Reminders:

Parameter – number that describes the population

Statistic – number that is computed from the sample data

Mean of population = µ Mean of sample = x

4.3 Sample Distributions

Statistical inference uses sample data to draw

conclusions about the entire population

Statistical estimation and the law of large numbers

As the number of observations drawn increases, the mean x of the observed values gets closer and closer to the mean of the population

Similar to flipping a coin “many times”

Foundation of gambling casinos and insurance companies

Sampling distributions

the sampling distribution of a statistic is the distribution (histogram) of values taken by the statistic in all possible samples of the same size from the same population

Hospital beds data Make a histogram of the data & calculate the

population mean Take 2 samples of ten and find the sample

means Make a histogram of all the sample means Compare the population mean to the

approximate mean on the histogram Take two more samples of ten and find the

sample means. Make histogram of all sample means and

compare to population mean

Central limit theorem

As the sample size increases, the distribution of x changes shape: it looks less like that of the population and more like a normal distribution (no matter what the population curve looks like)

When the sample is large enough, the distribution of x is very close to normal

Hospital beds data

Take 2 samples of 20 and find the sample means

Make a histogram of all the sample means

Compare the population mean to the approximate mean on the histogram

Compare the histogram made with 2 samples of twenty to the histogram made with 2 samples of ten

When we choose many SRS’ s from a population, the sampling distribution of the sample means. . .

is centered at the mean of the original population

is less spread out than the distribution of individual observations

The mean of the sampling distribution of x is μ and its standard deviation is n

Unbiased estimator – “correct on the average in many samples” because the mean of x is equal to μ,

we say that the statistic x is an unbiased estimator of the parameter μ

The standard deviation of the distribution of x gets smaller as we take larger samples

Engineers are redesigning fighter jet ejection seats to better accommodate women. In human engineering and product design, it is often important to consider people’s weights so that airplanes or elevators aren’t overloaded, chairs don’t break, etc. Given that the population of women has normally distributed weights with a mean of 143 lb and a standard deviation of 29 lb, find the probability that

If 1 woman is randomly selected, her weight is greater than 150 lb

If 36 different women are randomly selected, their mean weight is greater than 150 lb

Interpretation There is a 0.405 probability that a

woman will weigh more than 150 lb, but there is only a 0.0738 probability that 36 women will have a mean weight of more than 150 lb.

It is much easier for an individual to deviate from the mean than it is for a group of 36. A single extreme weight among the 36 weights will have reduced impact when it is averaged in with the other weights.

IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. If 25 people are randomly selected for an IQ test, find the probability that their mean IQ score is between 95 and 105.