1. Sample Means & Proportions
Week 7
Sample Means & Proportions

2. Variability of Summary Statistics
Variability in shape of distn of sample
Variability in summary statistics
Mean, median, st devn, upper quartile, …
Summary statistics have distributions

3. Parameters and statistics
Parameter describes underlying population
Constant
Greek letter (e.g. , , , …)
Unknown value in practice
Summary statistic
Random
Roman letter (e.g. m, s, p, …)
We hope statistic will tell us about corresponding parameter

4. Distn of sample vs Sampling distn of statistic
Values in a single random sample have a distribution
Single sample --> single value for statistic
Sample-to-sample variability of statistic is its sampling distribution.

5. Means Unknown population mean, 
Sample mean, X, has a distribution — its sampling distribution.
Usually x ≠ 
A single sample mean, x, gives us information about 

6. Sampling distribution of mean
If sample size, n, increases:
Spread of distn of sample is (approx) same.
Spread of sampling distn of mean gets smaller.
x is likely to be closer to 
x becomes a better estimate of 

7. Sampling distribution of mean
Population with mean , st devn 
Random sample (n independent values)
Sample mean, X, has sampling distn with:
Mean,
St devn,
(We will deal later with the problem that  and  are unknown in practice.)

8. Weight loss
Estimate mean weight loss for those attending clinic for 10 weeks
Random sample of n = 25 people
Sample mean, x
How accurate?
Let’s see, if the population distn of weight loss is:

9. Some samples Four random samples of n = 25 people:
Mean = 8.32 pounds, st devn = 4.74 pounds
Mean = 8.48 pounds, st devn = 5.27 pounds
Mean = 7.16 pounds, st devn = 5.93 pounds
N.B. In all samples, x ≠ 

10. Sampling distribution
Means from simulation of 400 samples
Theory:
mean =  = 8 lb, s.d.( ) = lb
(How does this compare to simulation? To popn distn?)

11. Errors in estimation From 70-95-100 rule Even if we didn’t know 
Population
Sampling distribution of mean
mean =  = 8 lb, s.d.( ) = lb
From rule
x will be almost certainly within 8 ± 3 lb
x is unlikely to be more than 3 lb in error
Even if we didn’t know 

12. Increasing sample size, n
If we sample n = 100 people instead of 25:
s.d.( ) = lb.
Larger samples  more accurate estimates

13. Central Limit Theorem If population is normal (, )
If popn is non-normal with (, ) but n is large
Guideline: n > 30 even if very non-normal

14. Other summary statistics
E.g. Lower quartile, proportion, correlation
Usually not normal distns
Formula for standard devn of samling distn sometimes
Sampling distn usually close to normal if n is large

15. Lottery problem Pennsylvania Cash 5 lottery
5 numbers selected from 1-39
Pick birthdays of family members (none 32-39)
P(highest selected is 32 or over)?
Statistic:
H = highest of 5 random numbers (without replacement)

16. Lottery simulation Theory? Fairly hard.
Simulation: Generated 5 numbers (without replacement) 1560 times
Highest number > 31 in about 72% of repetitions

17. Normal distributions Family of distributions (populations)
Shape depends only on parameters  (mean) &  (st devn)
All have same symmetric ‘bell shape’
= 65 inches,
s = 2.7 inches

18. Importance of normal distn
A reasonable model for many data sets
Transformed data often approx normal
Sample means (and many other statistics) are approx normal.

19. Standard normal distribution
Z ~ Normal ( = 0,  = 1) -3 -2 -1 1 2 3
Prob ( Z < z* )

20. Probabilities for normal (0, 1)
P(Z  -3.00) =
P(Z  −2.59) =
P(Z  1.31) =
P(Z  2.00) =
P(Z  -4.75) =
0.0013
Check from tables:

21. Probability Z > 1.31 P(Z > 1.31) = 1 – P(Z  1.31)
= 1 – = .0951

22. Prob ( Z between –2.59 and 1.31) P(-2.59  Z  1.31)
= P(Z  1.31) – P(Z  -2.59)
= – = .9001

23. Standard devns from mean
Normal (, )    
Heights of students
= 65 inches,
s = 2.7 inches

24. Probability and area X ~ normal ( = 65 , s = 2.7 )
P (X ≤ 67.7) = area

25. Probability and area (cont.)
Normal (, )    
Exactly rule
P(X within  of ) = approx 70%
P(X within 2 of ) = approx 95%
P(X within 3 of ) = approx 100%

