Calculating the Perfect Sample Size for your research

14/07/26 •0•0

Discover how to calculate the ideal sample size for your research project with our comprehensive guide. Ensure statistical accuracy and reliability in your data analysis.

작성자:Leon

번역자Editorial

개정자Editorial

Introduction

The process of choosing the individuals who will belong to a SAMPLE is called SAMPLING. The researcher seeks to generalize conclusions referring to the SAMPLE, extending them to the entire POPULATION from which this sample was drawn.

There is no doubt that a sample does not perfectly represent a population. In other words, the use of a sample implies the acceptance of a margin of error that we will call SAMPLING ERROR. This is the difference between a sample result and the true population result; such errors result from random sampling fluctuations.

Non-sampling errors occur when:

The sample data are collected, recorded, or analyzed incorrectly.

A defective instrument is used when taking measurements.

A questionnaire or form has questions worded in a biased way.

To avoid non-sampling errors check our lesson on sampling.

We cannot prevent SAMPLING ERROR from occurring, but we can limit its value by choosing a sample of adequate size. Obviously, SAMPLING ERROR and SAMPLE SIZE move in opposite directions: the larger the sample size, the smaller the error made, and vice versa.

Determining the Sample Size

Determining the size of a sample is a very important problem because: unnecessarily large samples lead to wasted time and money; and excessively small samples can lead to unreliable results.

In many cases, it is possible to determine the minimum size of a sample to estimate a statistical parameter, such as the POPULATION MEAN (µ).

But how do we calculate the sample size (n) from already known parameters?

Given a confidence interval [lower limit (a), upper limit (b)], we have:

\[ a = \bar{X} - z *\frac{\sigma}{\sqrt{n}} \]

\[ b = \bar{X} + z *\frac{\sigma}{\sqrt{n}} \]

It can be seen that the distance from the sample mean to the endpoint of the interval, which we can also call the margin of error of the interval, is given by:

\[ \text{Margin of error} = z *\frac{\sigma}{\sqrt{n}} \]

Which can also be written as:

\[ \sqrt{n} = z *\frac{\sigma}{\text{Margin of error}} \]

\[ n = [z *\frac{\sigma}{\text{Margin of error}}]^{2} \]

\[ n = [z *\frac{\sigma}{E}]^{2} \]

Therefore, if you want to know the sample size, you will need the following variables:

- Choose a given margin of error;

- Choose a given significance level;

- Know the population standard deviation.

Examples

An economist wants to estimate the average income for the first year of work of a law graduate. How many income values should be taken if the economist wants to be 95% confident that the sample mean is within R$500.00 of the true population mean? Suppose we know, from a previous study, that for such incomes, σ = R$ 6250.00.

$\sigma = 6250$

$E = 500$

95% significance, or z of 1.96.

\[ n = [z *\frac{\sigma}{E}]^{2} = 600.25 = 601 \]

We must therefore obtain a sample of at least 601 first-year incomes, randomly selected, from graduates of colleges who have completed a law degree. With such a sample, we will be 95% confident that the sample mean x differs by less than R$500.00 from the true population mean µ.

Based on the data from the previous example, use a larger margin of error, such as R$1,000.00, and determine what the required sample size would be in this situation.

Since the margin of error doubled and it divides the sample size squared, the previous sample size should therefore be divided by 2 squared, which is 4. Thus, the sample size is 151. Note that more precise studies may require a significant amount of resources.

What if σ is not known?

The equation for sample size requires that the population standard deviation σ be replaced by some value, but if it is unknown, we should be able to use a preliminary value obtained through processes such as the following:

Use the approximation σ ≈ range/4.
Conduct a pilot study, beginning the sampling process. Based on the first set of at least 31 randomly selected sample values, calculate the sample standard deviation S and use it instead of σ. This value can be refined by obtaining more sample data.
Use an estimate of the proportion of the population size of the item you would like to study. In this case, the standard deviation of the binomial is used: square root of p (1 – p).
If no estimated values are given for the proportion, but the proportion is discussed, in a sample with a range from 0 to 100%, use 50% for p. A p of 50% is what maximizes the function p (1 – p).

Examples

A social worker wants to know the sample size (n) needed to determine the proportion of the population served by a Health Unit that belongs to the municipality of Cariacica. No prior survey of the sample proportion was conducted and, therefore, its value is unknown. She wants to be 90% confident that her maximum estimation error (E) is ±5% (or 0.05). How many people need to be interviewed?

90% significance is equal to a z of 1.645;

σ of 0.25 because it is not known;

Error of 0.05.

\[ n = [z *\frac{\sigma}{E}]^{2} = 270.6 = 271 \]

We must therefore obtain a sample of 271 people to determine the proportion of the population served at the Health Unit that comes from the municipality of Cariacica.

Exercises

1) A researcher wants to estimate the proportion of rats in which a certain type of tumor develops when exposed to radiation. He wants his estimate not to deviate from the true proportion by more than 0.02 with a probability of at least 90%. How many animals does he need to examine to satisfy this requirement?

2) Using the data from the previous exercise, how would it be possible to reduce the sample size using the additional information that, in general, this type of radiation does not affect more than 20% of the rats?

3) Before an election, a certain party is interested in estimating the proportion of voters favorable to its candidate. Determine the sample size necessary so that the error made in the estimation is at most 0.01, with a probability of 80%.

4) If, from the previous exercise, it was observed that 55% of voters were in favor of the candidate, construct a confidence interval for the proportion of the candidate’s voters with a confidence coefficient of 0.95.

5) A scientist decides to estimate the proportion p of individuals with a certain illness in a region. He wants the probability that his estimate does not deviate from the true value of p by more than 0.02 to be at least 95%. What should the sample size be for these conditions to be satisfied? Another scientist discovers that the disease in question is related to the concentration of substance A in the blood and that any individual for whom concentration A is less than 1.488 mg/cm 3 is considered ill. It is known that the concentration of substance A in the blood has a normal distribution with a standard deviation of 0.4 mg/cm 3 and a mean greater than 2.0 mg/cm 3. Do you think this new information can be used by the first scientist to reduce the sample size? What is the ideal sample size?

6) An opinion research study center conducted a survey to evaluate the opinion of viewers in a region about a certain sports commentator. To do this, it interviewed 380 viewers, randomly selected from the region, and found that 180 wanted the commentator to be removed from TV. Determine a 90% confidence interval for p: proportion of viewers in favor of the commentator’s removal.