1. Mathematics & Statistics
Topic 7
Estimation

2. Topic Goals After completing this topic, you should be able to:
Distinguish between a point estimate and a confidence interval estimate
Construct and interpret a confidence interval estimate for a single population mean using both the Z and t distributions
Form and interpret a confidence interval estimate for a single population proportion
Form and interpret a confidence interval estimate for a single population variance

3. Estimation
Last week we looked at the distribution of sample statistics...
...given the value of a population parameter.
In real-life situations we don’t know the true value of the population parameter...
...so the question is:
Can we say something about the value of the population parameter...
...given an observed value of a sample statistic?

4. Confidence Intervals Content of this topic
Confidence Intervals for the Population Mean, μ
when Population Variance σ2 is Known (Section 8.2)
when Population Variance σ2 is Unknown (Section 8.3)
Confidence Intervals for the Population Proportion, p (large samples) (Section 8.4)
Confidence Intervals for the Population Variance, σ2 (Section 9.4)

5. Definitions An estimator of a population parameter is
a random variable that depends on sample information . . .
whose value provides an approximation to this unknown parameter
A specific value of that random variable is called an estimate

6. Point Estimates μ x p Mean Proportion s2 Variance σ2 We can estimate a
Population Parameter …
with a Sample Statistic
(a Point Estimate) μ x
Mean
Proportion p
Variance s2
Variance σ2

7. Unbiasedness
A point estimator is said to be an unbiased estimator of the parameter  if the expected value, or mean, of the sampling distribution of is ,
Examples:
The sample mean is an unbiased estimator of μ
The sample variance is an unbiased estimator of σ2
The sample proportion is an unbiased estimator of p

8. Unbiasedness
(continued)
is an unbiased estimator, is biased:

9. Most Efficient Estimator
Suppose there are several unbiased estimators of 
The most efficient estimator or the minimum variance unbiased estimator of  is the unbiased estimator with the smallest variance
Let and be two unbiased estimators of , based on the same number of sample observations. Then,
is said to be more efficient than if
The relative efficiency of with respect to is the ratio of their variances:

10. Point and Interval Estimates
A point estimate is a single number,
a confidence interval provides additional information about variability
Upper
Confidence
Limit
Lower
Confidence
Limit
Point Estimate
Width of
confidence interval

11. Confidence Intervals
How much uncertainty is associated with a point estimate of a population parameter?
An interval estimate provides more information about a population characteristic than does a point estimate
Such interval estimates are called confidence intervals

12. Confidence Interval Estimate
An interval gives a range of values:
Takes into consideration variation in sample statistics from sample to sample
Based on observation from 1 sample
Gives information about closeness to unknown population parameters
Stated in terms of level of confidence
Can never be 100% confident

13. Confidence Interval and Confidence Level
If P(a <  < b) = 1 -  then the interval from a to b is called a 100(1 - )% confidence interval of .
The quantity (1 - ) is called the confidence level of the interval ( between 0 and 1)
In repeated samples of the population, the true value of the parameter  would be contained in 100(1 - )% of intervals calculated this way.
The confidence interval calculated in this manner is written as a <  < b with 100(1 - )% confidence

14. Estimation Process Random Sample Population
I am 95% confident that μ is between 40 & 60.
Random Sample
Population
Mean
X = 50
(mean, μ, is unknown)
Sample

15. Confidence Level, (1-) Suppose confidence level = 95%
(continued)
Suppose confidence level = 95%
Also written (1 - ) = 0.95
A relative frequency interpretation:
From repeated samples, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter
A specific interval either will contain or will not contain the true parameter
The procedure used leads to a correct interval in 95% of the time...
...but this does not guarantee anything about a particular sample.

16. Point Estimate ± (Reliability Factor)(Standard deviation)
General Formula
The general formula for all confidence intervals is:
The value of the reliability factor depends on the desired level of confidence
Point Estimate ± (Reliability Factor)(Standard deviation)

17. Confidence Intervals Confidence Intervals Population Population
Variance
Population
Mean
Population
Proportion
σ2 Unknown
σ2 Known

18. Confidence Interval for μ (σ2 Known)
Assumptions
Population variance σ2 is known
Population is normally distributed...
....or large sample so that CLT can be used.
Confidence interval estimate:
(where z/2 is the normal distribution value for a probability of /2 in each tail)

19. Example
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.
Determine a 95% confidence interval for the true mean resistance of the population.

