1. Intro to Stats and Collecting Data

Intro to Stats

1. Intro to Stats and Collecting Data

Intro to Stats

4:54 minutes

Problem 1.R.25

Textbook Question

Obtaining a Systematic Sample A quality-control engineer wants to be sure that bolts coming off an assembly line are within prescribed tolerances. He wants to conduct a systematic sample by selecting every 9th bolt to come off the assembly line. The machine produces 30,000 bolts per day, and the engineer wants a sample of 32 bolts. Which bolts will be sampled?

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Understand that a systematic sample involves selecting items at regular intervals from a population. Here, the engineer wants to select every 9th bolt from the assembly line.

Calculate the total number of bolts produced per day, which is given as 30,000 bolts.

Since the engineer wants a sample of 32 bolts, and the sampling interval is every 9th bolt, confirm that selecting every 9th bolt will yield the desired sample size or adjust accordingly.

Identify the positions of the bolts to be sampled by starting from the first bolt selected (usually the 9th bolt) and then adding the interval repeatedly. This can be expressed as the sequence: \$9, 18, 27, \ldots$ up to the 32nd selected bolt.

Formulate the general term for the sampled bolts as \$9k$, where $k$ ranges from 1 to 32, representing the 1st to the 32nd bolt selected in the sample.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Systematic Sampling

Systematic sampling is a probability sampling method where elements are selected from an ordered population at regular intervals, called the sampling interval. In this case, every 9th bolt is chosen, starting from a randomly selected point within the first interval. This method ensures a spread-out sample across the entire population.

Sampling Interval

The sampling interval is the fixed distance between selected elements in systematic sampling. It is calculated by dividing the population size by the desired sample size. Here, the interval is 9, meaning every 9th bolt is selected to achieve a representative sample of 32 bolts from 30,000.

Sample Size Determination

Sample size refers to the number of observations or units chosen from the population for analysis. The engineer wants 32 bolts sampled, so the sampling interval and starting point must be chosen to ensure exactly 32 bolts are selected systematically from the 30,000 produced.

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