26. Finding approx probabilities
Ht of college woman, X ~ normal ( = 65 , s = 2.7 )
Prob (X ≤ 62 )?
Sketch normal density
Estimate area
P (X ≤ 62) = area
About 1/8

27. Translate question from X to Z
X ~ Normal (, )
Find P(X ≤ x*)    x* 
Translate to z-score:
Z ~ Normal ( = 0,  = 1) -3 -2 z* -1 1 2 3

28. Finding probabilities
Prob (height of randomly selected college woman ≤ 62 )?
About 13%.

29. Prob (X > value)
Ht of college woman, X ~ normal ( = 65 , s = 2.7 )
Prob (X > 68 inches)?

30. Finding upper quartile
Blood Pressures are normal with mean 120 and standard deviation 10. What is the 75th percentile?
Step 1: Solve for z-score
Closest z* with area of (tables)
z = 0.67
Step 2: Calculate x = z*s + m
x = (0.67)(10) = or about 127.

31. Probabilities about means
Blood pressure ~ normal ( = 120,  = 10)
8 people given drug
If drug does not affect blood pressure,
Find P(average blood pressure > 130)

32. P ( X > 130) ? X ~ normal ( = 120,  = 10) n = 8 prob = 0.0023
Very little chance!

33. Distribution of sum   X ~ distn with (, ) aX ~ distn with (a, a)
e.g. miles to kilometers 
Central Limit Theorem implies approx normal

34. Probabilities about sum
Profit in 1 day ~ normal (= $300, = $200)
Prob(total profit in week < $1,000)?
Total =
Prob =
Assumes
independence

35. Categorical data Most important parameter is  = Prob (success)
Corresponding summary statistic is
p = Proportion (success) ^
N.B. Textbook uses p and p

36. Number of successes
Easiest to deal with count of successes before proportion.
If…
1. n “trials” (fixed beforehand).
2. Only “success” or “failure” possible for each trial.
3. Outcomes are independent.
Prob (success), remains same for all trials, .
Prob (failure) is 1 – .
X = number of successes ~ binomial (n, )

37. Examples

38. Binomial Probabilities
for k = 0, 1, 2, …, n
You won’t need to use this!!
Prob (win game) = 0.2
Plays of game are independent.
What is Prob (wins 2 out of 3 games)?
What is P(X = 2)?

39. Mean & st devn of Binomial
For a binomial (n, )

40. Extraterrestrial Life?
50% of large population would say “yes” if asked, “Do you believe there is extraterrestrial life?”
Sample of n = 100
X = # “yes” ~ binomial (n = 100,  = 0.5)

41. Extraterrestrial Life?
Sample of n = 100
X = # “yes” ~ binomial (n = 100,  = 0.5)
rule of thumb for # “yes”
About 95% chance of between 40 & 60
Almost certainly between 35 & 65

42. Normal approx to binomial
If X is binomial (n , ), and n is large, then X is also approximately normal, with
Conditions: Both n and n(1 – ) are at least 10.
(Justified by Central Limit Theorem)

43. Number of H in 30 Flips
X = # heads in n = 30 flips of fair coin X ~ binomial ( n = 30, = 0.5)
Bell-shaped & approx normal.

44. Opinion poll n = 500 adults; 240 agreed with statement
If  = 0.5 of all adults agree, what P(X ≤ 240) ?
X is approx normal with
Not unlikely to see 48% or less, even if 50% in population agree.

45. Sample Proportion
Suppose (unknown to us) 40% of a population carry the gene for a disease, ( = 0.40).
Random sample of 25 people; X = # with gene.
X ~ binomial (n = 25 ,  = 0.4)
p = proportion with gene

46. Distn of sample proportion
X ~ binomial (n , )
Large n:
p is approx normal
(n ≥ 10 & n (1 – ) ≥ 10)

47. Examples
Election Polls: to estimate proportion who favor a candidate; units = all voters.
Television Ratings: to estimate proportion of households watching TV program; units = all households with TV.
Consumer Preferences: to estimate proportion of consumers who prefer new recipe compared with old; units = all consumers.
Testing ESP: to estimate probability a person can successfully guess which of 5 symbols on a hidden card; repeatable situation = a guess.

48. Public opinion poll Suppose 40% of all voters favor Candidate A.
Pollsters sample n = 2400 voters.
Propn voting for A is approx normal
Simulation 400 times & theory.

49. Probability from normal approx
If 40% of voters favor Candidate A, and n = 2400 sampled
Sample proportion, p, is almost certain to be between 0.37 and 0.43
Prob 0.95 of p being between 0.38 and 0.42