STAT1008 Lecture Notes - Lecture 26: Null Hypothesis, Confidence Interval, Minimax
STAT1008 Week 9 Lecture B
● Confidence Interval for mean:
○ The general formula for a confidence interval is statistic + or - z*.SE(z*.SE =
margin of error)
○ For means, replacing sigma with s causes us to use the t-distribution instead of
the standard normal
○ For means: statistic + or - t*.SE
○ Thus t* is quantiles form
○ If the population is approximately normal or the n is large (n > or equal to 30),
then a confidence interval for mu can be computed by mean + or - t*. s/sqrt(n)
○ E.g. Gribbles
■ Gribbles are small marine worms that bore through wood, and the
enzyme they secrete may allow us to turn inedible wood and plant waste
into biofuel
■ A sample of 50 gribbles finds an average length of 3.1mm with a standard
deviation of 0.72mm
■ Give a 90% confidence interval for the average length of gribbles
● Statistic + or - t*.SE where df (degrees of freedom) = n-1 = 49
● Mean + or - t*.SE = mean + or - t*. s/sqrt(n) thus 3.1 + or - 1.677 x
(0.72/sqrt(50)) = 2.93,3.27
● Margin of error:
○ ME = t*.s/sqrt(n)
○ You can choose your sample size in advance, depending on your desired margin
or error!
○ Given this formula for margin of error, solve for n.
■ N = (t*s/ME)2
■ Problem 1: For t*, need to know n.
● Solution: use z* instead ot t* (they are usually close)
■ Problem 2: For s, need data.
● Solution: Estimate s by using data from a previous study or similar
population, take a small pre-sample to estimate s and estimate the
range (max-min) and use s~ range/4 or make a reasonable guess
○ E.g. GPA
■ Suppose we want to estimate average GPA at a college (where GPAs go
from 0 - 4.0), with a margin of error of 0.1 with a 95% confidence
■ n = (Z*s/ME)2
■ How large a sample size do we need?
● S ~ max-min/4 = 4-0/4 = 1
● Thus n = (Z*s/ME)2 = (2 x 1/0.1)2 = 400
● Test for single mean:
○ E.g. Chips Ahoy!
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Document Summary
The general formula for a confidence interval is statistic + or - z*. se(z*. se = margin of error) For means, replacing sigma with s causes us to use the t-distribution instead of the standard normal. For means: statistic + or - t*. se. If the population is approximately normal or the n is large (n > or equal to 30), then a confidence interval for mu can be computed by mean + or - t*. s/sqrt(n) Gribbles are small marine worms that bore through wood, and the enzyme they secrete may allow us to turn inedible wood and plant waste into biofuel. A sample of 50 gribbles finds an average length of 3. 1mm with a standard deviation of 0. 72mm. Give a 90% confidence interval for the average length of gribbles. Statistic + or - t*. se where df (degrees of freedom) = n-1 = 49.
