Overview
This lecture explains the key results of the Central Limit Theorem (CLT), focusing on the normality of the sampling distribution and how to use this fact to find probabilities using the normalcdf function.
Central Limit Theorem Results
- The CLT states that the sampling distribution of sample proportions is normal in shape if certain conditions are met.
- The shape being normal is crucial for applying probability calculations in statistics.
- The normal shape allows us to use the normalcdf function to find probabilities related to sample proportions.
Application to Probability
- Probability questions (e.g., chance more than 30% of drivers text) rely on the normality of the sampling distribution.
- To use normalcdf, we need four inputs: lower bound, upper bound, mean (center), and standard deviation (spread).
- The center (mean) of the sampling distribution is the population proportion (P).
- The spread (standard deviation) is found using the standard error formula: sqrt[P(1-P)/n].
Addressing Unknown Population Proportion
- Often, the true population proportion (P) is unknown in real problems.
- If the sample is good, we can substitute the sample proportion (P-hat) in place of P.
- The sample proportion is calculated as the number of successes divided by the sample size.
Example Calculation
- Sample proportion (P-hat): 48 out of 200 drivers = 0.24.
- This 0.24 becomes the mean (center) of the normal curve for normalcdf.
- Standard deviation (spread): sqrt[0.24 × (1 − 0.24) / 200] ≈ 0.033.
- This value (0.033) is used as the standard deviation in the normalcdf function for probability calculations.
Key Terms & Definitions
- Central Limit Theorem (CLT) — States that the sampling distribution of sample proportions will be normal if conditions are met.
- Sampling Distribution — Distribution of sample statistics (e.g., proportions) from many samples.
- Population Proportion (P) — The actual proportion of a characteristic in the population.
- Sample Proportion (P-hat) — The observed proportion in your sample; used as an estimate of P.
- Standard Error — The standard deviation of a sampling distribution; calculated as sqrt[P(1-P)/n].
- normalcdf — Calculator function used to find probabilities for normal distributions; requires lower bound, upper bound, mean, and standard deviation.
Action Items / Next Steps
- Review section 6.2 for details on using normalcdf for probability.
- Write the standard error formula on your note sheet for quick reference.
- Practice substituting P-hat for P when the population proportion is unknown.