A researcher wants to know the average age of teacher in one city . If there are 250 elementary teachers, 220 secondary teachers and 340 tertiary level teachers in that city,

Question

A) Using the Slovin's formula to find the appropriate number of sample using 1% margin error

B) Show the type of sampling technique that can be used to determine the teacher samples from the different school levels.

Answer 1

Slovin vs Cochran

Well, I'll respond answering for the Slovin⁽²⁾ formula, even though it is a really bad corruption of the original Cochran's formula ⁽¹⁾ which is mathematically much more robust. (see notes below) It is identical to Yamane's formula ⁽³⁾ Note: Yamane/Slovin is an estimator for sample size when proportions (categorical data) are being sampled and should NEVER be used for estimating sample size where ratio (numerical) data such as age or weight are being sampled. This is so wrong on many levels. However - here are the calculations.

n = N/(1+N e² )

where

1. e = error tolerance limit / margin of error
2. n = number of samples to take from the population
3. N = Population size

OK, realizing that the three populations (Elementary, Secondary, and Tertiary) probably represent independent age groups we will still add them all up for the calculations, thus.

1) n = 810/(1+810*.01²) or 750 (rounding up)

2) To be able to do post hoc tests such as an ANOVA after sampling to determine differences between populations (elementary, etc.) a prorated Stratified Random Sample should be used where the same proportion of samples are taken from each group. (92.6% if we use the sample size calculated in 1 above. or 232, 204, 314) With random selection within the groups.

Comments:

The referencing of Slovin to the formula associated with him is questionable as his work is basically undocumented , being referenced by Sevilia⁽²⁾ but not listed in the biography.

In the Slovin/Yamane formula, e is quoted as margin of error. Which in standard definitions is (1 - degree of confidence)/2. as it refers to a two sided comparison (confidence interval) In this example with e (or moe) of 1% the degree of confidence would be 98%. HOWEVER! the students text may have other definitions because e in the Slovin formula is sometimes falsely referred to as (1 - confidence level) ⁽⁴⁾. THIS IS WRONG. Due to this irregularity in definition of parameters and the undocumented history of the so called 'Slovin's formula', it should be avoided in all circumstances and one should use either Cochran's formula for small to large populations and continuous data:⁽¹⁾⁽⁵⁾

n₀ = (t² * s²)/d²

where:

1. n0 = estimated sample size
2. t² = critical t value squared. (t is selected for confidence level and estimated sample size)
3. s² = estimated variance (square of standard deviation) from previous data
4. d² = square of acceptable margin of error (calculated from a percentage e of the expected range of the samples. )

or a modified Cochran's [n = N*n₀ / (n₀ + N – 1) where n₀ is the Cochran's estimate ], for very small populations.

Cochran's formula is arguably the most robust and vetted formula, based on work done by Dr William G Cochran (various publications) and published in his 1963 book, "Sampling Techniques" It is almost as easy to use as Slovin/Yamane. The main difference is determining the estimated range and variance, which usually is available from earlier or similar data/measurements.

References

--------------------------------------------------

1)Cochran, W. G. 1953. Sampling Techniques, 2nd Ed., New York: John Wiley and Sons, Inc.

2) Slovin - formula noted in Sevilla et. al., 1960: 182

3)Yamane, Taro. 1967. Statistics, An Introductory Analysis, 2nd Ed., New York: Harper and Row

4) https://prudencexd.weebly.com/

5) Bartlett, Kotrlk, Higgins, "Determining Appropriate Sample Size in Survey Research" https://www.academia.edu/5521356/Determining_appropriate_sample_size"