1. Chapter 7 Confidence Intervals (置信区间)
z-Based Confidence Intervals for a Population Mean: Known
t-Based Confidence Intervals for a Population Mean: Unknown
Sample Size Determination(样本量计算)
Confidence Intervals for a Population Proportion

2. Chapter 7 Confidence Intervals

3. Statistical Inferences
Estimation (估计)
Point estimation (点估计)
Confidence interval estimation
(置信区间的估计)
Hypothesis-testing (假设检验 )
Chapter 7 Confidence Intervals

4. Chapter 7 Confidence Intervals
Example
The Dean of the Business School wants to estimate the mean number of hours worked per week by students. A sample of 49 students showed a mean of 24 hours with a standard deviation of 4 hours. What is the population mean?
The value of the population mean is not known. Our best estimate of this value is the sample mean of 24.0 hours. This value is called a point estimate.
Chapter 7 Confidence Intervals

5. Chapter 7 Confidence Intervals
A point estimate is a single value (statistic) used to estimate a population value (parameter).
A confidence interval (CI) is a range of values within which the population parameter is expected to occur at a specified probability (confidence level).
The two confidence levels that are used extensively are the 95% and the 99%.
An Interval Estimate states the range within which a population parameter probably lies.
Chapter 7 Confidence Intervals

6. z-Based Confidence Intervals for a Population Mean
Consider the sampling distribution of the sample mean
Recall from Chapter 6 that if a population is normally distributed with mean m and standard deviation s, then the sampling distribution of is normal with mean
m = m and standard deviation
Even if the population being sampled is non-normal, when the sample size n is large, the sampling distribution of sample means is again approximately normal (by the Central Limit Theorem)
Chapter 7 Confidence Intervals

7. Chapter 7 Confidence Intervals
Recall the empirical rule, so…
68.26% of all possible sample means are within one standard deviation of the population mean
95.44% of all possible sample means are within two standard deviations of the population mean
99.73% of all possible sample means are within three standard deviations of the population mean
Chapter 7 Confidence Intervals

8. Chapter 7 Confidence Intervals
The Car Mileage Case
Example 7.1
Recall that the population of car mileages is normally distributed with mean m. We assume that the standard deviation is known : s = 0.8 mpg
Note that m is unknown and is to be estimated
Taking samples of size n = 5, the sampling distribution of sample mean mileages is normal with mean m = m (which is also unknown) and standard deviation
The probability is that will be within plus or minus 2s = 2 • = of m
Chapter 7 Confidence Intervals

9. Chapter 7 Confidence Intervals
The Car Mileage Case ＃2
Then there is a probability that will be a value so that interval [ ± ] contains m
In other words P( – ≤ m ≤ ) =
The interval [ ± ] is referred to as the 95.44% confidence interval for m.
We take three samples with means = 31.3, , and 32.5, respectively. Suppose actually m=31.5.
Chapter 7 Confidence Intervals

10. Chapter 7 Confidence Intervals
The Car Mileage Case ＃3
95.44% Confidence Intervals for m
Three intervals shown
Two contain m
One does not
Chapter 7 Confidence Intervals

11. Chapter 7 Confidence Intervals
According to the 95.44% confidence interval, we know even before we sample that of all the possible samples that could be selected …
… There is 95.44% probability that the sample mean that is calculated is such that the interval [ ± ] will contain the actual (but unknown) population mean m
In other words, of all possible sample means, 95.44% of all the corresponding intervals will contain the population mean m
Note that there is a 4.56% probability that the interval does not contain m
The sample mean is either too high or too low
Chapter 7 Confidence Intervals

12. Chapter 7 Confidence Intervals
Generalizing
In the example, we found the probability that m is contained in an interval of integer multiples of s
More usual to specify the (integer) probability and find the corresponding number of s
The probability that the confidence interval will not contain the population mean m is denoted by a
In the example, a =
Chapter 7 Confidence Intervals

13. Generalizing Continued
The probability that the confidence interval will contain the population mean m is denoted by 1 - a
1 – a is referred to as the confidence coefficient
(1 – a)  100% is called the confidence level
In the example, 1 – a =
Usual to use two decimal point probabilities for 1 – a
Here, focus on 1 – a = 0.95 or 0.99
Chapter 7 Confidence Intervals

14. General Confidence Interval
In general, the probability is 1 – a that the population mean m is contained in the interval
The normal point za/2 gives a right hand tail area under the standard normal curve equal to a/2
The normal point - za/2 gives a left hand tail area under the standard normal curve equal to a/2
The area under the standard normal curve between -za/2 and za/2 is 1 – a
Chapter 7 Confidence Intervals

