Confidence Intervals – Proportion and One Mean - Solutions
1 The term sampling frame refers to the group that actually had a chance to get into the sample. Ideally,
this is the same as the population of interest, but sometimes it isn’t. In the following situation, describe
the population, the sampling frame, the sample, the parameter of interest, and the statistic.
A Gallup Poll is done using random digit dialing to reach individuals in households with land-line
telephones. The purpose is to estimate the proportion of U.S. adults who favor stronger gun control laws.
One-thousand persons are sampled, and 63% favor stronger gun control.
a. Population = all U.S. adults
b. Sampling frame = adults in the households with land-line telephones
c. Parameter = proportion of U.S. adults favoring stronger gun control
d. Sample = the 1,000 surveyed
e. Statistic =63%, the sample percent
2 From the Data Sets folder open the Class Survey data and select a column of your choice to answer
the questions following the special notes.
Special Notes:
1. To perform a proportion interval your column can only have two options (e.g. Male/Female for Gender,
Yes/No for smoking, etc.) However, you can manipulate your data to create a column with just two
outcomes. For example, if you wanted to manipulate the GPA data to either ≥ 3.00 or < 3.00 you can use
Calc > Calculator, enter a new variable name in the text box “Store result in variable” and then in the
expression Window enter ‘GPA’ ≥ 3.00. This will create in your new variable name column a series of
‘1’ and ‘0’ where the 1’s are for those with a 3.00 GPA or higher and 0’s for those less than 3.00.
2. When Minitab calculates a proportion, if using variables with Text responses (e.g. Male/Female;
Yes/No) the proportion will be calculated on the second word listed alphabetically (i.e. Male or Yes).
You will see this in the output where it reads “Event = Male” or “Event = Yes”. So if you want Female
or No you will have to refer to Special Note 1.
a. What column of data did you select?
Answers vary
b. What would be your population of interest?
Answers vary, but some examples are “All PSU undergraduate students at University Park”; “All PSU
undergraduate students at University Park”; “All PSU Stat200 Students”. The population would NOT be
all stat 200 students in our section(s) since this survey would not be a sample but instead the population as
all students took the survey
c. How is the class survey representative of that population?
The class is probably similar to these populations in make-up regarding percent that are female,
international, high/low GPA, etc.
d. How would you best describe the sampling technique used in attaining the Class Survey?
Definitely not one that involves statistical methods as no random sampling was done. Most likely you
would consider this sample to be a convenient sample.
e. Define a parameter value of interest (e.g. from Activity 9 the U.S. Government reported that 23% of
US adults ages 18-24 smoked cigarettes). If you cannot easily find one think of one. For instance
1
(although you cannot use this parameter value now since I am giving it to you!), maybe you believe that
25% of college students cheat or have cheated on a significant other.
Answers will vary
f. Referring to condition 3 on page 297 regarding using the normal approximation for sample proportions
(i.e. is n p̂ ≥ 5 and n(1- p̂ ) ≥ 5) verify that this condition has been met thus allowing you to select in
Minitab the option “Use test and interval based on Normal Distribution”.
Just need to plug in the numbers to verify
g. Use Minitab to create a 90% 1-proportion confidence interval on the column you selected in part a.
What is your interval?
Answers will vary
h. For your interval, answer the following:
What is the sample proportion, p̂ ? Answers will vary
What is the z* multiplier used for your interval? 1.645 or 1.65
What is the standard error? Answers will vary
What is the margin of error? Answers will vary, but is found by multiplier*standard error.
Confidence intervals consist of statistic ± margin of error
i. What is the sample size? The number of students who submitted a survey was 226. If your sample size
is not 226 why do you believe your number differs? Answers will vary
j. List the value and expression for the following: values will vary but expressions are:
Parameter: p
Statistic: p-hat p̂
Sample Size: n
k. If the confidence level in part g were changed to 95% would your resulting interval be wider or
narrower? The interval would get wider since you increased your confidence from 90 to 95%.
Remember, the greater the confidence level the more “confident” you become that the interval contains
the true parameter value. A wider interval would increase your confidence since a wider interval would
give more possible outcomes. Mathematically this is also true because you will notice that at the
confidence level gets larger so does the multiplier.
3 For mean confidence intervals we call into use T-table which can be found in this week’s folder. The
first concept to understand is the idea of Degrees of Freedom (DF). For our activity today, since we are
only concerned with one mean (either from one sample of a difference between paired data), DF = sample
size – 1 (i.e. n – 1). When finding confidence intervals, T-table provides the t multiplier (t*) for the
confidence interval expression:
2
s
n if data is consists of only one sample
x  t* *
NOTE: you will notice that the DF column in this tables only increases by 1 from 1 to 30. After that
the increments vary. If your DF is NOT found in the table then conservatively use the CLOSEST
df in the table that does not exceed the DF of interest. For example, if the degrees of freedom were
37 then from the table use 30.
Find t-multipliers and DF from Table A2 for the following conditions:
Confidence Level 90%, n = 25: t* = 1.71
DF = 24
Confidence Level 95%, n = 25: t* = 2.06
DF = 24
Confidence Level 95%, n = 35: t* = 2.04
DF = 34
Confidence Level 99%, n = 35: t* = 2.75
DF = 34
What do you notice that happens to t* as the level of confidence increases for the same sample size?
The multiplier, t*, increases as the level of confidence increases
What do you notice that happens to t* as sample size increases for the same level of confidence?
The multiplier, t*, decreases as sample size increases
4 Using the Class Survey data, PSU wants to estimate the true SATM scores for its undergraduate
population. Assuming that our survey represents this population, calculate a 1-mean 95% confidence
interval to estimate the parameter. First we will do by hand and then using Minitab. To start, the sample
mean ( x ) is 599 and n = 216 and s = 85.34.
a. Calculate the standard error of the mean.
s
n = 85.34/√216 = 5.81
b. What are the DF and t* from T-table?
DF = 215
t* = 1.98
c. Calculate the 95% Confidence Interval and provide an interpretation of this interval.
x  t* *
s
n = 599 ± 1.98*5.81 = 599 ± 11.504 = 587.496 ≤ u ≤ 610.504
d. Now use Minitab to verify your results. Go to Stat > Basic Statistics > 1-Sample t. For “Sample in
columns” enter SATM. Click Options and be sure that 95 is entered as the Confidence Level and “Not
Equal” is the Alternative (if you don’t have Not Equal then Minitab will produce an interval with just
either a lower bound or upper bound.) Click OK twice and copy and paste your Minitab results. Do your
results by hand and those from Minitab roughly match?
One-Sample T: SATM
Variable N
Mean StDev SE Mean
95% CI
3
SATM
216 599.005 85.335 5.806 (587.560, 610.449)
The intervals by hand and Minitab are comparable.
e. Write a sentence that explains what this interval means.
We are 95% confident that the true mean SATM score for the 2005 incoming freshman class at Penn
State was between 587.56 and 610.449.
f. Based on the interval calculated, if Penn State claimed that the true mean SATM score for the 2005
incoming freshman was 610 would you believe them? Explain. Yes, since the score of 610 is in the
interval.
4
5