Section 6.3
Confidence Intervals for
Population Proportions
Point Estimate for Proportions
 The Population Proportion is called p
 The Point Estimate is the sample proportion is
called “p hat”
To find the Margin of Error, E
Confidence Intervals for the
Population Proportion

Construct a C.I. for the
Proportion
 1. Find n and x to find p-hat
 2. Make sure the normal approximation is
allowed
 3. Find the critical value zc that corresponds
with the given level of confidence.
 4. Find the margin of error, E.
 5. Find the left and right endpoints and form
the confidence interval.
EX from p 325-326
 14. In a survey of 4013 US adults, 722 say they
have seen a ghost. Construct a 99% C.I. for the
population proportion of US adults who have
seen a ghost.
 16. In a survey of 2303 US adults, 734 believe in
UFOs. Construct a 90% C.I. for the population
proportion of US adults who believe in UFOs.
To find minimum sample size
 20. You wish to estimate, with 95% confidence, the
population proportion of US adults who say
chocolate is their favorite ice cream flavor. Your
estimate must be accurate within 5% of the
population proportion.
 A) No preliminary estimate is available. Find the
minimum sample size needed.
 B) Find the minimum sample size needed, using a
prior study that found that 28% of US adults say that
chocolate is their favorite ice cream flavor.
 C) Compare results from parts (A) and (B)
Which Table do I use???
 Confidence Interval for MEAN:
 If σ is known, use the Z table.
 If σ is unknown, consider the sample size
n.
 If n > 30, use the Z table.
 If n < 30, use the T table.
 Confidence Interval for PROPORTION:
 Use the Z table.
 C.I. for VARIANCE or STANDARD
DEVIATION: use CHI-Squared Table.
Section 6.4
Confidence Intervals for
Variance & Standard
Deviation
Point Estimates
 Population variance is σ2
 The point estimate for variance is s2
 Population standard deviation is σ
 The point estimate for standard
deviation is s.
The Chi-Square Distribution
2
(table #6) Chi-Square = X
 Use for sample sizes n > 1
 All X2 > 0
 Uses Degrees of Freedom: d.f. = n – 1
 Area under the curve = 1
 Chi-Square distributions are positively (or
right) skewed.
 NOTE: The shaded area is to the RIGHT!
Finding Critical Values for X2
 X2R is the RIGHT hand critical value. To
find this, use ½ of c as α on the table.
 X2L is the LEFT hand critical value. To
find this, use the COMPLEMENT of ½ of c
as α on the table.
 Keep in mind that X2L < X2R
Find the critical values X2L & X2R
 7. c = 0.95
n = 20
 8. c = 0.80
n = 51
Confidence Interval for
Variance
 Remember… intervals are always left to right,
smaller to larger!
To find Confidence Intervals
 1. Verify the population has a normal
distribution.
 2. Find degrees of freedom: d.f. = n – 1
 3. Find point estimate s2, use StatCrunch if
needed.
 4. Find critical values using chi-square table.
 5. Find the left and right endpoints for the C.I.
for the population VARIANCE.
 6. Square root to find the left and right
endpoints for the C.I. for the population
STANDARD DEVIATION.
Examples from p 334-5
 10. The volumes (in fluid ounces) of the
contents of 15 randomly selected bottles of
cough syrup are listed. Construct a 90% C.I. for
the population variance and the standard
deviation. (Assume the population is normally
distributed.)
4.211
4.264
4.269
4.241
4.260
4.293
4.189
4.248
4.220
4.239
4.253
4.209
4.300
4.256
4.290
18. A magazine includes a report on the prices of
subcompact digital cameras. The article states
that 11 randomly selected subcompact digital
cameras have a sample standard deviation of
$109. Assume the population is normally
distributed. Construct a CI for the population
variance and standard deviation using an 80%
level of confidence.