15. Chapter 7 Confidence Intervals

16. z-Based Confidence Intervals for a Mean with s Known
If a population has standard deviation s (known),
and if the population is normal or if sample size is large (n  30), then …
… a (1-a)100% confidence interval for m is
Chapter 7 Confidence Intervals

17. Chapter 7 Confidence Intervals
95% Confidence Level
For a 95% confidence level,
1 – a = 0.95
a = 0.05
a/2 = 0.025
For 95% confidence, need the normal point z0.025
The area under the standard normal curve between -z0.025 and z0.025 is 0.95
Then the area under the standard normal curve between 0 and z0.025 is 0.475
From the standard normal table, the area is for z = 1.96
Then z0.025 = 1.96
Chapter 7 Confidence Intervals

18. The Effect of a on Confidence Interval Width
za/2 = z0.005 = 2.575
za/2 = z0.025 = 1.96
Chapter 7 Confidence Intervals

19. Chapter 7 Confidence Intervals
The 95% confidence interval is
Chapter 7 Confidence Intervals

20. Chapter 7 Confidence Intervals
For 99% confidence, need the normal point z0.005
Reading between table entries in the standard normal table, the area is for z0.005 = 2.575
The 99% confidence interval is
Chapter 7 Confidence Intervals

21. Chapter 7 Confidence Intervals
The Car Mileage Case
Example 7.2
Given: = mpg
s = 0.8 mpg
n = 49
95% Confidence Interval:
99% Confidence Interval:
Chapter 7 Confidence Intervals

22. Chapter 7 Confidence Intervals
The 99% confidence interval is slightly wider than the 95% confidence interval
The higher the confidence level, the wider the interval
Reasoning from the intervals:
The target mean mileage should be at least 31 mpg
Both confidence intervals exceed this target
According to the 95% confidence interval, we can be 95% confident that the mileage is between and mpg
So we can be 95% confident that, on average, the mean mileage exceeds the target by at least 0.33 mpg and at most 0.78 mpg
Chapter 7 Confidence Intervals

23. Section 7.2 t-Based Confidence Intervals for a Population Mean
If s is unknown (which is usually the case), we can construct a confidence interval for m based on the sampling distribution of
If the population is normal, then for any sample size n, this sampling distribution is called the t distribution
(t分布)
Chapter 7 Confidence Intervals

24. The t Distribution(t 分布)
The curve of the t distribution is similar to that of the standard normal curve
Symmetrical and bell-shaped
The t distribution is more spread out than the standard normal distribution
The spread of the t is given by the number of degrees of freedom(自由度)
Denoted by df
For a sample of size n, there are one fewer degrees of freedom, that is,
df = n – 1
Chapter 7 Confidence Intervals

25. Degrees of Freedom and the t-Distribution
As the number of degrees of freedom increases, the spread of the t distribution decreases and the t curve approaches the standard normal curve
Chapter 7 Confidence Intervals

26. The t Distribution and Degrees of Freedom
For a t distribution with n – 1 degrees of freedom,
As the sample size n increases, the degrees of freedom also increases
As the degrees of freedom increase, the spread of the t curve decreases
As the degrees of freedom increases indefinitely, the t curve approaches the standard normal curve
If n ≥ 30, so df = n – 1 ≥ 29, the t curve is very similar to the standard normal curve
Chapter 7 Confidence Intervals

27. t and Right Hand Tail Areas
Use a t point denoted by ta
ta is the point on the horizontal axis under the t curve that gives a right hand tail equal to a
So the value of ta in a particular situation depends on the right hand tail area a and the number of degrees of freedom
df = n – 1
a = 1 – a , where 1 – a is the specified confidence coefficient
Chapter 7 Confidence Intervals

28. Chapter 7 Confidence Intervals

29. Using the t Distribution Table
Rows correspond to the different values of df
Columns correspond to different values of a
See Table 7.3, Tables A.4 and A.20 in Appendix A and the table on the inside cover
Table 7.3 and A.4 gives t points for df 1 to 30, then for df = 40, 60, 120, and ∞
On the row for ∞, the t points are the z points
Table A.20 gives t points for df from 1 to 100
For df greater than 100, t points can be approximated by the corresponding z points on the bottom row for df = ∞
Always look at the accompanying figure for guidance on how to use the table
Chapter 7 Confidence Intervals

