Consider the following to answer the question(s) below:
In a metal fabrication process, metal rods are produced that have an average length of 20.5 metres with a standard deviation of 2.3 metres. A quality control specialist collects a random sample of 30 rods and measures their lengths.
-Suppose the resulting sample mean is 19.5 metres. Which of the following statements is true?
A) This sample mean is 2.38 standard deviations above what we expect.
B) This sample mean is more than 3 standard deviations away from the population mean.
C) This sample mean is 2.38 standard deviations below what we expect.
D) This sample mean is only 1 standard deviation above the population mean.
E) This sample mean is only 1 standard deviation below the population mean.

Question

Quizplus · Accepted Answer

To calculate the number of standard deviations that the sample mean deviates from the population mean, we use the formula: 
z = (x̄ - μ) / (σ / √n) 
where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size. 
Plugging in the values given in the question, we get: 
z = (19.5 - 20.5) / (2.3 / √30) ≈ -2.38 
This means that the sample mean is 2.38 standard deviations below the population mean, so the correct answer is C.

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