# Chapter 5 Summary - Grinnell College

```Chapter 6
California SAT Data
There are 250,000 California High School Students
We did a SRS of 500 to take the test. With a mean of X = 461
What if we had selected a different 500 students?
 Would the sample mean and variance be the same?
 Would the population mean and variance be the same?
 How good is our estimate of the population mean?
 How sure are we that the population mean is less than 500?
Assume x = 100 (is this realistic?) Can we use the normal
distribution? What is  X ?
P(  X - 1.96 X &lt; X &lt;  X + 1.96 X ) =
P( X - 1.96 X &lt;  X &lt; X + 1.96 X ) =
X +/- 1.96 X is the 95% Confidence Interval for  X
If we took many SRS of 500, 95% of the CIs would include the true mean.
We are 95% confident that the population (true) mean is within
X +/- 1.96 X
95% CI is an interval calculated from sample data that is guaranteed
to capture the true population parameter in 95% of all samples.
Confidence Intervals gives estimates of parameters (population means)
They also tell us how confident we can be that our answers are correct.
They assume unbiased sampling!!!!
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2 sided
Confidence Level
Table Value
70%
80%
90%
95%
98%
99%
99.9%
1.04
1.28
1.64
1.96
2.33
2.58
3.29
In Example 6.1
X = 461  X = 100/sqrt(500) = 4.5
X - 1.96 X =
X + 1.96 X =
So a 95% CI for the true population mean is (
,
)
Margin of error =
How big should n be to get a margin of error = 1
1.96*/sqrt(n) = 196/ sqrt(n) = 1
n=
n is an integer ALWAYS ROUND UP!
99% CI for the true population mean?
X - ____ X
X + ____ X
=
=
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Section 6.2 Hypothesis Testing
How much evidence is there to support a claim about a population.
Is there enough evidence to show a claim is true?
California high school students average SAT scores higher than 485
California high school students perform better than the national
average
California high school students perform better than Iowa high school
students
Living near power lines increases your chances of getting cancer.
Taking an aspirin immediately after a heart attack will increase your
chances of living
Hypothesis Test -- Testing the population mean (  )
1) State a Hypothesis
Ho:  = o Vs
H1:   o
Ho:   o Vs
H1:  &gt; o
Ho:   o Vs
H1:  &lt; o
H1 is always what you want to prove, {In this class Ho always
includes = }
x
Z

2) Create a test statistic
x
3) Compare to appropriate table value
Z( / 2)
{2-sided hypothesis test}
Z or -Z {2-sided hypothesis test}
4) Reject or Fail to Reject Ho
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Example
1) California High School students don't average 475 on there SATs
Ho:  = 475 Vs
H1:   475
461475
 311
.
2) Z 
45
.
3) Since |Z| &gt; 2.58, we are 99% confident that the average SAT score
is not 475
99% 99.9%
2.58 3.29
4) We Reject Ho in favor of H1 at the 99% confidence level
[ at the 99.9% Confidence level we fail to reject .001 &lt; p-value &lt; .01]
p. 445 To test the hypothesis Ho : &micro; = &micro;o at significance level , based on a
SRS of size n, from a population with unknown mean &micro; and known ,
compute the test statistic
1) state the hypothesis
a) H1: &micro; ≠ &micro;o
b) H1: &micro; &gt; &micro;o
c) H1: &micro; &lt; &micro;o
2)
z
x
x
3) find p-value
a) 2*P(Z &gt; |z| )
b) P(Z &gt; z )
c) P(Z &lt; z)
 is determined before the test is conducted
p-values are based on sample data
4) Reject Ho if p-value is less than 
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The Survey of Study Habits and Attitudes (SSHA) is a phychological test that
measures student’s study habits and attitudes toward school. Scores range
from 0 to 200. The mean score for college students is about 115 and the
standard deviation is about 30. A teacher suspects that the mean  for for
older students higher than 115. She gives the SSHA to a SRS of 25 students
who are at least 30 years old. Suppose we know the scores in the population
of older students are Normally distributed with standard deviation  = 30.
We seek evident against the claim that  = 115. What is Ho and H1?
If Ho is true, what is the sampling distribution? {Distribution of X }
Sketch this distribution.
Consider the following 2 options X =118.6 or X = 125.8, is either good
evidence that the true mean is greater than 115?
1) H1: &micro; &gt; 115
2)
z
Ho: &micro;  115
{Ho: &micro; = 115}
x
x
3) find p-value
b) P(Z &gt; z )
for  = .05
for  = .025
4) Reject Ho if p-value is less than 
The data must be a SRS from the population
Pshyc 101 class as SRS for vision
Soc 101 class as SRS to test attitudes towards the poor
Different methods are needed for different designs
SRS from Normal data,
Shape of the population distribution and sample distribution
 known
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Outliers can distort the result
Margin of error doesn’t cover all errors
How small a P is convincing?
Statistical significance VS. Practical significance
6.4 Assessing performance
Confidence intervals
Confidence Level – how often the method succeeds (contains &micro;)
MOE - how close (narrow) the interval is
Hypothesis Tests
RARELY reject Ho when it is really true [=.05 or =.01]
USUALLY reject Ho when Ho is not true (i.e. Ha true) [power]
Power = P(reject Ho when a specific value of Ha is true)
reject Ho
Ho True
Ha true
Type 1 error
Correct!
P(Type 1 error) = 
1-P(Type II error) = Power
=P(Correct given Ha true)
fail to reject
Ho
Correct!
Type II error
P(Correct given Ho True) = 1- 
P(Type II error) = 1- Power
sum = 1
sum = 1
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```