Sampling Distributions for Proportions

1
Sampling Distributions for Proportions

• Allow us to work with the proportion of successes
  rather than the actual number of successes in
  binomial experiments.

2
Sampling Distribution of the Proportion

• n number of binomial trials
• r number of successes
• p probability of success on each trial
• q 1 - p probability of failure on each trial

3
Sampling Distribution of the Proportion

• If np gt 5 and nq gt 5 then p-hat r/n can be
  approximated by a normal random variable (x)
  with

4
The Standard Error for
5
Continuity Correction

• When using the normal distribution (which is
  continuous) to approximate p-hat, a discrete
  distribution, always use the continuity
  correction.
• Add or subtract 0.5/n to the endpoints of a
  (discrete) p-hat interval to convert it to a
  (continuous) normal interval.

6
Continuity Correction

• Since n 20,
• .5/n 0.025
• 5/8 - 0.025 0.6
• 6/8 0.025 0.775
• Required x interval is 0.6 to 0.775

• If n 20, convert a p-hat interval from 5/8 to
  6/8 to a normal interval.
• Note 5/8 0.625
• 6/8 0.75
• So p-hat interval is 0.625 to 0.75.

7
Suppose 12 of the population is in favor of a
new park.

• Two hundred citizen are surveyed.
• What is the probability that between10 and 15
  of them will be in favor of the new park?

8
Is it appropriate to the normal distribution?

• 12 of the population is in favor of a new park.
  
• p 0.12, q 0.88
• Two hundred citizen are surveyed.
• n 200
• Both np and nq are greater than five.

9
Find the mean and the standard deviation
10
What is the probability that between 10 and
15of them will be in favor of the new park?

• Use the continuity correction
• Since n 200, .5/n .0025
• The interval for p-hat (0.10 to 0.15) converts to
  0.0975 to 0.1525.

11
Calculate z-score for x 0.0975
12
Calculate z-score for x 0.1525
13
P(-0.98 lt z lt 1.41)

• 0.9207 -- 0.1635 0.7572
• There is about a 75.7 chance that between 10
  and 15 of the citizens surveyed will be in favor
  of the park.

14
Control Chart for Proportions

• P-Chart

15
Constructing a P-Chart

• Select samples of fixed size n at regular
  intervals.
• Count the number of successes r from the n
  trials.
• Use the normal approximation for r/n to plot
  control limits.
• Interpret results.

16
Determining Control Limits for a P-Chart

• Suppose employee absences are to be plotted.
• In a daily sample of 50 employees, the number of
  employees absent is recorded.
• p/n for each day number absent/50.For the
  random variable p-hat p/n, we can find the mean
  and the standard deviation.

17
Finding the mean and the standard deviation
18
Is it appropriate to use the normal distribution?

• The mean of p-hat p 0.12
• The value of n 50.
• The value of q 1 - p 0.88.
• Both np and nq are greater than five.
• The normal distribution will be a good
  approximation of the p-hat distribution.

19
Control Limits

• Control limits are placed at two and three
  standard deviations above and below the mean.

20
Control Limits
The center line is at 0.12. Control limits are
placed at -0.018, 0.028, 0.212, and 0.258.
21
Control Chart for Proportions
Employee Absences 0.3 3s 0.258
0.2 2s 0.212 0.1 mean 0.12
0.0 -2s 0.028 -0.1 -3s
-0.018