1. Chapter 8
Confidence Intervals

2. Confidence Intervals
8.1 z-Based Confidence Intervals for a Population Mean: σ Known
8.2 t-Based Confidence Intervals for a Population Mean: σ Unknown
8.3 Sample Size Determination
8.4 Confidence Intervals for a Population Proportion
8.5 Confidence Intervals for Parameters of Finite Populations (Optional)
8.6 A Comparison of Confidence Intervals and Tolerance Intervals (Optional)

3. z-Based Confidence Intervals for a Mean: σ Known
The starting point is the sampling distribution of the sample mean
Recall that if a population is normally distributed with mean m and standard deviation σ, then the sampling distribution of x is normal with mean mx = m and standard deviation
Use a normal curve as a model of the sampling distribution of the sample mean
Exactly, because the population is normal
Approximately, by the Central Limit Theorem for large samples

4. The Empirical Rule
68.26% of all possible sample means are within one standard deviation of the population mean
95.44% of all possible observed values of x are within two standard deviations of the population mean
99.73% of all possible observed values of x are within three standard deviations of the population mean

5. Example 8.1: The Car Mileage Case
Assume a sample size (n) of 5
Assume the population of all individual car mileages is normally distributed
Assume the population standard deviation (σ) is 0.8
The probability that x with be within ±0.7 of µ is

6. Example 8.1 The Car Mileage Case Continued
Assume the sample mean is 31.3
That gives us an interval of [31.3 ± 0.7] = [30.6, 32.0]
The probability is that the interval [x ± 2σ] contains the population mean µ

7. Example 8.1: The Car Mileage Case #3
That the sample mean is within ± of µ is equivalent to…
x will be such that the interval [x ± ] contains µ
Then there is a probability that x will be a value so that interval [x ± ] contains µ
In other words P(x – ≤ µ ≤ x ) =
The interval [x ± ] is referred to as the 95.44% confidence interval for µ

8. Example 8.1: The Car Mileage Case #4
Three 95.44% Confidence Intervals for m
Thee intervals shown
Two contain µ
One does not

9. Example 8.1: The Car Mileage Case #5
According to the 95.44% confidence interval, we know before we sample that of all the possible samples that could be selected …
There is 95.44% probability the sample mean is such the interval [x ± ] will contain the actual (but unknown) population mean µ
In other words, of all possible sample means, 95.44% of all the resulting intervals will contain the population mean µ
Note that there is a 4.56% probability that the interval does not contain µ
The sample mean is either too high or too low

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

11. 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
Usual to use two decimal point probabilities for 1 – a
Here, focus on 1 – a = 0.95 or 0.99

12. 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

13. Sampling Distribution Of All Possible Sample Means

14. z-Based Confidence Intervals for a Mean with σ 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

15. 95% Confidence Level
For a 95% confidence level, 1 – a = 0.95, so a = 0.05, and a/2 = 0.025
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

16. 95% Confidence Interval
The 95% confidence interval is

17. 99% Confidence Interval
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

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

19. Example 8.2: Car Mileage Case
Given
x = 31.56
σ = 0.8
n = 50
Will calculate
95 Percent confidence interval
99 Percent confidence interval

20. Example 8.2: 95 Percent Confidence Interval

21. Example 8.2: 99 Percent Confidence Interval

22. Notes on the Example
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 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
We can be 95% confident that, on average, the mean exceeds the target by at least 0.33 and at most 0.78 mpg

23. t-Based Confidence Intervals for a Mean: s Unknown
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

24. The t Distribution
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
Denoted by df
For a sample of size n, there are one fewer degrees of freedom, that is,
df = n – 1

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

26. The t Distribution and 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

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

28. t and Right Hand Tail Areas

29. Using the t Distribution Table
Rows correspond to the different values of df
Columns correspond to different values of a
See Table 8.3, Tables A.4 and A.20 in Appendix A and the table on the inside cover
Table 8.3 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

