OPTION 3 Confidence Interval on the Mean – one sample (T-Interva.docx

OPTION 3: Confidence Interval on the Mean – one sample (T-
Interval)
A) Introduction – introduce the topic and yourself: 5% of score
Explicitly identify the following:
1) Parameter of interest: __________________________
2) Population of interest (be as specific as necessary):
___________________________
3) Variable of interest: _____________________________
4) Then write a paragraph or two of introduction to your paper.
In the paragraph, include at least the following information:
Your research objective, your interest in the topic.
Describe in detail where you found your sample and
specifically how you chose which individuals to survey.

B) Data Sets – Present your organized data and sample
statistics: 15%
1) DATA SET A: a) Show the frequency distribution of your
full data set. Call it Data Set A.
It must have at least 80 individuals: n > 80. You can use either
a table or a histogram to show the distribution:
Identify b) sample size: __________ c) mean
:___________ d ) standard deviation: ____________
e) Estimated Standard Error of the means (SE):
______________________. Round each to four decimal places

if needed.
2) DATA SET B: Select some of the data points from Data Set
A to create smaller data set. Call this Data Set B. It should
have about half of the values from Data Set A.
a) Show the frequency distribution of Data Set B. You can
use either a table or a histogram to show the distribution. USE
THE SAME GROUPING (CLASSES) FOR EACH DATA SET:

if needed.
f) Describe specifically how you chose the individuals from
Data Set A to create Data Set B.
3) DATA SET C: Select between 10 and 20 data values from
Data Set A to create a still smaller data set. Call this Data Set
C.
a) Show the frequency distribution of Data Set C. You can
THE SAME GROUPING (CLASSES) FOR EACH DATA SET

if needed.
Data Set A to create Data Set C.
C) The Intervals – Construct the Confidence Intervals for three
different sample sizes and three different levels of confidence:
45%
1) Degrees of freedom (df) – determine the degrees of freedom
for each Data Set.
Data Set A: df = __________ Data Set B: df =
__________ Data Set C: df = __________
2) Choose three confidence levels, between 80% and 99%.
C-level 1: __________ C-level 2: __________
C-level 3: __________
3) Determine for each level of confidence you chose.
C-level 1: = __________ C-level 2: =
__________ C-level 3: = __________
4) Critical Values – Each sample size together with a level of
confidence will have a critical value.
a) Determine the critical values for each of the nine
confidence intervals. Put the critical values in the chart below
organized by either sample size or confidence level. Round each
to three decimal places if needed.
b) To find the critical values, did you use the t-table or the
calculator?

5) Show the formula for constructing confidence intervals for
the mean by hand:
6) Confidence Intervals– Create a confidence interval for each
of the three confidence levels for each of the three samples, for
a total of nine confidence intervals. Put the intervals in the
chart below. Round each to three decimal places if needed.
7) Determine the Margin of Error for each of the nine
confidence intervals. Put the Margin of Errors in the chart
below. Round each to three decimal places if needed.
8) Fill in the rest of the chart using values found in your results
above.
n
df
s
SE
C-level
Critical value
Confidence Interval (CI)
E

Put the information in the chart organized by either
sample size or confidence level for ease of comparison.
9) Interpret at least three of the confidence intervals in the
context of the problem. Include population, parameter and
variable.

D) Written Analysis – discuss validity and explain the results of
your findings: 30%
1) Validity of the method
State the three requirements for valid inference on the mean
using our methods
a) Population size:
b) Sampling Method:
c) Shape of the Sampling Distribution:
2) a) Estimate the population size for your population of
interest. Explain your reasoning:
b) Verify whether the population size requirement is met
(from 1a above) (Show your work):
3) Describe your sampling method (how you gathered the data)
and explain if you think it might be biased or not. Explain
whether or not this meets the requirement (from 1b above)
4) For Data Sets A and B, explain whether the shape of the
sampling distribution requirement is met. Show your reasoning.
5) For Data Set C, a) display the boxplot and b) normal
probability plot.

c) Explain why it is necessary to examine these graphs.
d) Interpret the graphs and tell whether the method is valid
for your small data set (based on the graphs).
6) Effect of sampling and sample size
a) Identify which values from the confidence intervals
(point estimate, standard error, critical value, etc.) were
impacted by sample size and which ones weren’t, based on your
observations.
b) Apply the Law of Large Numbers to your tests and
explain which one of the three results should be most trusted.

E) Summary – summarize your results: 5% -
Write a paragraph summarizing your paper: Include:
restate your research objective, an overall estimate of the
parameter based on your findings, any possible flaws in your
research (sampling method? sample size? population size?).
Did the results concur for each sample? Why, why not? Note
any interesting observations you had or possible extensions of
this research.
x
x
OPTION 2: Hypothesis Testing on the Proportion – one sample
(1-Prop Z-Test)
1) Parameter of interest: ____________________
___________________________
3) Characteristic of interest (define a “success”):
_____________________________
Your research objective, Your interest in the topic.

statistics: 15%
1) DATA SET A: Select a sample that includes data from at
least 150 individuals. Call it Data Set A: n > 150.
a) Display a frequency and relative frequency table of
those with and without the characteristic of interest.
Identify b) sample size: __________ c) number of
successes :___________ d ) sample proportion: ____________
e) Standard Error of the proportions (SE):
if needed.

if needed.
C.
if needed.

