Chapter 7: Inferences Based on a Single Sample: Estimation with Confidence Interval Statistics.
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Transcript of Chapter 7: Inferences Based on a Single Sample: Estimation with Confidence Interval Statistics.
Chapter 7: Inferences Based on a Single Sample: Estimation with Confidence Interval
Statistics
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
2
Where We’ve Been
Populations are characterized by numerical measures called parameters
Decisions about population parameters are based on sample statistics
Inferences involve uncertainty reflected in the sampling distribution of the statistic
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
3
Where We’re Going
Estimate a parameter based on a large sample
Use the sampling distribution of the statistic to form a confidence interval for the parameter
Select the proper sample size when estimating a parameter
7.1: Identifying the Target Parameter
The unknown population parameter that we are interested in estimating is called the target parameter.
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
4
Parameter Key Word or Phrase Type of Data
µ Mean, average Quantitative
p Proportion, percentage, fraction, rate
Qualitative
7.2: Large-Sample Confidence Interval for a Population Mean
A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used to estimate the population parameter.
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
5
7.2: Large-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
6
Suppose a sample of 225 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours. What can we conclude about all college
students’ television time?
7.2: Large-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
7
Assuming a normal distribution for television hours, we can be 95%* sure that
31.128)67(.96.128
2251096.128
96.1
n
x
*In the standard normal distribution, exactly 95% of the area under the curve is in the interval
-1.96 … +1.96
7.2: Large-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
8
An interval estimator or confidence interval is a formula that tell us how to use sample data to calculate an interval that estimates a population parameter.
xzx
7.2: Large-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
9
The confidence coefficient is the probability that a randomly selected confidence interval encloses the population parameter.
The confidence level is the confidence coefficient expressed as a percentage.(90%, 95% and 99% are very commonly used.)
95% sure
xzx
7.2: Large-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
10
The area outside the confidence interval is called α
95 % surexzx
So we are left with (1 – 95)% = 5% = α uncertainty about µ
7.2: Large-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
11
Large-Sample (1-α)% Confidence Interval for µ
If is unknown and n is large, the confidence interval becomes
nzxzx axa
2/2/
nszxszx axa 2/2/
7.2: Large-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
12
If n is large, the sampling distribution of the sample mean is normal, and s is a good estimate of
7.3: Small-Sample Confidence Interval for a Population Mean
Large Sample Sampling Distribution on
is normal Known or large n
Standard Normal (z) Distribution
Small Sample
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
13
Sampling Distribution on is unknown
Unknown and small n
Student’s t Distribution (with n-1 degrees of freedom)
nzx a
2/nstx a 2/
7.3: Small-Sample Confidence Interval for a Population Mean
Large Sample Small Sample
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
14
nzx a
2/nstx a 2/
7.3: Small-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
15* If not, see Chapter 14
7.3: Small-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
16
Suppose a sample of 25 college students watch an average of 28 hours of television per week, with a standard deviation of 10 hours. What can we conclude about all college
students’ television time?
7.3: Small-Sample Confidence Interval for a Population Mean
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
17
Assuming a normal distribution for television hours, we can be 95% sure that
128.428)67(.064.2282510064.228
064.2
nsx
7.4: Large-Sample Confidence Interval for a Population Proportion
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
18
p̂
npq
p ˆ
13ˆ0 ˆ pp
pq 1
7.4: Large-Sample Confidence Interval for a Population Proportion
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
19
Sampling distribution of
7.4: Large-Sample Confidence Interval for a Population Proportion
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
20
nqpzp
npqzpzpp p
ˆˆˆˆˆ 2/2/ˆ2/
nxp ˆ pq ˆ1ˆ where and
We can be 100(1-α)% confident that
7.4: Large-Sample Confidence Interval for a Population Proportion
A nationwide poll of nearly 1,500 people … conducted by the syndicated cable television show Dateline: USA found that more than 70 percent of those surveyed believe there is intelligent life in the universe, perhaps even in our own Milky Way Galaxy.
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
21
What proportion of the entirepopulation agree, at the 95% confidence level?
023.70.)012)(.96.1(70.
1500)30)(.70(.96.170.
ˆˆˆ 2/
pp
p
nqpzpp a
7.4: Large-Sample Confidence Interval for a Population Proportion
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
22
If p is close to 0 or 1, Wilson’s adjustment for estimating p yields better results
where
42~
4)~1(~~
2/
nxp
nppzp
7.4: Large-Sample Confidence Interval for a Population Proportion
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
23
032.0289.4100
)0289.1(0289.96.10289.
0289.410021
42~
4)~1(~~
2/
p
p
nxp
nppzpp
Suppose in a particular year the percentage of firms declaring bankruptcy that had shown profits the previous year is .002. If 100 firms are sampled and one had declared bankruptcy, what is the 95% CI on the proportion of profitable firms that will tank the next year?
7.5: Determining the Sample Size
To be within a certain sampling error (SE) of µ with a level of confidence equal to 100(1-α)%, we can solve
for n:
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
24
SEn
z
2/
2
222/ )(SEzn
7.5: Determining the Sample Size
The value of will almost always be unknown, so we need an estimate: s from a previous sample approximate the range, R, and use R/4
Round the calculated value of n upwards to be sure you don’t have too small a sample.
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
25
7.5: Determining the Sample Size
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
26
Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test?
7.5: Determining the Sample Size
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
27
Suppose we need to know the mean driving distance for a new composite golf ball within 3 yards, with 95% confidence. A previous study had a standard deviation of 25 yards. How many golf balls must we test?
26778.26632596.1
)(
2
22
2
222/
n
n
SEzn
7.5: Determining the Sample Size
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
28
For a confidence interval on the population proportion, p, we can solve
for n:
SEnpqz 2/
2
22/ )()(SE
pqzn
To estimate p, usethe sample proportion from a prior study, oruse p = .5.
Round the value of n upward to ensure the sample size is large enough to produce the required level of confidence.
7.5: Determining the Sample Size
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
29
How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within .01 with 90% confidence, if an initial estimate of .10 is used for p?
7.5: Determining the Sample Size
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample:
Estimation by Confidence Intervals
30
How many cellular phones must a manufacturer test to estimate the fraction defective, p, to within .01 with 90% confidence, if an initial estimate of .10 is used for p?
24364.2435)01(.
)9)(.1(.)645.1(
)()()(
2
2
2
22/
2/
n
n
SEpqzn
npqzSE