8.2 Estimating Proportions
Uses:
• Unemployment rates (proportion unemployed)
• Student drop rates (proportion college freshmen failing)
• Company's claims (proportion light bulbs lasting as long
as claimed)
Activity: Proportion of pennies older than 10 years (before 2002)
1. Take a sample of 25 pennies.
2. Find p­hat = proportion older than 10 years
3. Assume your p­hat is equal to true proportion, p. Find a 95% confidence interval. Draw your interval on the board.
Challenge: Find a 90% confidence interval.
Did we meet all the conditions in the penny sampling?
RANDOM: We took an SRS of 50 pennies.
NORMAL: Check np and n(1-p)... (or...#successes/failures)
INDEPENDENT: (10% condition) -- 10(25) = 250
There are more than 250 pennies in the total population
of pennies, so it is okay to draw without replacement
and assume independence.
Back to that 90% confidence interval... CRITICAL VALUE
Now, find the CRITICAL VALUE (z*) for these
different confidence levels:
C = 80%
C = 96%
(Why do we use p-hat instead of p?)
C = 99.7%
Example: The Pennies
Ms. Smith’s class took an SRS of 102 pennies and discovered that 57 of the pennies were more than 10 years old. (a) Calculate and interpret a 99% confidence interval for p = the true proportion of pennies from the collection that are more than 10 years old.
p = 57/102 = 0.559 z* = invNorm(0.995) = 2.576 The 99% confidence interval is: 0.559 ± (2.576)
= (0.432, 0.686)
We are 99% confident that the interval from 0.432 to 0.686 captures the actual proportion of pennies in the collection that are more than 10 years old. (b) Is it plausible that exactly 60% of all the pennies in the collection are more than 10 years old? Explain. Yes, since 0.6 is included in the confidence interval, it is plausible that 60% of all the pennies in the collection are more than 10 years old.