C) The Tests – Perform the hypothesis Tests using the two
approaches: 45%
Step 1 – a) Write a sentence making a claim.
_________________________________
b) Explain how you decided on this claim.
c) Form the hypotheses using the symbol notation
used in class.
Step 2 – Determine the level of significance and the rejection
region
A) Describe a Type I Error generally speaking:
_______________________________
Describe a Type I Error specific to this
situation:
B) Describe a Type II Error generally speaking:
_______________________________
Describe a Type II Error specific to this
situation:
C) Tell any consequences (positive or negative) of
making either type of error, and explain which error you think
is worse, in this specific case.

D) Select your level of significance based on C) and
explain why you chose the value you chose.
E) Determine the critical value(s) and identify the
rejection region.
F) Draw a z-distribution for each data set, showing
the rejection region and critical value, clearly labeled.
Step 3 – a) Show the formula for calculating the test statistic by
hand:
b) Determine the value of the test statistic for each data
set:
Data Set A: test statistic = __________ Data
Set B: test statistic = __________

Data Set C: test statistic = __________
c) Show the probability notation for finding the P-value of
the calculated test statistic specific to your research.
d) Determine the P-value for each data set:
Data Set A: P = __________ Data Set B: P =
__________ Data Set C: P = __________
e) In the z-distributions you drew above in Step 2– F,
clearly indicate the locations of the test statistic and P-value.
f) Fill in the table using your results from above.
Record your results in order of sample size. Round all
calculated values to three decimal places if needed.
n
x
Standard Error, SE
Critical value(s)
Test statistic,
P-value

Step 4 – Use the Decision Rule to either reject or not reject the
null hypothesis
Fill in the charts for each approach with the values
from above and show the comparisons used to make your
decisions.
P-Value Approach:
n
P-value
Compare P to
Decision (reject or do not reject H0)

Classical Approach
n
Critical Value(s) /Rejection Region
Test Statistic
Is z in rejection region?
Step 5 – Write the conclusion - For each sample size, give
the full conclusion in “plain English”, in the context of the
problem.

your findings: 30%
State the three requirements for valid inference on proportions
using our methods
a) Population size:
b) Sampling Method:

4) For each Data Set, explain whether the shape of the sampling
distribution requirement is met. Show your reasoning.
a) Identify which values from the hypothesis test (P-value,
test statistic, critical value, etc.) were impacted by sample size
and which ones weren’t, based on your observations in the test.
Write a paragraph summarizing your paper.
Include: Restatement your research objective,
Any possible flaws in your research (sampling
method? sample size? population size?).
Was the claim supported or not?
Did the results concur for each sample? Why,
why not?
Note any interesting observations you had or

possible extensions of this research.
p̂
ˆ
p
zp̂
z
ˆ
p
OPTION 1: Hypothesis Testing on the Mean – one sample (T-
Test)
1) Parameter of interest: __________________________
___________________________
3) Variable of interest: _____________________________
Your research objective, your interest in the topic.

statistics: 15%
1) DATA SET A: a) Show the frequency distribution of your
full data set. Call it Data Set A.
It must have at least 80 individuals: n > 80. You can use either
a table or a histogram to show the distribution:
Identify b) sample size: __________ c) mean :___________ d )
standard deviation: ____________
e) Standard Error of the means (SE):

if needed.
a) Show the frequency distribution of Data Set B. You can
THE SAME GROUPING (CLASSES) FOR EACH DATA SET:
if needed.
C.
a) Show the frequency distribution of Data Set C. You can
THE SAME GROUPING (CLASSES) FOR EACH DATA SET

if needed.
C) The Tests – Perform the hypothesis Tests using the two
approaches: 45%
Step 1 – a) Write a sentence making a claim.
_________________________________
b) Explain how you decided on this claim.
c) Form the hypotheses using the symbol notation
used in class.
Step 2 – Determine the level of significance and the rejection
region
A) Describe a Type I Error generally speaking:
_______________________________

Describe a Type I Error specific to this
situation:
B) Describe a Type II Error generally speaking:
_______________________________
Describe a Type II Error specific to this
situation:
C) Tell any consequences (positive or negative) of
making either type of error, and explain which error you think
is worse, in this specific case.
D) Select your level of significance based on C) and
explain why you chose the value you chose.
E) For each data set, determine the critical value(s)
and identify the rejection region.
Data Set A: critical value(s) = __________
Data Set B: critical value(s) = __________
Data Set C: critical value(s) = __________
F) Draw a t-distribution for each sample size,
showing the rejection region and critical value, clearly labeled

