Sampling Distributions for Proportions Allow us to work with the proportion of successes rather than the actual number of successes in binomial experiments.

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

Sampling Distribution of the Proportion If np > 5 and nq > 5 then p-hat = r/n can be approximated by a normal random variable (x) with:

The Standard Error for

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.

Continuity Correction If n = 20, convert a p-hat interval from 5/8 to 6/8 to a normal interval. Note: 5/8 = /8 = 0.75 So p-hat interval is to Since n = 20,.5/n = / = 0.6 6/ = Required x interval is 0.6 to 0.775

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?

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. Is it appropriate to the normal distribution?

Find the mean and the standard deviation

What is the probability that between 10 % and 15%of 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 to

Calculate z-score for x =

Calculate z-score for x =

P(-0.98 < z < 1.41) = There is about a 75.7% chance that between 10% and 15% of the citizens surveyed will be in favor of the park.

Control Chart for Proportions P-Chart

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.

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.

Finding the mean and the standard deviation

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 = Both np and nq are greater than five. The normal distribution will be a good approximation of the p-hat distribution.

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

Control Limits The center line is at Control limits are placed at , 0.028, 0.212, and

Control Chart for Proportions Employee Absences s = s = mean = s = s =