Download Sampling Distribution of a Sample Proportion The sampling

Sampling Distribution of a Sample Proportion
The sampling distribution of a sample proportion p̂ is described when all possible simple random samples
of size n are taken from a population where the probability of an individual success is p.
The mean of the sample proportion p̂ is  p̂  p .
The standard deviation of the sample proportion p̂ is  pˆ 
p(1  p)
n
Example: In a large high school of 2,500 students, 21% of them are seniors. A sample random sample of 150
students is taken and the proportion of seniors is calculated. What are the mean and standard deviation of the
sample proportion, p̂ ?
Note: The value of  p̂ assumes the probability of a success remains constant throughout the sampling process.
When sampling from a population without replacement, this is not the case, but the formula is reasonably
accurate when the sample size is no more than 10% of the population size. (this is VERY important when
doing inference tests) In the previous example above, the sample size of 150 is clearly less than 10% of 2,500.
Example: A tossed tack lands “point up” with probability of 0.42. If the tack is tossed 50 times, what are the
man and standard deviation of p̂ , the proportion of times the tack lands “point up”?
The distribution of sample proportions is closely related to the binomial distribution. In fact, the mean and
standard deviation of the sample proportion, p̂ , are simply the mean and standard deviation of the binomial
random variable X when divided by the sample size of n. Because of this, the sampling distribution of sample
proportions can be modeled with a normal distribution like the binomial distribution if
np  10 and n(1  p )  10.
Example: In a large high school of 2,500 students, 21% of them are seniors. A simple random sample of 150
students is taken and the proportion of seniors is calculated. What is the probability that the sample will contain
less than 15% seniors?
Check that the binomial conditions are met:
You must actually show the calculations of
np  10 and n(1  p )  10. If these are met, draw the normal curve with values and continue on with the
calculations.
Example: A tossed tack lands “point up” with probability of 0.42. If the tack is tossed 50 times, what
proportions of successes correspond to the highest 10% of all possible outcomes? (use some information from
the previous “tack” example to draw the normal curve if the conditions are met.