Download Randomness and Probability

Confidence Intervals:
Estimating Population
Proportion
Section 8.2
Reference Text:
The Practice of Statistics, Fourth Edition.
Starnes, Yates, Moore
Objectives
• Conditions For estimating p, why we use 𝒑
– Standard deviation
-Standard Error
• Calculating Confidence Intervals for p
– statistic ± (critical value) · (standard deviation of statistic)
– How do we find the critical Value
• One Sample Z interval for a Population
Proportion
– Formula, by hand
--Calculator efficiency
• Conditions of Constructing Confidence Intervals
• Putting it all together: The 4 steps to ALWAYS
cover.
• Choosing the sample size
So…
• In 8.1, we saw that a confidence interval can be used
to estimate an unknown population parameter. We
are often interested in estimating the proportion p of
some outcome in the population. Some examples:
• What proportion of US adults are unemployed right
now?
• What proportion of high school students cheated on
a test?
• What proportion of a company’s lap top batteries last
as long as the company claims?
• This section shows us how to construct and interpret a confidence interval
for a population proportion.
Why We can Use 𝒑
• In practice, of course, we don’t know the value
of our population proportion (p). If we did, we
wouldn’t need to construct a confidence
interval for it!
• So we use 𝒑 to estimate for p.
• In large samples, 𝒑 will be close to p. so we
can replace p with 𝒑!
• Lets look at our normal conditions and
standard deviation of p, and replace it with 𝒑.
Normal Conditions and
Standard Deviation
• Pop Proportion  Sample Proportion
•
np≥10

n𝒑≥ 10
•
n(1-p)≥10
 n(1-𝒑)≥10
• σ𝒑 =
𝑝(1−𝑝)
𝑛
 σ𝒑 =
𝒑(1−𝒑)
𝑛
• Again, since we don’t know the value of p we replace it with 𝒑 .
When the standard deviation of a statistic is estimated from
data, the result is called the standard error. Since we are
replacing a value, this creates some error. We are ok with this.
Conditions for Constructing A
Confidence Interval
• Random: The data should come from a welldesigned random sample or randomized experiment.
• Normal: n𝒑≥ 10
AND
n(1-𝒑)≥10
• Independent: the procedure of calculating
confidence intervals assume that individual
observations are independent and the 10% condition
1
is met.
n≤ 𝑁
10
• Run through these conditions before
constructing an interval!!!
Check Your Understanding
• In each of the following settings, check whether the
conditions for calculating a confidence interval for the
population proportion p are met.
• Conditions: Random, Normal, Independent
• 1. An AP Statistics class at a large high school conducts a survey.
They ask the first 100 students to arrive at school one morning
whether or not they slept at least 8 hours the night before. Only 17
students say “Yes.”
• 2.A quality control inspector takes a random sample of 25 bags of
potato chips from the thousands of bags filled in an hour. Of the bags
selected, 3 had too much salt.
Calculating a Confidence
interval for p.
statistic ± (critical value) · (standard deviation of statistic)
𝒑 ±𝑧
∗
𝒑(1−𝒑)
𝑛
How do I find the critical value 𝑧 ∗ ?
To find a level C (say…95%?) confidence interval, we
need to catch the central area C under the standard
Normal curve.
∗
How do I find the critical value 𝑧 ?
• Finding the critical value for a 95% confidence
interval, we look at the area under the curve
that’s NOT being accounted for.
100%-95% =5% / 2 tails = 2.5% each tail
Example:
• PROBLEM: Use Table A to find the critical value z* for an 80%
confidence interval. Assume that the Normal condition is met.
Example Cont…
• SOLUTION: For an 80% confidence level, we need to catch the central
80% of the standard Normal distribution. In catching the central 80%, we
leave out 20%, or 10% in each tail. So the desired critical value z* is the
point with area 0.1 to its right under the standard Normal curve. The table
shows the details in picture form:
• Search the body of Table a to find the point −z* with area 0.1 to its left. The
closest entry is z = −1.28. (See the excerpt from Table A above.) So the
critical value we want is z* = 1.28.
