M303 Sample Size and Statistical Theory II
Where we are
Define the population, sampling frame, and sampling procedure.
Distinguish population parameters from sample statistics.
Use confidence intervals and sampling error to choose a defensible sample size.
Learning Objectives
Use a sample statistic to reason about an unknown population parameter.
Construct and interpret confidence intervals for a population mean.
Determine the sample size needed for a desired confidence level and sampling error.
Use graphical representations to make sample evidence clear.
Previous-class review
A larger sample is automatically more representative than a smaller sample.
A population parameter describes the full population; a sample statistic describes the observed sample.
Greater population variability generally requires a larger sample for the same precision.
How do we infer the population mean μ from the sample mean x̄?
Remember: x̄ ≠ μ.
We use the sample mean x̄ to learn about the population mean μ.
We usually cannot know μ exactly, but we can construct an interval around x̄ that has a high confidence level—such as 90% or 95%—of capturing μ.
How do we construct confidence intervals for μ?
CI for μ = x̄ ± E = (x̄ − E, x̄ + E)
E = zσ / √n
Use z = 2 for 95% confidence and z = 5/3 for 90% confidence. If σ is unknown, use the sample standard deviation S.
Example: x̄ = 0, S = 1.41, and n = 9. What are the 95% and 90% confidence intervals for μ?
How do we determine sample size statistically?
E = zσ / √n
n = z²σ² / E²
If σ is unknown, use S from a pilot or test sample.
Desired sampling error E = 0.30 and estimated variance S² = 2.
Graphical Representations
In addition to sample statistics, graphical charts can be used for preliminary data analysis such as
Bar charts
Pie charts
Line charts
Example: Graphical Representations
In choosing a computer, which factor is more important to you? (N = 300)
Design factor · Performance · Both equally| Preference | College students | College graduates | Total |
|---|---|---|---|
| Design factor | 100 | 44 | 144 |
| Performance | 60 | 86 | 146 |
| Both equally | 5 | 5 | 10 |
| Total | 165 | 135 | 300 |
Example: Grouped Bar and Line Charts
Grouped bar chart
Students vs. graduates
Example: Pie Charts by Group
College students · n = 165
- 100 Design · 60.6%
- 60 Performance · 36.4%
- 5 Both · 3.0%
College graduates · n = 135
- 44 Design · 32.6%
- 86 Performance · 63.7%
- 5 Both · 3.7%
Total · N = 300
- 144 Design · 48.0%
- 146 Performance · 48.7%
- 10 Both · 3.3%
Practice question 1
Practice Question 1
A population of 25,000 design engineers was asked: “How important is it that the machine be capable of working with both hard and soft metals?”
What are the population mean, variance, and standard deviation?
| Scale | Description | Frequency | Percentage |
|---|---|---|---|
| 5 | Extremely important | 5,000 | 20 |
| 4 | Important | 8,000 | 32 |
| 3 | Desirable | 6,000 | 24 |
| 2 | Only a small plus | 2,000 | 8 |
| 1 | Of no consequence | 4,000 | 16 |
| Total | 25,000 | 100 | |
Practice question 1 — solution
| Scale | Description | Frequency | Percentage |
|---|---|---|---|
| 5 | Extremely important | 5,000 | 20 |
| 4 | Important | 8,000 | 32 |
| 3 | Desirable | 6,000 | 24 |
| 2 | Only a small plus | 2,000 | 8 |
| 1 | Of no consequence | 4,000 | 16 |
| Total | 25,000 | 100 | |
Practice Question 2
Suppose we want to understand fast-food consumption among college students.
We ask:
“On average, how often do you eat at—or take out from—fast-food restaurants in a month?”
a. What are the sample mean, variance, and standard deviation?
b. What is the 95% confidence interval estimate for the population mean?
c. At 95% confidence, if the desired margin of error is 1.1, what sample size is required?
| Respondent | Fast-food visits |
|---|---|
| 1 | 5 |
| 2 | 6 |
| 3 | 7 |
| 4 | 3 |
| 5 | 5 |
| 6 | 4 |
Practice question 2 — solution
| Respondent | Fast-food visits |
|---|---|
| 1 | 5 |
| 2 | 6 |
| 3 | 7 |
| 4 | 3 |
| 5 | 5 |
| 6 | 4 |
To sum up
Factors determining sample size
How to determine the sample size by ad hoc techniques
Parameters and statistics
How to calculate parameters and statistics of central tendency and dispersion
Calculating confidence intervals for the population mean
Determining the sample size with statistical techniques
Basic graphical illustrations of data such as bar, pie and line charts