Statistical Inference – Single Mean PDF

Statistical Inference – Single Mean  2 Conditions for Inference about a Mean We continue with statistical inference on a mean with unknown σ (more realistic condition). REMINDER: Conditions when testing for a...

Statistical Inference – Single Mean  2 Conditions for Inference about a Mean We continue with statistical inference on a mean with unknown σ (more realistic condition). REMINDER: Conditions when testing for a mean 1. Random sample of size n from the population of interest 2. Population has a Normal distribution N(µ, σ) in the population OR if skewed population, n is large enough (CLT) 3. The value of σ is assumed to be known. In practice, we do not know σ. 🡪 We need to estimate it. 🡪 Can we still use the z procedures previously learned?  3 Standard Error for Unknown σ  4 One-sample t statistic  5 Properties of t-distribution Comparison with standard normal distribution The t density curve is similar in shape to the standard Normal curve. They are both symmetric about 0 and bell-shaped. The variability of the t distributions is a bit greater than that of the standard Normal curve (i.e., the t curve is slightly “fatter”). When n is very large (s ≈ σ), the t-distribution will get closer to standard Normal distribution.  6 Conditions when using the t-procedures 1. Except in the case of small samples, the assumption that the data are a random sample from the population of interest is more important than the assumption that the population distribution is Normal. 2. Depending on sample size: Sample size less than 15: Use t procedures if the data appear close to Normal (symmetric, single peak, no outliers). If the data are clearly skewed or if outliers are present, do not use t. Sample size at least 15: The t procedures can be used except in the presence of outliers or strong skewness in the data. Large samples: The t procedures can be used even for clearly skewed distributions when the sample is large, roughly n ≥ 40.  7 Examples: Can or Cannot we use t procedure? b) This stem plot shows the c) This histogram shows the distribution of a) This histogram shows the percent force required to pull apart word lengths in a sample Shakespeare’s of each state’s residents who are 20 random pieces of Douglas plays (80 random words). Hispanic in the USA. Note that this fir. Can use t. The data is skewed right, but is a census data, not from a Cannot use t. The data are there are no outliers. We can use the t sample. strongly skewed to the left, so we procedures since n ≥ 40. There is no need for t. cannot trust the t procedures for n Census – study of every unit, = 20. everyone/everything in the population. Everything about this population is known including population mean, etc.