Unit 6: Sample Size Determination
- Normal Distribution
- Characteristics: mean = median = mode, symmetry about the center, 50% of values less than and greater than the mean.
- Understanding Standard Deviation: It is crucial because it allows us to say that any value is:
- Likely within 1 SD (68 out of 100 should be).
- Very likely within 2 SDs (95 out of 100 should be).
- Almost certainly within 3 SDs (99.7 out of 100 should be).
- Example: Student heights between 1.1m and 1.7m; Mean = 1.4m; SD = 0.15m.
- Z-Score Example
- Friend's height: 1.85m is 3 SDs from the mean of 1.4m, giving a Z-score of 3.0.
- What is a Z-Score: The number of standard deviations a data point is from the mean. Calculated as (Value - Mean) / SD.
- Standardizing Process:
- Value → Subtract Mean → Divide by SD → Z-Score.
- Sampling Distribution of Mean
- According to the Central Limit Theorem, arithmetic means of large random samples will form a normal distribution curve, matching the population mean.
- Standard Error Formula
- Format: Sample mean, Frequency, Deviation from assumed mean (d), d/0.5, fd, d², fd².
- This format helps in the calculation of mean and standard deviation.
- Important Terms
- Parameter: A fixed characteristic or measure of the target population.
- Statistic: A characteristic or measure of the sample.
- Finite Population Correction (fpc): Corrects overestimation for sample sizes ≥ 5% of the population.
- Sample Size Determination
- Considerations: Allowed error, desired confidence level, estimated population SD.
- Formula
- Determining Sample Size with Known Standard Deviation
- Examples of determining the necessary sample size for estimating average amounts per invoice in populations of sales invoices with specified SDs and allowable errors.
- Stratified Sample Size
- Comparing proportionate vs. disproportionate stratified sample sizes for market research in luxury retail.
- Stratified Sampling Process:
- Identify Strata → Determine SD and Size per Stratum → Calculate Sample Size.
- Formula
- Sample Mean Frequency Format
- Format: Sample mean, Frequency, Deviation from assumed mean (d), d/0.5, fd, d², fd².
- This format helps in the calculation of mean and standard deviation.