20. Example
(continued)
A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is .35 ohms.
Solution:

21. Interpretation
We are 95% confident that the true mean resistance is between and ohms
Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean

22. Margin of Error The confidence interval, Can also be written as
where ME is called the margin of error
The interval width, w, is equal to twice the margin of error

23. Finding the Reliability Factor, z/2
Consider a 95% confidence interval:
z = -1.96
z = 1.96
Z units:
Lower
Confidence
Limit
Upper
Confidence
Limit
X units:
Point Estimate
Find z.025 = 1.96 from the standard normal distribution table

24. Common Levels of Confidence
Commonly used confidence levels are 90%, 95%, and 99%
Confidence Coefficient,
Confidence Level
Z/2 value
80%
90%
95%
98%
99%
99.8%
99.9%
.80
.90
.95
.98
.99
.998
.999
1.28
1.645
1.96
2.33
2.58
3.08
3.27

25. Intervals and Level of Confidence
Sampling Distribution of the Mean x
Intervals extend from to x1
100(1-)% of intervals constructed contain μ;
100()% do not. x2
Confidence Intervals

26. Summary:Finding a confidence interval for μ (σ known)
Choose confidence level 1-α (e.g. .95).
Find an interval [a,b] such that P(a<μ<b)=1-α.
a and b are determined by
How to find zα/2?
Look in table for value such that P(Z> zα/2)=α
e.g. if 1-α=.95, then zα/2 = 1.96.

27. Finding the Reliability Factor, z/2
Consider a 95% confidence interval:
z = -1.96
z = 1.96
Z units:
Lower
Confidence
Limit
Upper
Confidence
Limit
X units:
Point Estimate
Find z.025 = 1.96 from the standard normal distribution table

28. Example
Assume that the calorie contents per 100 ml of Guiness is normally distributed.
A sample of 11 pints has a mean calorie content per 100 ml of We know from past testing that the population standard deviation is 2.35.
Determine a 95% confidence interval for the true mean calorie content per 100 ml Guiness.

29. Example
(continued)
A sample of 11 pints from a large normal population has a mean calorie content per 100 ml of We know from past testing that the population standard deviation is 2.35 calories per 100 ml.
Solution:

30. Interpretation
We are 95% confident that the true calorie content per 100 ml is between and
Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean

31. Example
Suppose a second sample of 11 pints has a mean calorie content per 100 ml of 35.9.
A 95% confidence interval for this sample is:

32. Reducing the Margin of Error
The margin of error can be reduced if
the population standard deviation can be reduced (σ↓)
The sample size is increased (n↑)
The confidence level is decreased, (1 – ) ↓

33. How is the formula obtained?
Recall the formula for the confidence interval:
How is it obtained?
A 100(1-α)% confidence interval is an interval [a,b] such that P(a<μ<b) = 1-α.
We use the fact that

34. Derivation

35. Large Samples If the population is not normal....
....and the variance is not known....
The same confidence interval can still be used...
...if the sample is large.
For in that case the Central Limit Theorem tells us that the sample mean is approximately normal with mean μ and standard deviation σ/√n,...
...and s2 ≈ σ2.

36. Example
For a sample of 200 tea boxes you observe that the average weight is grams with a standard deviation of 2.78 grams.
Determine a 99% confidence interval for the population mean.
Solution. Note that the sample is large, so:
σ2 ≈ s2
the Central Limit Theorem says that the sample mean is approximately normal with mean μ and standard deviation σ/√n≈2.78/√200=.197

37. Example
So, we can proceed as if the sample mean is normal with known variance and apply
Therefore:

38. Confidence Intervals Confidence Intervals Population Population
Variance
Population
Mean
Population
Proportion
σ2 Unknown
σ2 Known

39. Confidence Interval for μ (σ2 Unknown)
If the population standard deviation σ is unknown, we can substitute the sample standard deviation, s
This introduces extra uncertainty, since s is variable from sample to sample
Therefore we use the t distribution instead of the normal distribution

40. Student’s t Distribution
Consider a random sample of n observations
with mean x and standard deviation s
from a normally distributed population with mean μ
Then the variable
follows the Student’s t distribution with (n - 1) degrees of freedom

41. Student’s t Distribution
The t is a family of distributions
The t-value depends on degrees of freedom (d.f.)
Number of observations that are free to vary after sample mean has been calculated
d.f. = n - 1

42. Student’s t Distribution
Note: t Z as n increases
Standard Normal
(t with df = ∞)
t (df = 13)
t-distributions are bell-shaped and symmetric, but have ‘fatter’ tails than the normal
t (df = 5) t

43. Confidence Interval for μ (σ Unknown)
(continued)
Assumptions
Population standard deviation is unknown
Population is normally distributed
Use Student’s t Distribution
Confidence Interval Estimate:
where tn-1,α/2 is the critical value of the t distribution with n-1 d.f. and an area of α/2 in each tail:

