Question

(a) a consultant wants to ask workers at a factory about the workers job satisfaction. which of the following best describes a stratified sample of workers?the consultant forms 6 groups of workers based on the lengths of time the workers have worked at the factory. then, he randomly chooses 6 workers from each group.the consultant forms 6 groups of workers based on the workers shifts. then, he selects 13 workers at random from each group. then, the consultant assigns each worker a different number. using a random number table, he draws 78 of those numbers at random. then, the consultant selects the workers assigned to the drawn numbers. every set of 78 workers is equally likely to be drawn using the random number table.(b) a manager at a bus company wants to survey employees about the new uniforms. which of the following best describes a systematic sample of employees?the manager forms 6 groups of employees based on the departments the employees are in. then, she selects 9 employees at random from each group.the manager asks for volunteers until 54 employees have volunteered. she selects these volunteers because these employees are easily accessible.the manager takes a list of the employees and selects every 6th employee until 54 employees are selected.(c) a facilities supervisor at a sports stadium wants to rate the condition of the seats at the stadium. which of the following best describes a random sample of seats?the supervisor takes a list of the seats and selects every 4th seat until 80 seats are selected.the supervisor uses a computer program to draw 80 seats at random and selects these seats. every set of 80 seats is equally likely to be drawn by the computer program.the supervisor forms groups of 20 seats based on the sections the seats are in. then, she selects all of the seats in 4 randomly chosen groups.

Explanation:

Part (a)

Stratified sampling involves dividing the population into groups (strata) based on a characteristic and then sampling from each group. Here, the consultant forms 6 groups based on workers' shifts (a characteristic) and then selects 13 workers from each group, which matches stratified sampling. The other options: first option is cluster sampling (selecting groups and then all in group), last is simple random sampling (using random numbers for all).

Step1: Recall sampling methods

Stratified: divide into groups, sample from each. Cluster: select groups, sample all in group. Simple random: random selection of individuals.

Step2: Analyze options

• Option 1: Cluster (select groups, sample all).
• Option 2: Stratified (divide by shift, sample from each).
• Option 3: Simple random (assign numbers, random selection).

Systematic sampling involves selecting every \( k^{\text{th}} \) element from a list. The manager takes a list of employees and selects every 6th employee until 54 are selected, which is systematic (fixed interval). Other options: first is stratified (divide into groups, sample from each), second is convenience (select volunteers).

Step1: Recall sampling methods

Systematic: select every \( k^{\text{th}} \) item. Stratified: sample from groups. Convenience: use easy-to-reach samples.

Step2: Analyze options

• Option 1: Stratified (divide by department, sample from each).
• Option 2: Convenience (select volunteers).
• Option 3: Systematic (select every 6th employee).

Simple random sampling means every set of the same size has an equal chance of being selected. The option "The supervisor uses a computer program to draw 80 seats at random. Every set of 80 seats is equally likely to be drawn by the computer program" matches simple random sampling. The first is systematic (select every 4th), last is stratified (divide into groups, sample all in chosen groups).

Step1: Recall sampling methods

Simple random: equal chance for any set of size \( n \). Systematic: every \( k^{\text{th}} \) item. Stratified: sample from groups.

Step2: Analyze options

• Option 1: Systematic (select every 4th seat).
• Option 2: Simple random (random draw, equal chance for any 80 - seat set).
• Option 3: Cluster (divide into groups, sample all in chosen groups).

Answer:

The consultant forms 6 groups of workers based on the workers' shifts. Then, he selects 13 workers at random from each group.

Part (b)