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AP Statistics Chapter 19 Notes. “Confidence Intervals for Sample Proportions”. - the population’s proportion - the sample’s proportion In the last chapter, we knew the population proportion - PowerPoint PPT Presentation
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AP Statistics Chapter AP Statistics Chapter 19 Notes19 Notes
“Confidence Intervals for Sample Proportions”
pp ˆ vs.◦ - the population’s proportion
◦ - the sample’s proportion
◦In the last chapter, we knew the population proportion
◦In this chapter, we DO NOT know the population proportion and are going to use the sample proportion to ESTIMATE it
p
p̂
Standard Error (not standard Standard Error (not standard deviation this time)deviation this time)
Standard Error =
We call it a “standard error” here instead of a “standard deviation” because we don’t know the population proportion “p” and we are trying to ESTIMATE it with a certain amount of room for error.
n
qp ˆˆ
Example 1Example 1Police set up an auto checkpoint at which drivers
are stopped and their cars inspected for safety problems. They find that 14 of 134 cars stopped have at least one safety violation. They want to estimate the percentage of all cars that may be unsafe.
What is the population?
What is the sample size?
What does p represent?
What does represent?p̂
All cars in the US
The 134 cars at this checkpoint
The % of all cars in the US with at least one safety violation
The 10.4% of cars with at least one safety violation found in this sample
More about this exampleMore about this exampleBased on this sample, we want to
estimate the true percentage of cars that are unsafe. Draw the normal model up to three “standard errors” away from the mean.
Make a prediction about the true value of p with 95% confidence.
Mean = 10.4%
SE = %6.2134
%6.89%4.10ˆˆ
n
qp10.4%13%7.8%
15.6%5.2%18.2%2.6%
Based on this sample, we are 95% confident that between 5.2% and 15.6% of all US cars have a safety violation.
Confidence IntervalsConfidence Intervals
When the true proportion p is unknown, we use to predict it to some level of confidence. This is called a confidence interval.
A confidence interval is the interval of values that you are C% confident contains the true value of p. ( +/- margin of error)
p̂
p̂
Margin of Error:Margin of Error: The extent of the interval on either side of is called margin
of error As the confidence level improves (e.g. 90% … 95% … 99%),
the margin of error widens. As the margin of error tightens up, the confidence level will
decrease. The balance between certainty and precision is somewhat
subjective, but a 90% or 95% confidence interval is usually standard
Know that the larger the sample size, the smaller the standard error and therefore the smaller the margin of error.
p̂
ExampleExample A consumer group hoping to estimate the percentage of
college students who have cell phones surveyed 2883 students as they entered a football stadium. 2781 indicated they had a cell phone.
Based on this sample, find a 95% confidence interval for the true proportion of all cell-phone-carrying college students.
What is the margin of error for your confidence interval?
Mean = 96.5%
SE = %34.02883
%5.3%5.96ˆˆ
n
qp96.5%96.84%96.16%
97.18%95.82%
97.52%95.48%
96.5% - 95.82% = 0.68 or 96.5% - 97.18% = -0.68 so the ME = +/- 0.68%
Confidence Intervals in the Confidence Intervals in the CalculatorCalculatorSTATTESTSA. 1 – PropZInt
◦x = number of successes in your sample◦n = the sample size◦C – level = the confidence level you want
Press “Calculate”
For margin of error – subtract either end of the confidence interval from p̂
2781
2883
.95
What we can say and what we What we can say and what we cannot saycannot say
Say this:◦ Based on this sample, we can be ___% confident
that the true proportion of college students carrying cell phones is between ___% and ____%.
Don’t say this:Between ___% and ___% of all college students carry cell phones. (We don’t have 100% certainty.)
95%
97.1895.82
97.1895.82
Don’t Forget the ConditionsDon’t Forget the ConditionsIndependence – We assume the data values do
not affect each otherRandom sample – We assume the data are
sampled randomly and adequately represent the population
Population must be at least 10 times the size of the sample
10 Successes/10 Failures condition – np > 10 and nq > 10 to ensure that the sample size is big enough.