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Statistics 101
Chapter 10
Section 10-1
We want to infer from the sample data
some conclusion about a wider
population that the sample represents.
Inferential Statistics allows us this
Statistical inference
Provides methods for drawing
conclusions about a population form
sample data.
We use probability to express
strength of our conclusions
Confidence Intervals and tests of
Estimating with confidence
SAT Math Scores in California
Want to test population of 350,000
SRS of 500
x = 461
What can you say about the mean score
μ in the population of all 350,000
Law of large numbers
Distribution close to normal
Mean of sample = mean of population
Standard deviation of x is σ / √500, let
σ be 100 for known sd. Then, the
standard deviation of x will be 4.5
Statistical Confidence
The 68-95-99.7 rule says that in 95%
of all samples, the mean score x for
the sample will be within two standard
deviations of the population mean.
Whenever x is within + 9 points of the
unknown μ, this happens 95% of all
What we can say
We are 95% confident that the
unknown mean SAT Math score for all
California high school seniors lies
between 452 and 470.
How to find interval
Interval = estimate + margin of error
We use the symbol C to represent
confidence interval.
Question: Can you choose a different
confidence interval instead of 95%.
10.2, 10.3
Confidence interval for a
population mean
The construction of a confidence
interval for a population mean μ is
appropriate when
The data come from an SRS from the
population of interest
The sample distribution of x is
approximately normal
Finding z*
To find 80% confidence interval, we
catch the central 80%.
We leave off 10% on both tails.
So z* is point with area 0.1 to its right
(and 0.9 to its left) under the normal
Search Table A to find the point with
area 0.9 to its left.
Critical values
The number z* with
probability p lying
to its right under
the standard
normal curve is
called the upper p
critical value of
the standard
normal distribution
Work through Example 10.5
10.5 – 10.7
Confidence Interval Behavior
High confidence says that our method
almost always gives correct answers.
Small margin of error says that we
have pinned down the parameter quite
Margin of error = z* σ / √ n
z* gets smaller : to obtain a smaller margin
of error from the same data, you must be
willing to accept lower confidence.
σ gets smaller: measures the variation in the
population. Think of the variation among
individuals in the population as noise that
obscures the average value of μ.
n gets larger: reduces margin of error.
Since the n appears under the root sign, we
must take four times as many observations
in order to cut the margin of error in half.
10.8 – 10.11
Choosing the sample size.
Example 10.7 on pages 551 and 552
Exercises 10.13 and 10.14