1. Statistics
Chapter 7: Inferences Based on a Single Sample: Estimation with Confidence Interval

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
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

4. 7.1: Identifying the Target Parameter
The unknown population parameter that we are interested in estimating is called the target parameter.
Parameter
Key Word or Phrase
Type of Data
Mean, average
Quantitative p
Proportion, percentage, fraction, rate
Qualitative
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
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

6. 7.2: Large-Sample Confidence Interval for a Population Mean
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?
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

7. 7.2: Large-Sample Confidence Interval for a Population Mean
Assuming a normal distribution for television hours, we can be 95%* sure that
*In the standard normal distribution, exactly 95% of the area under the curve is in the interval
-1.96 … +1.96
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

8. 7.2: Large-Sample Confidence Interval for a Population Mean
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.
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

9. 7.2: Large-Sample Confidence Interval for a Population Mean
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
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

10. 7.2: Large-Sample Confidence Interval for a Population Mean
The area outside the confidence interval is called 
95 % sure
So we are left with (1 – 95)% = 5% =  uncertainty about µ
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

11. 7.2: Large-Sample Confidence Interval for a Population Mean
If  is unknown and n is large, the confidence interval becomes
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

12. 7.2: Large-Sample Confidence Interval for a Population Mean
For the confidence interval to be valid …
the sample must be random and …
the sample size n must be large.
If n is large, the sampling distribution of the sample
mean is normal, and s is a good estimate of 
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

13. 7.3: Small-Sample Confidence Interval for a Population Mean
Large Sample
Small Sample
Sampling Distribution on  is normal
Known  or large n
Standard Normal (z) Distribution
Sampling Distribution on  is unknown
Unknown  and small n
Student’s t Distribution (with n-1 degrees of freedom)
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

14. 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

15. 7.3: Small-Sample Confidence Interval for a Population Mean
For the confidence interval to be valid …
the sample must be random and …
the population must have a relative frequency distribution that is approximately normal*
* If not, see Chapter 14
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

16. 7.3: Small-Sample Confidence Interval for a Population Mean
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?
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

17. 7.3: Small-Sample Confidence Interval for a Population Mean
Assuming a normal distribution for television hours, we can be 95% sure that
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

18. 7.4: Large-Sample Confidence Interval for a Population Proportion
Sampling distribution of
The mean of the sampling distribution is p, the population proportion.
The standard deviation of the sampling distribution is
where
For large samples the sampling distribution is approximately normal. Large is defined as
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

19. 7.4: Large-Sample Confidence Interval for a Population Proportion
Sampling distribution of
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

20. 7.4: Large-Sample Confidence Interval for a Population Proportion
We can be 100(1-)% confident that
where and
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

21. 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.
What proportion of the entire
population agree, at the 95%
confidence level?
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

22. 7.4: Large-Sample Confidence Interval for a Population Proportion
If p is close to 0 or 1, Wilson’s adjustment for estimating p yields better results
where
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

23. 7.4: Large-Sample Confidence Interval for a Population Proportion
Suppose in a particular year the percentage of firms declaring bankruptcy that had shown profits the previous year is 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?
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

24. 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-)%, we can solve
for n:
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

25. 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

26. 7.5: Determining the Sample Size
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?
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

27. 7.5: Determining the Sample Size
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?
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

28. 7.5: Determining the Sample Size
To estimate p, use
the sample proportion
from a prior study, or
use p = .5.
Round the value of n
upward to ensure the
sample size is large
enough to produce the
required level of
confidence.
For a confidence interval on the population proportion, p, we can solve
for n:
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

29. 7.5: Determining the Sample Size
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?
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals

30. 7.5: Determining the Sample Size
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?
McClave, Statistics, 11th ed. Chapter 7: Inferences Based on a Single Sample: Estimation by Confidence Intervals