Step 3 – a) Show the formula for calculating the test statistic by
hand:
b) Determine the value of the test statistic for each data
set:
Data Set A: test statistic = __________ Data
Set B: test statistic = __________
Data Set C: test statistic = __________
c) Show the probability notation for finding the P-value of
the calculated test statistic specific to your research.
d) Determine the P-value for each data set:
Data Set A: P = __________ Data Set B: P =
__________ Data Set C: P = __________
e) In the t-distributions you drew above in Step 2– F,
clearly indicate the locations of the test statistic and P-value.
f) Fill in the table using your results from above.
Record your results in order of sample size.
Round all calculated values to 4 decimal places if needed.
n

Observed value,
s
Standard Error, SE
Critical value(s)
Test statistic,
P-value
Step 4 – Use the Decision Rule to either reject or not reject the
null hypothesis
Fill in the charts for each approach with the values

from above and show the comparisons used to make your
decisions.
P-Value Approach:
n
P-value
Compare P to
Classical Approach
n
Critical Value(s) /Rejection Region
Test Statistic
Is t in rejection region?

Step 5 – Write the conclusion - For each sample size, give
the full conclusion in “plain English”, in the context of the
problem.
your findings: 30%
State the three requirements for valid inference on the mean
using our methods
a) Population size:
b) Sampling Method:

4) For Data Sets A and B, explain whether the shape of the
sampling distribution requirement is met. Show your reasoning.
5) For Data Set C, a) display the boxplot and b) normal
probability plot.
c) Explain why it is necessary to examine these graphs.
d) Interpret the graphs and tell whether the method is valid
for your small data set (based on the graphs).

a) Identify which values from the hypothesis test (P-value,
test statistic, critical value, etc.) were impacted by sample size
and which ones weren’t, based on your observations in the test.
Write a paragraph summarizing your paper: Include:
restate your research objective, any possible flaws in your
research (sampling method? sample size? population size?).
Was the claim supported or not? Did the results concur for each
sample? Why, why not? Note any interesting observations you
had or possible extensions of this research.
tx
t
x
x
x

Statistics Project #2 – Inference
· We will be doing observational studies: Observational studies
measure the characteristics of a population by studying
individuals in a sample, but do not try to manipulate or
influence the variables.
This assignment counts as 25% of your grade. Think of it as an
exam on Inferential Statistics.
Project Overview: You will either construct confidence
intervals or conduct hypothesis tests using the two approaches
learned in class: Classical Approach and P-value Approach.
You will use three different samples to examine the role of
sample size on analysis.
You must analyze a sample from the population.
You have four options for your study of inference
1) Test a claim about a population mean (T-Test). You
must have a sample of at least 80 (n > 80)
2) Test a claim about a population proportion (1-Prop Z-
Test). You must have a sample of at least 150 (n > 150)
3) Estimate the value of a population mean (T-Interval).
You must have a sample of at least 80 (n > 80)
4) Estimate the value of a population proportion (1-Prop Z
-Interval).You must have a sample of at least 150 (n > 150)
The Completed Assignment is due Tuesday, March 10 at the
beginning of class.
To complete the assignment, you can either fill in the
blanks and answer the questions on the appropriate form you
choose, or you can write it in a paper, being sure to include all
of the parts requested.
In either case, The Completed Project should be written in
sections in this order:
Introduction (5%) – Introduce the topic and yourself

Data sets(15%) – Present your organized data and sample
statistics
The Inference Method(45%) – Show the process of the method
of the inference you choose
Written Analysis(30%) – Give a written analysis of your
research
Summary (5%) – summarize your findings and conclusions
Automatic Deductions from your grade
Project is not in the order above – deduct ½ grade step
You have data from more than half of the entire population
– deduct full grade
You will receive no higher than a D if the proper test is
not conducted according to the data.
· You may turn in work early. It must be a hard copy. See me
if this is a problem for you.
· I will be happy to check what you are doing before you turn it
in. This must be a hard copy.
· Each person needs to turn in a complete and unique paper.
· Check Moodle for websites that have data available.
· All work must be non-offensive to the general public and
within University guidelines of expected behavior.
For inference on the mean - Extra Credit opportunity -
Explore Outliers in your Data Sets
Sketch a boxplot of each of the three data sets to determine
presence of outliers.
If there are outliers, remove them and redo the test
If there are no outliers, add one by changing one data point or
by adding a data point and redo the test
For each altered data set
Show the new boxplot, clearly identifying your adjustment of
the data set.
Identify the sample size, mean and standard error
Present the findings for both original and altered data sets

in an organized fashion for ease of comparison.
Discuss how the outlier impacted the results.
For test on proportions - Extra Credit opportunity - Find P-
values using Binomial Distribution
Give the notation for determining P-value using the value
of x, the number of successes in the sample.
Give the binomial formula for determining P-values of
small samples (see pages 337-339 in textbook or I can help you
with it)
Calculate the P-value using the binomial function for each
of the three tests
Present the P-values using both the binomial function and the
normal approximation in an organized fashion for ease of
comparison.
Discuss how the sample size impacted the closeness of the
approximations.

OPTION 3 Confidence Interval on the Mean – one sample (T-Interva.docx

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