Statistical Inference

1. Statistical Inference • Statistical Inference is the process of making judgments about a population based on properties of the sample • intuition only goes so far towards making decisions of this nature • experts can offer conflicting opinions using the same data

2. Methods of Statistical Inference • Estimation • Predict the value of an unknown parameter with specified confidence • Decision Making • Decide between opposing statements about the population (parameter)

3. Estimation • Estimating a Population Mean (m) • Point Estimate • Mean, median, mode, etc. • Easy to calculate and use • Random in value (changes from sample to sample) • Interval Estimate • Range of values containing parameter • Unknown accuracy within range • Confidence Interval • Interval with known probability of containing truth

4. Estimating m (s known) • (1-a) Confidence Interval for μ (n≥30): • The central limit theorem provides a sampling distribution for the sample mean in cases of sufficient sample size (n ≥ 30) that can be used to find a confidence interval for μ with margin of error E is: • An alternative form is:

5. Confidence Intervals • The level of confidence and sample size both effect the width of the confidence interval. • Increasing the level of confidence results in a wider confidence interval. • Increasing the sample size results in a narrower confidence interval. • Setting the level of confidence too high results in a confidence interval that is too wide to be of any practical use. • i.e. 100% confidence intervals are from -∞ to ∞

6. Necessary Sample Size for Estimating the Mean (m) • The sample size necessary to estimate the mean (μ) with a margin of error E and (1-α) level of confidence is:

7. What if s isunknown? • William Gossett- a chemist for Guiness Brewery in the early 1900's discovered that substituting s for σ in the margin of error formula, resulted in a confidence interval that was too narrow for the desired level of confidence (1- α). • Resulted in increased error rate in statistical inference. • Error rate particularly noticeable for small samples

8. Student t Distribution • Gossett discovered that the statistic, has a Student t distribution with degree of freedom equal to n-1. The t distribution: • is symmetric about 0 • has heavier tails than the normal distribution • converges to the normal distribution as n∞.

9. Estimating m (s unknown) • If the sample data is from a normal distribution or if the sample is of sufficient size (n≥30), then the (1-a) Confidence Interval for μ is:

10. Why settle for small sample size? • Can’t you just collect more data? • Samples can be expensive to obtain. • shuttle launch, batch run • Samples can be difficult to obtain. • rare specimen, chemical process • Samples can be time consuming to obtain. • cancer research, effects of time • Ethical questions can arise. • medical research can't continue if initial results look bad

11. Estimating Population Proportion (p) • Can be thought of as the binomial probability of success if randomly sampling from the population. • Let p be the proportion of the population with some characteristic of interest. The characteristic is either present or it is not present, so the number with the characteristic is binomial.

12. Estimating Population Proportion (p) • The central limit theorem applies to a binomial random variable if the expected number of successes (np) and expected number of failures (nq) are at least 5. • The number of successes (X) is normally distributed with mean np and variance npq. • The proportion of interest is also normally distributed, with a mean of p and a variance of pq/n.

13. (1-a) Confidence Interval for a Population Proportion (p) • The point estimate for proportion is: • The (1-a) Confidence Level for p is:

14. Necessary Sample Size for Estimating the Proportion (p) • The sample size necessary to estimate the proportion (p) a margin of error E and (1-a) level of confidence (a) with prior knowledge of p and q and (b) no prior knowledge of p and q is.