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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
opportunity
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
significance
Estimating with confidence

SAT Math Scores in California




Want to test population of 350,000
seniors
SRS of 500
x = 461
What can you say about the mean score
μ in the population of all 350,000
seniors?
Results




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
samples.
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%.
Exercises

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
curve.
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


Exercises
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
precisely.
Margin of error = z* σ / √ n
Behavior



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.
Exercises

10.8 – 10.11
Choosing the sample size.

Example 10.7 on pages 551 and 552

Exercises 10.13 and 10.14