44. Student’s t Table .05 2 t /2 = .05 2.920 Upper Tail Area df .10 .025
Let: n = df = n - 1 =  = /2 =.05 df
.10
.05
.025 1
3.078
6.314
12.706 2
1.886
2.920
4.303
/2 = .05 3
1.638
2.353
3.182
The body of the table contains t values, not probabilities t
2.920

45. With comparison to the Z value
t distribution values
With comparison to the Z value
Confidence t t t Z
Level (10 d.f.) (20 d.f.) (30 d.f.) ____
Note: t Z as n increases

46. Example
A sample of 11 pints has a mean calorie content per 100 ml of 35.1, with a standard deviation of 2.35.
Determine a 95% confidence interval for the true mean calorie content per 100 ml Guiness if the calorie content is normal.

47. Example
(continued)
A sample of 11 pints from a large normal population has a mean calorie content per 100 ml of 35.1, with a standard deviation of 2.35 calories per 100 ml.
Solution:

48. Interpretation
We are 95% confident that the true calorie content per 100 ml is between and
Compare with the previous example (variance known)...
...where we were 95% confident that the true calorie content per 100 ml is between and
The interval is wider because there is additional uncertainty over σ.

49. About Student Student is the pseudonym of William Sealy Gosset...
...who worked for the Guiness brewery in Dublin...
...and needed procedures for quality control of the brewing process.
He derived the distribution of
which is still named after him

50. Confidence Intervals Confidence Intervals Population Population
Variance
Population
Mean
Population
Proportion
σ2 Known
σ2 Unknown

51. Confidence Intervals for the Population Proportion, p
An interval estimate for the population proportion ( p ) can be calculated by adding an allowance for uncertainty to the sample proportion ( )
Sample has to be large because we will use the normal approximation to the binomial

52. Confidence Intervals for the Population Proportion, p
(continued)
Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation
We will estimate this with sample data:

53. Confidence Interval Endpoints
Upper and lower confidence limits for the population proportion are calculated with the formula
where
z/2 is the standard normal value for the level of confidence desired
is the sample proportion
n is the sample size

54. Example A random sample of 100 people shows that 25 are left-handed.
Form a 95% confidence interval for the true proportion of left-handers

55. Example
(continued)
A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers.

56. Interpretation
We are 95% confident that the true percentage of left-handers in the population is between
16.51% and 33.49%.
Although the interval from to may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.

57. General Example
Trinity news, Issue 9, vol. 54 reports the results of a poll with 430 respondents on the SU president election. It is reported that: “Moore [has] the slightest lead of 31.7% to [...] Donohue’s 29.2% [...] and Reilly’s 27.2%. Given the margin of error the narrow gap [...] means any one [can win]”.
Compute a 95% confidence interval for the proportion of votes for Moore.
What do you think about the paper’s claim referring to the margin of error?

58. First question We have a large sample: n = 430
We are asked to compute a 95% confidence level: 1-α=.95 and – thus – α=.05.
The quantity of interest is a population proportion.
Let p denote the proportion of votes for Moore.
So, the appropriate confidence interval is

59. Obtain necessary info and apply
Sample proportion:
Use Table 1 or 8 to find
So:
We are 95% confident that the true poportion of votes
for Moore is between 27.3% and 36.1%

60. Second question We are 95% confident that the true poportion of votes
for Moore is between 27.3% and 36.1%
The score for Donohue is 29.2% and lies inside this
95% confidence interval, Reilly’s 27.2%, however, does not. So, given the chosen level of confidence and the resulting margin of error, the race could be deemed “too close to call” between Moore and Donohue.
NB The article does not specify the confidence level.
So, we don’t know what “margin of error” it refers to.
Bad statistical practise.

61. Probability Question
In a particular city, 20% of mobile phones are owned by people younger than 15. In addition, 52% of people with a mobile phone have a “Pay As You Talk” deal. Among those younger than 15, 12% have a “Pay As You Talk” deal.
What is the probability that a randomly chosen mobile phone is “Pay As You Talk” and belongs to someone younger than 15?

62. Set up the analysis Give the events of interest a name:
Let A be the event that the phone belongs to a person younger than 15.
Let B be the event that the phone is “Pay As You Talk”
What probabilties are given?
P(A) = .2
P(B) = .52
P(B|A) = .12
What are we asked?
Joint probability of A and B: P(A∩B)

63. Find Solution
We know that:
Therefore,

64. Exam Tips Move on to next question if stuck
Method more important than outcome
don’t worry too much about rounding (not too many digits)
computation errors have minor penalty
Write down all the steps you take
avoids mistakes
I can’t give marks if it’s not clear what you’re doing
Before you answer a question think about your “plan of attack” before you start writing.
Be relaxed when you walk in (no cramming).