Calculator to find my Critical
Value!
We are AP, and AP is pressed for time!
How to find the critical value on your calc:
2nd>Vars (distr) >3: invNorm( >Enter > fill in
the following invNorm(.05,0,1)
.05 = area to the left of the curve, 0 and 1 are
just to state it’s a standard Normal curve.
The Calculator makes Table A obsolete!
One Sample Z Interval for a
Population Proportion
• Choose a SRS of size n from a large population that contains
an unknown proportion p of successes. An approximate level
C confidence interval for p is :
𝒑 ±𝑧
∗
𝒑(1−𝒑)
𝑛
Where 𝑧 ∗ is the critical value for the standard Normal
curve with area C between −𝑧 ∗ 𝑎𝑛𝑑 𝑧 ∗ .
Use this interval only when the numbers of successes and
failures in the sample are both at least 10 and the population is
at last 10 times as large as the sample.
Example of a 1-prop Z interval
• PROBLEM: Mr. Vignolini’s class took an SRS of beads from
the container and got 107 red beads and 144 white beads,
making a total of 251 beads. We are using p = red beads.
• (a) Calculate and interpret a 90% confidence interval for p.
• (b) Mr. Vignolini claims that exactly half of the beads in the
container are red. Use your result from (a) to comment on this
claim.
Example Cont…
• 90% confidence interval:
• Lets calculate 𝒑, and find our critical value.
Then plug it in!
Calculator!
• We are AP, and AP is pressed for time!
How to construct an confidence interval for population
proportion:
TI 83/84
STAT> TESTS> A:1-PropZInt> enter
information in X, n, and C-level> highlight
Calculate> Enter
TI 89
Statistics/List editor>2nd>F2 (F7) > 5:1propZint
Calculator Practice:
• We’ll demonstrate using the example of n = 439
teens surveyed, X = 246 said they thought that
young people should go to college. Use your
calculator to calc the interval:
TI 83/84
STAT> TESTS> A:1-PropZInt> enter information in
X, n, and C-level> highlight Calculate> Enter
TI 89
Statistics/List editor>2nd>F2 (F7) > 5:1-propZint
AP Warning!
• AP EXAM TIP You may use your calculator to
compute a confidence interval on the AP exam. But
there’s a risk involved. If you just give the calculator
answer with no work, you’ll get either full credit for the
“Do” step (if the interval is correct) or no credit (if it’s
wrong). We recommend showing the calculation with
the appropriate formula and then checking with your
calculator. If you opt for the calculator-only method,
be sure to name the procedure (e.g., one-proportion
z interval) and to give the interval (e.g., 0.514 to
0.606).
Putting it all Together:
A Four-Step Process
• State: what parameter do you want to
estimate, and at what confidence level?
• Plan: Identify the appropriate inference
method. Check conditions.
• Do: If the conditions are met, preform
calculations.
• Conclude: Interpret your interval in the context
of the problem.
• AP expects ALL FOUR, do not skip step 1!
Example:
See Handout for article:
Teens Say Sex Can Wait
Choosing the Sample Size
• To determine the sample size n that will yield a level C
confidence interval for a population proportion p with a
maximum margin of error ME, solve the following inequality
for n:
𝑧∗
𝒑(1−𝒑)
≤
𝑛
ME
• Where 𝒑 is a guessed value for the sample proportion. The
margin of error will always be less than or equal to ME if
you take the guess of 𝒑 to be 0.5.
Objectives
• Conditions For estimating p, why we use 𝒑
– Standard deviation
-Standard Error
• Calculating Confidence Intervals for p
– statistic ± (critical value) · (standard deviation of statistic)
– How do we find the critical Value
• One Sample Z interval for a Population
Proportion
– Formula, by hand
--Calculator efficiency
• Conditions of Constructing Confidence Intervals
• Putting it all together: The 4 steps to ALWAYS
cover.
• Choosing the sample size
Homework
Worksheet