Central Limit Theorem & Statistical Inference

1. Central Limit Theorem & Statistical Inference

2. Recapping the CLT The Central Limit Theorems for a sample proportion and for a sample mean state conditions under which a sample statistic follows a Normal distribution very closely. Both versions of CLT are very similar in specifying that: The shape of the sampling distribution is either exactly or approximately Normal The center (mean) of the distribution equals the value of the population parameter The variability of the sampling distribution decreases by a factor of 1/√n, as the sample size n increases CLT is powerful because it enables us, under certain conditions, to predict the long-term pattern of variability of a sample statistic. Since the statistic will follow a Normal distribution, we can use the Normal model to make probability predictions about the statistic’s expected values relative to the value of a parameter of interest for the sampled population.

3. Summary of the specifics for each version of CLT Applying CLT: 1) Determine whether the underlying variable of interest is quantitative or categorical, as that will dictate which version of CLT to use. 2) Determine whether the technical conditions of the version being considered are met.

4. Activity 1: Kissing Couples Question 10 (using standard Normal model) Question 11 (using standard Normal model)

5. Activity 1 has introduced the concept of statistical significance. You determined the statistical significance of a sample statistic by exploring the sampling distribution of the statistic, investigating how frequently an observed sample result occurs simply by random chance. Informally speaking, a sample result is said to be statistically significant if it is unlikely to occur (i.e., has low probability) due to random sampling variability alone.

6. Activity 2: Christmas Shopping Question 5b Question 6b

7. Activity 2 has introduced the concept of statistical confidence, involving this line of reasoning: 1. Although a sample statistic cannot be used to determine the exact value of a corresponding population parameter, we can be reasonably confident that the sample statistic will fall within a certain distance of the parameter’s value. 2. We can therefore reason “in reverse” from the observed value of a sample statistic to confidently infer that the population parameter is within that distance of the observed statistic. That distance depends on the level of confidence desired and on the sample size.