Review Confidence Intervals Sample Size Confidence Intervals The

Review • Confidence Intervals • Sample Size

Confidence Intervals The Confidence Interval is expressed as: E is called the margin of error. For samples of size > 30,

Sample Size The sample size needed to estimate m so as to be (1 -a)*100 % confident that the sample mean does not differ from m more than E is: …round up

Small Samples What happens if n is small (n < 30)? Our formulas from the last section no longer apply.

Small Samples What happens if n is small (n < 30)? Our formulas from the last section no longer apply. There are two main issues that arise for small samples: 1) s no longer can be approximated by s 2) The CLT no longer holds. That is the distribution of the sampling means is not necessarily normal.

t- distribution If we have a small sample (n < 30) and wish to construct a confidence interval for the mean we can use a t-statistic, provided the sample is drawn from a normally distributed population.

t-distributions s is unknown so we use s (the sample standard deviation) as a point estimate of s. We convert the nonstandard t-distributed problem to a standard t-distributed problem through the use of the standard t-score

t-distributions • Mean 0 • Symmetric and bell-shaped • Shape depends upon the degrees of freedom, which is one less than the sample size. df = n-1 • Lower in center, higher tails than normal. • See Table inside front cover in text

Example In n=15 and after some calculation a/2=0. 025, we use the table and n -1 = 14 degrees of freedom to deduce t 0. 025 = 2. 145

Confidence Interval for the mean when s is unknown and n is small The (1 - a)*100% confidence interval for the population mean m is The margin of error E, is in this case N. B. The sample is assumed to be drawn from a normal population.

Confidence Intervals for a small sample population mean The Confidence Interval is expressed as: The degrees of freedom is n-1.

Example The following are the heat producing capabilities of coal from a particular mine (in millions of calories per ton) 8, 500 8, 330 8, 480 7, 960 8, 030 Construct a 99% confidence interval for the true mean heat capacity. Solution: sample mean is 8260. 0 sample Std. Dev. is 251. 9 degrees of freedom = 4 a = 0. 01 7741. 4 m 8778. 6

Confidence intervals for a population proportion The objective of many surveys is to determine the proportion, p, of the population that possess a particular attribute. . Example: Determine the fraction of Canadians who support gun control. If the size of the population is N, and X people have this attribute, then as we already know, is the population proportion.

Confidence intervals for a population proportion If the size of the population is N, and X people have this attribute, then as we already know, is the population proportion. The idea here is to take a sample of size n, and count how many items in the sample have this attribute, call it x. Calculate the sample proportion, . We would like to use the sample proportion as an estimate for the population proportion.

The confidence interval for the population proportion is :