Transcript 8.1 Sampling Distributions

Statistical Reasoning
for everyday life
Intro to Probability and
Statistics
Mr. Spering – Room 113
8.1 Sampling Distributions
NOTATION AND SYMBOL REVIEW:
n…
x
SAMPLE
MEAN
Sample size
μ…
Population mean
σx…
Population standard deviation
Sx…
Sample standard deviation
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8.1 Sampling Distributions
NOTATION AND SYMBOL REVIEW:
z…
Z-score (value compared with mean and σ)
r…
Correlation coefficient
∑x…
Sum of x values
P(x)…
Probability of x
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8.1 Sampling Distributions
Sampling error…
• The error that is introduced when a sample is used to estimate
a population. It does not include other sources of error, such as
those due to biased sampling, bad survey questions, or
recording mistakes.
• Would you expect the sampling error to increase or decrease if
the sample size were increased?
•
The sampling error is the observed differences in such things
as means and standard deviations, or proportions, from the
population from which the sample is taken. If error is truly
random it should decrease with sample size.
C.L.T
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•
If systematic the errors should not change regardless of
sample.
8.1 Sampling Distributions
Distribution of Sample Means… (Samples > 30)
The larger the sample size, the more closely the sampling
distribution approximates a normal distribution. In all cases, the
mean of the sampling distribution equals the population mean.
If only one sample is available, its sample mean, is the best
estimate for the population mean. (C.L.T.)
EXAMPLE:
Texas has roughly 225,000 farms. The mean farm size is 582 acres. Suppose that
for random samples of 100 farms, the distribution of sample means has a mean of
582 acres and a standard deviation of 15 acres. You select a random sample of
100 farms which have a mean size of 600. What is the probability of selecting
another sample with a mean greater than 600 acres?
600  582
z
 1.2  88th percentile
15
 P > 600 is about 12 %
8.1 Sampling Distributions
SAMPLE PROPORTIONS → PERCENTAGES & FRACTIONS
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Population proportions →
More notations:
Sample proportions →
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Distribution of sample proportions…
The distribution of sample proportions results when we find the
proportions in all possible samples of a given size. The larger the
sample, the more closely the sampling distribution approximates a
normal distribution. In all cases, the mean of the sampling distribution
equals the population proportion. If only one sample is available it is
the best estimate for the population proportion.
8.1 Sampling Distributions
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Class work:
pg 343 # 1-15 all
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