From the Data at Hand to the World at Large Chapters 19, 23 Confidence Intervals

1
From the Data at Hand to the World at
LargeChapters 19, 23Confidence Intervals

• Estimation of population parameters
• an unknown population proportion p
• an unknown population mean ?

2
Concepts of Estimation

• The objective of estimation is to estimate the
  unknown value of a population parameter, like the
  mean ?, on the basis of a sample statistic
  calculated from sample data.
• ? e.g., NCSU housing office may want to estimate
  the mean distance ? from campus to hometown of
  all students
• There are two types of estimates
• Point Estimate
• Interval estimate

3
What do we frequently need to estimate?

• An unknown population proportion p
• An unknown population mean ?

?? p?
4
Point Estimates

• The sample mean is the best point estimate
  of the population mean ?
• p , the sample proportion of x successes
  in a sample of size n, is the best point estimate
  of the population proportion p

5
Example Estimating an unknown population
proportion p

• Is Herb Sendek's departure good or bad for
  State's men's basketball team? (Technician
  opinion poll not scientifically valid!!)
• In a sample of 1000 students, 590 say that
  Sendeks departure is good for the bb team.
• p 590/1000 .59 is the point estimate of the
  unknown population proportion p that think
  Sendeks departure is good.

6
Example Estimating an unknown mean ?

• In an effort to improve drive-through service, a
  Burger King records the drive-through service
  times of 52 randomly selected vehicles.
• The sample mean service time 181.3 seconds
  is the point estimate of the unknown mean service
  time ?

7
Shortcoming of Point Estimates

• 181.3 seconds, best estimate of mean
  service time ?
• p 590/1000 .59, best estimate of population
  proportion p
• BUT
• How good are these best estimates?
• No measure of reliability

8
Interval Estimator
A confidence interval is a range (or an
interval) of values used to estimate the unknown
value of a population parameter
. http//abcnews.go.com/US/PollVault/
9
95 Confidence Interval for p
10
Standard Normal
P(-1.96 ? z ? 1.96) . 95
11
Sampling distribution of
.95
12
Standard Normal
P(-1.96 ? z ? 1.96) . 95
13
Example (Gallup Polls)
http//abcnews.go.com/US/PollVault/story?id145373
page1
14
Medication side effects (confidence interval for
p)
Arthritis is a painful, chronic inflammation of
the joints. An experiment on the side effects of
pain relievers examined arthritis patients to
find the proportion of patients who suffer side
effects.
What are some side effects of ibuprofen? Serious
side effects (seek medical attention
immediately) Allergic reaction (difficulty
breathing, swelling, or hives), Muscle cramps,
numbness, or tingling, Ulcers (open sores) in
the mouth, Rapid weight gain (fluid
retention), Seizures, Black, bloody, or tarry
stools, Blood in your urine or vomit, Decreased
hearing or ringing in the ears, Jaundice
(yellowing of the skin or eyes), or Abdominal
cramping, indigestion, or heartburn, Less serious
side effects (discuss with your
doctor) Dizziness or headache, Nausea,
gaseousness, diarrhea, or constipation, Depressio
n, Fatigue or weakness, Dry mouth,
or Irregular menstrual periods
15
440 subjects with chronic arthritis were given
ibuprofen for pain relief 23 subjects suffered
from adverse side effects.
Calculate a 90 confidence interval for the
population proportion p of arthritis patients who
suffer some adverse symptoms.
What is the sample proportion ?
For a 90 confidence level, z 1.645.
? We are 90 confident that the interval (.034,
.070) contains the true proportion of arthritis
patients that experience some adverse symptoms
when taking ibuprofen.
16
Tool for Constructing Confidence Intervals for ?
The Central Limit Theorem

• If a random sample of n observations is selected
  from a population (any population), then when n
  is sufficiently large, the sampling distribution
  of x will be approximately normal.
• (The larger the sample size, the better will be
  the normal approximation to the sampling
  distribution of x well use n ? 30)

17
Estimating the Population Mean ? when the
Population Standard Deviation is Known

• How is an interval estimator produced from a
  sampling distribution?
• To estimate m, a sample of size n is drawn from
  the population, and its mean is calculated.
• Under certain conditions, is normally
  distributed (or approximately normally
  distributed by the CLT).

18
Confidence Interval for a population mean ?
19
Standard Normal
P(-1.96 ? z ? 1.96) . 95
20
EXAMPLE
21
Sampling distribution of x
.95
22
Standard Normal
23
98 Confidence Intervals
24
Four Commonly Used Confidence Levels

• Confidence Level Multiplier
• .90 1.645
• .95 1.96
• .98 2.33
• .99 2.58

25
Example (cont.)
26
Example (cont.)
27
Example Summary

• 90 (30.06, 30.74)
• 95 (29.995, 30.805)
• 98 (29.919, 30.881)
• 99 (29.867, 30.933)
• The higher the confidence level, the wider the
  interval
• Increasing the sample size n will make a
  confidence interval with the same confidence
  level narrower (i.e., more precise)

28
Example (cont.)
29
Example

• Find a 95 confidence interval for p, the
  proportion of small businesses in favor of a tax
  increase to decrease the national debt, if a
  random sample of 1000 found the number of
  businesses in favor of increased taxes was 50.

30
Example (solution)
31
Interpreting Confidence Intervals

• Previous example .05.014?(.036, .064)
• Correct We are 95 confident that the interval
  from .036 to .064 actually does contain the true
  value of p. This means that if we were to select
  many different samples of size 1000 and construct
  a 95 CI from each sample, 95 of the resulting
  intervals would contain the value of the
  population proportion p. (.036, .064) is one such
  interval. (Note that 95 refers to the procedure
  we used to construct the interval it does not
  refer to the population proportion p)
• Wrong There is a 95 chance that the population
  proportion p falls between .036 and .064. (Note
  that p is not random, it is a fixed but unknown
  number)