Two ways to ensure it can be used:
1. sample size(n) is >= 30.
2. is Normal Distribution(Bell-shaped)
The mean of the sampling distribution is the mean of the population.
Standard error:
1. Tells us about the width of the distribution of sample means.
2. The standard deviation of a sample statistic is called its standard error
σ2 = σ / √n
Where:
X̄ is the sampling distribution of the sample means
~ means “follows the distribution”
N is the normal distribution
µ is the mean of the population
σ is the standard deviation of the population
n is the sample size
Example:
Assume that men’s weight are normally distributed
with a mean given by µ = 172 lb and a standard deviation of 29 lb (based on data from the
National Health Survey).
1. (a) If 1 man is randomly selected, find the probability that his weight is greater than 180
lb.
x=180 µ = 172 sd = 29
Same as Z score. Z = (180 - 172) / 29 Z-Scores
(b) If 100 men are randomly selected, find the probability that they have a mean weight greater
than 180 lb.
This is done differently. The formula is (x-µ)/(sd / √n)
x=180 µ = 172 sd = 29 n = 100
Answer: Z = 2.758 Round up to 2.76 look up on table = .9971
P(Mean %lt 180) = 1-0.9971 = 0.0029
2. If 4 men are randomly selected, find the probability that they have a mean weight
between 160 lb and 180 lb
To solve this problem we must find the Z of both 160 and 180. P(160 < x̅ < 180)
plugged in x = 160 and n = 4 output: -0.827 (Round to -0.83)
plugged in x = 180 and n = 4 output: 0.5517 (Round to -0.55)
P(-0.83 < Z < 0.55 ) Charted: 0.55 => 0.7088 and -0.83 => 0.2033
0.7088 - 0.2033 = .5055 OR %50 chance
Note: the sample size is less than 30. Because the sample is from a normally distributed population.
Example of 1(b)
results
Imagining an experiment may help you to understand sampling distributions:
Suppose that you draw a random sample from a population and calculate a statistic for the sample, such as the mean.
Now you draw another random sample of the same size, and again calculate the mean.
You repeat this process many times, and end up with a large number of means, one for each sample.