30. Using the t Distribution
Example: Find t for a sample of size n=15 and right hand tail area of 0.025
For n = 15, df = 14
 = 0.025
Note that a = corresponds to a confidence level of 0.95
In Table 8.3, along row labeled 14 and under column labeled 0.025, read a table entry of 2.145
So t = 2.145

31. Using the t Distribution Continued

32. t-Based Confidence Intervals for a Mean: s Unknown
If the sampled population is normally distributed with mean , then a (1­a)100% confidence interval for  is
ta/2 is the t point giving a right-hand tail area of /2 under the t curve having n­1 degrees of freedom

33. Example 8.4 Debt-to-Equity Ratios
Estimate the mean debt-to-equity ratio of the loan portfolio of a bank
Select a random sample of 15 commercial loan accounts
x = 1.34
s = 0.192
n = 15
Want a 95% confidence interval for the ratio
Assume all ratios are normally distributed but σ unknown

34. Example 8.4 Debt-to-Equity Ratios Continued
Have to use the t distribution
At 95% confidence, 1 –  = 0.95 so  = 0.05 and /2 = 0.025
For n = 15, df = 15 – 1 = 14
Use the t table to find t/2 for df = 14, t/2 = t0.025 = 2.145
The 95% confidence interval:

35. Sample Size Determination (z)
If s is known, then a sample of size so that x is within B units of , with 100(1-)% confidence

36. Sample Size Determination (t)
If σ is unknown and is estimated from s, then a sample of size so that x is within B 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

37. Example 8.6: The Car Mileage Case
What sample size is needed to make the margin of error for a 95 percent confidence interval equal to 0.3?
Assume we do not know σ
Take prior example as preliminary sample
s =
z/2 = z0.025 = 1.96
t/2 = t0.025 = based on n-1 = 4 d.f.

38. Example 8.6: The Car Mileage Case Continued

39. Example 8.7: The Car Mileage Case
Want to see that the sample of 50 mileages produced a 95 percent confidence interval with a margin or error of ±0.3
The margin of error is 0.227, which is smaller than the 0.3 desired

40. Confidence Intervals for a Population Proportion
If the sample size n is large, then a (1­a)100% confidence interval for ρ is
Here, n should be considered large if both
n · p̂ ≥ 5
n · (1 – p̂) ≥ 5

41. Example 8.8: The Cheese Spread Case
Would like a confidence interval for ρ, the proportion of purchases who would stop buying if new spout used
Wish to see if n=1,000 is large enough
Point estimate p̂ is 0.63
np̂ = 1,000(0.063) = 63 > 5
n(1-p̂) = 1,000(0.937) = 937 > 5

42. Example 8.8: The Cheese Spread Case Continued
Upper limit is less than 0.10
Strong evidence that true proportion ρ is less than 0.10

43. Determining Sample Size for Confidence Interval for ρ
A sample size given by the formula… will yield an estimate p̂, precisely within B units of ρ, 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.

44. Confidence Intervals for Population Mean and Total for a Finite Population
For a large (n ≥ 30) random sample of measurements selected without replacement from a population of size N, a (1- )100% confidence interval for μ is
A (1- )100% confidence interval for the population total is found by multiplying the lower and upper limits of the corresponding interval for μ by N

45. Confidence Intervals for Proportion and Total for a Finite Population
For a large random sample of measurements selected without replacement from a population of size N, a (1- )100% confidence interval for ρ is
A (1- )100% confidence interval for the total number of units in a category is found by multiplying the lower and upper limits of the corresponding interval for ρ by N

46. 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 μ, and the confidence level expresses how sure we are that this interval contains μ
Often confidence level is set high (e.g., 95% or 99%)
Such a level is considered high enough to provide convincing evidence about the value of μ

47. Example 8.15 Car Mileage Case
Tolerance intervals shown:
[x ± s] contains 68% of all individual cars
[x ± 2s] contains 95.44% of all individual cars
[x ± 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

48. Selecting an Appropriate Confidence Interval for a Population Mean