Sample size without standard Deviation

In summary, the conversation discusses how to find the required sample size to estimate the mean annual income of natives in New York with a 99% confidence level and a margin of error of $1000, without knowing the standard deviation. Several methods are suggested, including using the Chebyshev's inequality and the approximation of standard deviation using the range. It is also mentioned that the process becomes more complicated when the standard deviation is not known and may require iteration.
  • #1
Fear_of_Math
7
0
Hello again,

I have a question here that asks me to find how large a sample size is, but I have no Standard deviation. How would you tackle this>

How large a sample size do we need to estimate the mean annual income of natives in New York, correct to within $1000 with probability 0.99? No information is available to us about the standard deviation of their annual income. We guess that nearly all of the incomes fall between $0 and $120,000 and that this distribution is approximately normal.

Here's what I see:
1 - alpha = 0.99 therefore alpha =0.01 /2 = 0.005
This gives a Z* of 2.575 (because it states normal distribution)
The 99% CI is (0, 120000).

I know that n = [Z*s/m]squared, buty I have neither s, nor m...

As always, the feedback and guidance is appreciated =)
 
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  • #2
Practically the endpoints of the distribution are 0 and 120K. Normal dist. is symmetric, so you can figure out the mean. As for standard dev., I would assume 99% of the people are within 0 to 120K, and find out how many standard deviations it would take to get 99% of people (within ___ standard deviations around the mean).
 
  • #3
I don't think this question can be answered without further information. Suppose that the income is distributed with a mean of $60,000, and a standard deviation of $1. After a small number of observations we would learn that the std. dev. is small, and realize we don't need to take many more samples.

On the other hand suppose that the income is distributed with a mean of $60,000 and a standard deviation of $20,000. In that case we'd have to take a much larger number of samples to achieve the same confidence.
 
  • #4
The statement "correct to within $1000 with probability 0.99" implies a standard deviation by Chebyshev's inequality: The probability an observation is with k standard deviations of the mean is less than [itex]1/k^2[/itex]. The largest k that has [itex]1/k^2< .99[/itex] is 2 so 1000 must be no more than 2 standard deviations. The smallest standard deviation that will work is $500.
 
  • #5
You can also try use the (very crude) approximation that

[tex]
\sigma \approx \frac{\text{Range}}{4}
[/tex]

presented in some texts. I suggest to students to use 6 rather than 4.
 
  • #6
HallsofIvy said:
The statement "correct to within $1000 with probability 0.99" implies a standard deviation by Chebyshev's inequality: The probability an observation is with k standard deviations of the mean is less than [itex]1/k^2[/itex]. The largest k that has [itex]1/k^2< .99[/itex] is 2 so 1000 must be no more than 2 standard deviations. The smallest standard deviation that will work is $500.

Chebyshev's inequality says the probability an observation is _not_ within k std. dev. of the mean is <= 1/k^2.
 
  • #7
Had the true std. dev. (σ) been known, you'd use N = (zσ/x)^2, where x is the margin of error = $1,000 (or x = 1 if you express everything in $1,000). When σ is unknown the process is more complicated and you may have to iterate. This page explains how.
 
Last edited:

1. What is sample size without standard deviation?

Sample size without standard deviation refers to the number of individuals or observations used in a study or experiment, without taking into account the variability or spread of the data. It is simply the number of subjects or data points included in a sample.

2. Why is sample size without standard deviation important?

Sample size without standard deviation is important because it determines the power and precision of a study. A larger sample size can increase the likelihood of detecting a significant effect, while a smaller sample size may not have enough statistical power to accurately represent the population.

3. How is sample size without standard deviation calculated?

Sample size without standard deviation can be calculated using a formula, such as the minimum sample size needed to achieve a certain level of statistical power. This formula takes into account the desired power, significance level, and effect size of the study.

4. What are the limitations of using sample size without standard deviation?

One limitation of using sample size without standard deviation is that it does not account for the variation or spread of the data. This means that a sample size with a large number of subjects may still have a high degree of variability, making it difficult to draw accurate conclusions. Additionally, sample size without standard deviation may not be representative of the entire population.

5. How can sample size without standard deviation be improved?

To improve sample size without standard deviation, researchers can conduct a pilot study to estimate the standard deviation of the population. This can help determine a more accurate sample size for the main study. Additionally, using random sampling techniques and increasing the sample size can also improve the accuracy and representation of the sample.

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