# 4.Sampling

# 4.4 Sample Size

### Determining the Sample Size

The **sample size does not depend on the size of the population** being studied, but rather it depends on qualitative factors of the research.

- desired precision of estimates
- knowledge of population parameters
- number of variables
- nature of the analysis
- importance of the decision
- incidence and completion rates
- resource constraints

### Sample Sizes Used in Marketing Research Studies

Type of Study |
Minimum Size |
Typical Size |

Problem identification research (e.g., market potential) |
500 |
1,000 – 2,000 |

Problem solving research (e.g., pricing) |
200 |
300 – 500 |

Product tests |
200 |
300 – 500 |

Test-market studies |
200 |
300 – 500 |

TV/Radio/Print advertising (per commercial ad tested) |
150 |
200 – 300 |

Test-market audits |
10 stores |
10 – 20 stores |

Focus groups |
6 groups |
10 – 15 groups |

### Margin of Error Approach to Determining Sample Size

### Margin of Error Approach to Determining Sample Size

The** Margin of Error** is the measure of accuracy of a survey. The smaller the margin of error, the more accurate are the estimates of a survey.

*How accurate is this statistic? What is the margin of error?*

**Means**

use this formula when evaluating estimates of population means

z = z-value for a given level of confidence*
*σ = standard deviation of a population parameter

*n = sample size*

**Proportions**

use this when evaluating estimates of proportions

z = z-value for a given level of confidence

π = estimate of the proportion in the population

n = sample size

**z-values**

z = 1.96

for 95% level of confidence

z = 2.58

for 99% level of confidence

**maximum margin of error ****for 95% level of confidence**

### Margin of Error Approach to Determining Sample Size

*How accurate is this statistic? What is the margin of error?*

Margin of Error = 1/√n

48,804 people in sample

√48,804=220.916

1/221 = 0.0045

*100 = 0.45%

⇒ *x *= 61% ± 0.45%

**⇒ ****60****.****55% ****to** **61****.****45% **** **

*calculations are approximate values for 95% level of confidence*

**How large should the sample be taking margin of error of ±1% into account?**

**Sample Size = (1/Margin of Error) ^{2}**

*calculations are approximate values for 95% level of confidence*

## Corrections needed, when sample size exceeds 10% of the population

### Sample Size = (1/Margin of Error)^{2}

^{2}

Sample Size does not depend on population.

n±1%= (1/0.01)^{2 }= (100)^{2} = 10,000

What if the population under study consists of only **100** elements? (e.g., firms producing cars)

**Correction of the Sample Size**

**Margin of Error 1%**

What if the population under study consists of only **100** elements? (e.g., firms producing cars)

**Margin of Error 5%**

What if the population under study consists of only **100** elements? (e.g., firms producing cars)

**Margin of Error 10%**

What if the population under study consists of only **100** elements? (e.g., firms producing cars)

### A Note on Confidence Interval

**Confidence Interval & Level of Confidence**

A **confidence interval **estimate is an interval of numbers, along with a measure of the likelihood that the interval contains the unknown parameter.

The **level of confidence **is the expected proportion of intervals that will contain the parameter if a large number of samples is maintained.

*Suppose we’re wondering what the average number of hours that people at Siemens spend working. We might take a sample of 30 individuals and find a sample mean of 7.5 hours. If we say that we’re 95% confident that the real mean is somewhere between 7.2 and 7.8, we’re saying that if we were to repeat this with new samples, and gave a margin of ±0.3 hours every time, our interval **would** contain the actual mean 95% of the time.*

### Confidence Interval, Margin of Error, and Sample Size

The higher the confidence we need, the wider the confidence interval and the greater the margin of error will be

**z-values**

z = 1.96

for 95% level of confidence

z = 2.58

for 99% level of confidence

**maximum margin of error ****for 99% level of confidence
**

smaller margins of error

require larger samples

higher levels of confidence

require larger samples