Download A sampling distribution is a distribution in which each data value is a

A sampling distribution is a distribution in which each data value is a statistic for a sample from
a population.
In here, the statistic of the sampling distributions will be a mean or a proportion.
If time permits, the statistic will be a standard deviation.
To form a sampling distribution:
1. Create all samples of a specific size from the population.
The number of samples is NCn.
2. Calculate the desired statistic for each sample.
A sampling distribution is a population.
Why?
Example
Let the population be {1, 2, 3, 4, 5, 6, 7, 8}.
Find all the samples of size five.
How many samples are there?
a. Find the mean of each sample.
Sampling distribution of means.
b. Find the standard deviation of each sample.
Sampling distribution of standard deviations.
c. Assuming an even number represent flipping a head, find the proportion of heads for each
sample.
Sampling distribution of proportions.
Describing a sampling distribution
Histogram
Mean
Denoted μ# where # represents the statistic.
Standard deviation
Denoted σ# where # represents the statistic.
Example
a. Sampling distribution of means description. What is the mean and standard deviation of the
population?
b. Sampling distribution of standard deviations description.
c. Sampling distribution of proportions description. What is the mean and standard deviation of
the population?
Sampling Distribution of Means
Given a population with mean, μ, and standard deviation, σ, the sampling distribution of means
of size n has the following properties.
1. The sampling distribution of means will be approximately normal provided n > 30.
This is the Central Limit Theorem.
If the original population is normal, the sampling distribution will be normal regardless of
the sample size.
2. The mean of the sampling distribution of means equals the mean of population, x
.
3. The standard deviation of the sampling distribution of means is approximately equal to the
standard deviation of the population divided by square root of sample size,
If n/N > 0.05, then
x
n
x
n
.
N n
.
N 1
N n
is the finite correction factor.
N 1
Generally, this correction is not necessary.
Examples
The GPA’s of all students at a large university are approximately normal with a mean of 3.02
and a standard deviation of 0.29. Assume a random sample of twenty students.
a. Find the probability the mean GPA of these students is 3.10 or higher.
b. Find the probability the mean GPA of these students is lower than 2.90.
c. Find the probability the mean GPA of these students is between 2.95 and 3.11.
d. What mean would put them in the top 5%?
The amounts of phone bills for all households in a city have a skewed distribution with a mean
of$90 and a standard deviation of $25. Assume a random sample of 75 households.
a. Find the probability the mean of these households will be between $83 and $88.
b. Find the probability the mean of these households will be above $85.
c. Find the probability the mean of these households will be within $6 of the population mean.
d. Find the probability the mean of these households will be less than the population mean by
$3.
e. What mean would put them in the lowest 15%?
Sampling Distribution of Proportions
Given a population with proportion, p, and n < 0.05N (for independence), the sampling
distribution of proportions of size n has the following properties.
1. The sampling distribution of proportions will be approximately normal provided npq > 10.
2. The mean of the sampling distribution of proportions equals the proportion of population,
p.
p̂
3. The standard deviation of the sampling distribution of proportions is approximately equal to
npq
pq
the standard deviation of the population divided by the sample size, p̂
.
n
n
n
Example
In the recent past, it was reported that 26% of adults have no credit cards. Suppose a random
sample of 500 hundred adults is chosen.
a. Find the probability that less than 24% of the sample has no credit cards.
b. Find the probability that between 25% and 30% of the sample has no credit cards.