Topics 16 - 18
Unit 4 – Inference from Data: Principles
TOPIC 16
CONFIDENCE INTERVALS: PROPORTION
Topic 16 - Confidence Interval: Proportion
The purpose of confidence intervals is to use the sample statistic to construct an
interval of values that you can be reasonably confident contains the actual, though
unknown, parameter.
^


p1  p 


n
^
The estimated standard deviation of the sample
statistic pˆ is called the standard error of pˆ.
Confidence Interval for a population proportion :
^


p 1  p 


n
^
^
est imat e margin of error  p  Z *
where n . P^ >= 10 and n (1-p^)>= 10
Z * Critical value-Z is calculated based on level of confidence
When running for example 95% Confidence Interval:
95% is called Confidence Level and
we are allowing possible 5% for error, we call this alpha (α )= 5% where α is the significant
level
Topic 16 - Confidence Interval: Proportion
Click on STAT, TESTS and scroll down to
1-PropZint…
To calculate Confidence Interval
You need to have x, n and C-Level
x and n comes from the sample
Please note if you have p-hat and n
calculate x = p-hat * n, round your
answer
Exercise: 16-12: Credit Card Usage - Page 347
Exercise: 16-13: Responding to Katrina – Page 347
Watch Out
• A confidence interval is just that— an interval— so it includes all
values between its endpoints.
• Do not mistakenly think that only the endpoints matter or that only
the margin- of- error matters.
• The midpoint and actual values within the interval matter.
The margin- of- error is affected by several factors
primarily
• A higher confidence level produces a greater margin- of- error
( a wider interval).
• A larger sample size produces a smaller margin- of- error
( a narrower interval).
• Common confidence levels are 90%, 95%, and 99%.
• Always check the technical conditions before applying this
procedure.
• The sample is considered large enough for this procedure to be
valid as long as npˆ>= 10 and n(1 –pˆ) >=10. If this condition is not
met, then the normal approximation of the sampling distribution is
not valid and the reported confidence level may not be accurate.
• Always consider how the sample was selected to determine the
population to which the interval applies.
Choosing the sample size
The confidence interval for the a Normal population will have a
specified margin of error m when the sample size is
2

z  ˆ
n    P 1  Pˆ
m
*

If n is not a whole number then round up.
Example: Activity 16-8: Cursive Writing
2

z  ˆ
n    P 1  Pˆ
m
*

a. The number of essays needed for a 99% CI is
0.01 = 2.576 √[ (.15)(.85) /n];
n = (2.576 /.01)2 (.15)(.85) = 8460.614;
n = 8461 Remember to round UP
b. You could use a lower confidence level (95% or 90% confidence, for example), or you
could use a wider margin-of-error, say .02. Either of these choices would allow you to
select a smaller (random) sample.
Activity 16-11: Penny Activities - Page 347
2

z  ˆ
n    P 1  Pˆ
m
*

TOPIC 17 - TESTS OF SIGNIFICANCE:
PROPORTIONS
Topic 17 – Test of Significant: Proportion
A sample result that is very unlikely to occur by random chance alone is said to be statistically
significant. We now formalize this process of determining whether or not a sample result
provides statistically significant evidence against a conjecture about the population
parameter. The resulting procedure is called a test of significance.
A significance test is designed to assess the strength of evidence against the null hypothesis.
Step 1: Identify and define the parameter.
Step 2: we initiate hypothesis regarding the
question – we can not run test of significant
without establishing the hypothesis
H 0 :
H :
 a








 0
 0
 0
or
or
 0
Step 3: Decide what test we have to run, in case of proportion, we use Z-test in

proportion
p 
Z 
0
 0 (1   0 )
n
Topic 17 – Test of Significant: Proportion
Step 4: Run the test from calculator
Step 5: From the calculator write down the p-value and Z-test
Step 6: Compare your p-value with α – alpha – Significant Level
If p-value is smaller than α
we “reject” the null hypothesis, then it is statistically significant based on data.
If p-value is greater than the α
we “Fail to reject” the null hypothesis, then it is not statistically significant based on data.
Last step: we write conclusion based on step 6 at significant level α
•
•
•
•
•
p- value > 0.1: little or no evidence against H0
0.05 < p- value <= 0.10: some evidence against H0
0.01 < p- value <= 0.05: moderate evidence against H0
0.001 < p- value <= 0.01: strong evidence against H0
p- value <= 0.001: very strong evidence against H0
Topic 17 – Test of Significant: Proportion
Click on STAT, TESTS and scroll down to
1-PropZTest…
To calculate One Sample Proportion
Z-Test
You need to have P0 , x, n and
Alternative Hypothesis
P0 is π0 from Null Hypothesis
x and n comes from the sample
Please note if you have p-hat and n
calculate x = p-hat * n, round your
answer
Prop is the alternative hypothesis
Exercise 17-6: Properties of p-value – Page 371
Exercise 17-7: Properties of p-value – Page 371
Exercise 17-8: Wonderful Conclusions– Page 371
Exercise 17-12: Kissing Couples – Page 372
Exercise: 17-26: Employee Sick Days–Page 375
Exercise: 17-27: Stating Hypothesis –Page 375
TOPIC 18
MORE INFERENCE CONSIDERATION
Watch Out
• Alpha = α
A Type I error is sometimes referred to as a false alarm because the
researcher mistakenly thinks that the parameter value differs from
what was hypothesized.
• Beta = β
a Type II error can be called a missed opportunity because the
parameter really did differ from what was hypothesized, yet the
researchers failed to realize it.
• 1–β
The power of a statistical test is the probability that the null
hypothesis will be rejected when it is actually false ( and therefore
should be rejected). Particularly with small sample sizes, a test may
have low power, so it is important to recognize that failing to reject
the null hypothesis does not mean accepting it as being true.