 1)\times15=8 + 45=53$
For $n = 5$, $a_5=8+(5 - 1)\times15=8+60 = 68$
For $n = 6$, $a_6=8+(6 - 1)\times15=8+75 = 83$
For $n = 7$, $a_7=8+(7 - 1)\times15=8+90 = 98$
For $n = 8$, $a_8=8+(8 - 1)\times15=8 + 105=113$
For $n = 9$, $a_9=8+(9 - 1)\times15=8+120 = 128$
For $n = 10$, $a_{10}=8+(10 - 1)\times15=8+135 = 143$

Answer:

a) $k = 15$
b) The positions of the 10 students are 8, 23, 38, 53, 68, 83, 98, 113, 128, 143.