USE OF THE t DISTRIBUTION Footnote: Who was “Student”? A pseudonym for William Gosset The t is often thought of as a small-sample technique But, STRICTLY SPEAKING, the t should be used whenever the population standard deviation σ is NOT KNOWN Some practitioners use z whenever the sample is large –Central Limit Theorem –There isn’t much difference between t and z

Notes: For large samples with σ unknown, different practitioners may proceed differently. Some argue for using a z, appealing to CLT. Others use a t since it gives a less precise estimate. For this course: use a t whenever the population standard deviation is not known. Small samples from non-normal populations are beyond the scope of this course

Confidence intervals for the population proportion  Sample proportion p = x/n E(p) =  and In general  is not know, so must be estimated with p and we use

Then the confidence interval is p  z C  s p Note that proportion problems always use a z value –Normal approximates binomial EXAMPLE: Of 112 students in a sample, 70 have paying jobs. Calculate a 95% confidence interval for the proportion in the population with paying jobs.

p = 70/112 =  1.96 * etc  or  0.09 We are 95% confident that 0.54    0.71

EXAMPLE: In a sample of 320 professional economists, 251 agreed that “offshoring” jobs is good for the American economy. Calculate a 90% confidence interval for the proportion in the population of professional economists who hold this view.

Finding the Right Sample Size The error in the estimate is given by z C  σ p or, substituting Solving for n yields:

In general  is not known Two solutions: –Assume  = 0.5 Result is the largest sample that would ever be needed –Conduct a pilot study and use the resulting p as an estimate of  May give a somewhat smaller sample size if p is much different from 0.5 Saves sampling cost

Example: Above we had a 95% confidence interval with n = 112 of  0.09 or a 9% error. Suppose we require a maximum error of 3%. Approach 1: let  = 0.5

Approach 2: assume  = The difference is more dramatic if p is much different from 0.5. In a random sample of 300 students in NC, 30 have experienced “study” abroad. A 95% confidence interval for the population proportion is 10%  3.4%. Suppose we require a maximum error of 2%. Approach 1 gives _______ and approach 2 gives _________.