30. Chapter 7 Confidence Intervals
Example 7.3
Find ta for a sample of size n = 15 and right hand tail area of 0.025
For n = 15, df = 14
a = 0.025
Note that a = corresponds to a confidence level of 0.95
In Table 7.3, along row labeled 14 and under column labeled 0.025, read a table entry of 2.145
So ta = 2.145
Chapter 7 Confidence Intervals

31. t-Based Confidence Intervals for a Mean: s Unknown
If the sampled population is normally distributed with mean , then a (1-a)100% confidence interval for m is
ta/2 is the t point giving a right-hand tail area of /2 under the t curve having n – 1 degrees of freedom
Chapter 7 Confidence Intervals

32. Chapter 7 Confidence Intervals
Debt-to-Equity Ratio
Example 7.4
Estimate the mean debt-to-equity ratio of the loan portfolio of a bank
Select a random sample of 15 commercial loan accounts
Box plot is given in figure below
Know: = 1.34
s = 0.192
n = 15
Want a 95% confidence interval for the ratio
Assume all ratios are normally distributed but s unknown
Chapter 7 Confidence Intervals

33. Chapter 7 Confidence Intervals
Have to use the t distribution
At 95% confidence,
1 – a = 0.95 so a = 0.05 and a/2 = 0.025
For n = 15,
df = 15 – 1 = 14
Use the t table to find ta/2 for df = 14
ta/2 = t0.025 = for df = 14
The 95% confidence interval:
Chapter 7 Confidence Intervals

34. Section 7.3 Sample Size Determination
If s is known, then a sample of size
Letting E denote the desired margin of error, so that is within E units of , with 100(1-)% confidence.
Chapter 7 Confidence Intervals

35. Chapter 7 Confidence Intervals
If s is unknown and is estimated from s, then a sample of size
so that is within E units of , with 100(1-)% confidence. The number of degrees of freedom for the ta/2 point is the size of the preliminary sample minus 1
Chapter 7 Confidence Intervals

36. Chapter 7 Confidence Intervals
Car Mileage Case
Example 7.5
We wish to find the sample size that is needed to make the margin of error in a 95% confidence interval for mean equal to 0.3.
We regard the previously discussed sample of five mileages as a preliminary sample, and calculated the s= from 5 sample cars.
We find that the appropriate sample size is:
Rounding up, we employ a sample of size 50.
Chapter 7 Confidence Intervals

37. Chapter 7 Confidence Intervals
Car Mileage Case
Example 7.5
Given: = mpg
s = mpg
n = 49
t- based 95% Confidence Interval:
where ta/2 = t0.025 = for df = 49 – 1 = 48 degrees of freedom
Note: the error bound B = mpg, within the maximum error of 0.3 mpg
Chapter 7 Confidence Intervals

38. Section 7.4 Confidence Intervals for a Population Proportion
If the sample size n is large*, then a (1-a)100% confidence interval for p is
* Here n should be considered large if both
Chapter 7 Confidence Intervals

39. Chapter 7 Confidence Intervals
Phe-Mycin Side Effects
Example 7.6
Given: n = 200, 35 patients experience nausea.
Note:
so both quantities are > 5
For 95% confidence, za/2 = z0.025 = 1.96 and
Chapter 7 Confidence Intervals

40. Determining Sample Size for Confidence Interval for p
A sample size
will yield an estimate , precisely within E units of p, with 100(1-)% confidence
Note that the formula requires a preliminary estimate of p. The conservative value of p = 0.5 is generally used when there is no prior information on p
Chapter 7 Confidence Intervals

41. A Comparison of Confidence Intervals and Tolerance Intervals
A tolerance interval contains a specified percentage of individual population measurements
Often 68.26%, 95.44%, 99.73%
A confidence interval is an interval that contains the population mean m, and the confidence level expresses how sure we are that this interval contains m
Often confidence level is set high (e.g., 95% or 99%)
Because such a level is considered high enough to provide convincing evidence about the value of m
Chapter 7 Confidence Intervals

42. Chapter 7 Confidence Intervals
Car Mileage Case
Example 7.7
Tolerance intervals shown:
[ ± s] contains 68% of all individual cars
[ ± 2s] contains 95.44% of all individual cars
[ ± 3s] contains 99.73% of all individual cars
The t-based 95% confidence interval for m is [31.32, 31.78], so we can be 95% confident that m lies between and mpg
Chapter 7 Confidence Intervals

43. Chapter 7 Confidence Intervals
Summary: Selecting an Appropriate Confidence Interval for a Population Mean
Chapter 7 Confidence Intervals