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Chapter 6: Problem 10

Consider an \(x\) distribution with standard deviation \(\sigma=12\). (a) If specifications for a research project require the standard error of the corresponding \(\bar{x}\) distribution to be 2, how large does the sample size need to be? (b) If specifications for a research project require the standard error of the corresponding \(\bar{x}\) distribution to be 1, how large does the sample size need to be?

Short Answer

Expert verified
(a) 36, (b) 144

Step by step solution

01

Understand the relationship between standard deviation, standard error, and sample size

The standard error of the mean (SE) is given by the formula:\[SE = \frac{\sigma}{\sqrt{n}}\]where \(\sigma\) is the population standard deviation and \(n\) is the sample size. We need \(n\) so that the standard error is as specified.
02

Rearrange the formula for sample size calculation

We need to solve for \(n\) in the standard error formula:\[\sqrt{n} = \frac{\sigma}{SE}\]\[ = \left(\frac{\sigma}{SE}\right)^2\]This equation will help us find the required sample size.
03

Solve for sample size for SE = 2

Plug in \(\sigma = 12\) and \(SE = 2\) into the formula:\[n = \left(\frac{12}{2}\right)^2 = \left(\frac{12}{2}\right)^2 = 36\]The sample size required when SE = 2 is \(n = 36\).
04

Solve for sample size for SE = 1

Plug in \(\sigma = 12\) and \(SE = 1\) into the formula:\[n = \left(\frac{12}{1}\right)^2 = \left(12\right)^2 = 144\]The sample size required when SE = 1 is \(n = 144\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Deviation
Standard deviation is a measure that provides insights into the variability or dispersion of a data set. It tells us how much the individual data points deviate from the mean of the dataset. When we discuss standard deviation, we often refer to the symbol \(\sigma\) (the Greek letter sigma), which represents the population standard deviation.
Here鈥檚 how it works:
  • If the data points are close to the mean, then the standard deviation is small.
  • If the data points are spread out over a wider range of values, the standard deviation is larger.
As an example, consider a set of test scores. A low standard deviation would indicate that the scores are clustered around the average, whereas a high standard deviation means they are spread out over a wide range. This property of standard deviation makes it an essential metric in statistics to understand data spread.
Standard Error
Standard error (SE) helps us understand how accurately we are estimating the true mean of a population with our sample mean. It is a key concept in statistics, crucial for tasks such as confidence interval calculation and hypothesis testing.
In more simple terms, while standard deviation describes the variability in a set of data, standard error measures how much the sample mean would vary if you took multiple samples from the same population.
The formula for the standard error of the mean is:
  • \(SE = \frac{\sigma}{\sqrt{n}}\)
Where \(\sigma\) is the population standard deviation, and \(n\) is the sample size.
A smaller SE indicates that the sample mean is a more accurate reflection of the actual population mean, which is why researchers often seek to minimize it through adequate sampling.
Sample Size Calculation
Calculating the appropriate sample size is an essential part of designing a study. It ensures the results will be statistically significant and makes the estimation of the population parameters more accurate.
To determine the necessary sample size to achieve a desired standard error, we use the formula:
  • \(n = \left(\frac{\sigma}{SE}\right)^2\)
Using this formula, it's possible to calculate how many data points are needed in order to achieve a particular standard error, given the known standard deviation of the population.
In practice, consider the comparison given in the problem statements:
  • For a standard error of 2, the sample size needed is 36.
  • For a standard error of 1, the sample size required increases to 144.
This showcases how increasing demand for precision (smaller SE) necessitates a larger sample size.

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Most popular questions from this chapter

Do you try to pad an insurance claim to cover your deductible? About \(40 \%\) of all U.S. adults will try to pad their insurance claims! (Source: Are You Normal?, by Bernice Kanner, St. Martin's Press.) Suppose that you are the director of an insurance adjustment office. Your office has just received 128 insurance claims to be processed in the next few days. What is the probability that (a) half or more of the claims have been padded? (b) fewer than 45 of the claims have been padded? (c) from 40 to 64 of the claims have been padded? (d) more than 80 of the claims have not been padded?

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put 75 tons of coal into each car. The actual weights of coal loaded into each car are normally distributed, with mean \(\mu=75\) tons and standard deviation \(\sigma=0.8\) ton. (a) What is the probability that one car chosen at random will have less than \(74.5\) tons of coal? (b) What is the probability that 20 cars chosen at random will have a mean load weight \(\bar{x}\) of less than \(74.5\) tons of coal? (c) Suppose the weight of coal in one car was less than \(74.5\) tons. Would that fact make you suspect that the loader had slipped out of adjustment? Suppose the weight of coal in 20 cars selected at random had an average \(\bar{x}\) of less than \(74.5\) tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?

Does a raw score less than the mean correspond to a positive or negative standard score? What about a raw score greater than the mean?

Suppose \(5 \%\) of the area under the standard normal curve lies to the left of \(z\). Is \(z\) positive or negative?

(a) If we have a distribution of \(x\) values that is more or less mound-shaped and somewhat symmetrical, what is the sample size needed to claim that the distribution of sample means \(\bar{x}\) from random samples of that size is approximately normal? (b) If the original distribution of \(x\) values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means \(\bar{x}\) taken from random samples of a given size is normal?
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