This document provides examples for a statistics lab involving random variables, probability distributions, and confidence intervals. [1] It asks whether rolling a die is a discrete or continuous random variable and to calculate the mean and standard deviation of rolling a 4-sided die. [2] The mean of rolling the 4-sided die is calculated as 2.5 and the standard deviation is calculated as 1.118. [3] It then has the student calculate descriptive statistics like the mean and median of various data samples and compares how centered they are around the true parameter.

1. B Heard
(Don’t copy or post without my permission,
students may download a copy for personal use)

2.  Thefollowing are examples of the Week 6
Lab, please note that I CHANGED the data!

3. 1. When rolling a die, is this an example of a
discrete or continuous random variable?
Explain your reasoning.
 You should be able to answer this
 Read about discrete and continuous variables

4. 2. Calculate the mean and standard deviation
of the probability distribution created by
rolling a four sided die. Either show work
or explain how your answer was calculated.
 For the sake of example, I am going to use a
four-sided die (your lab deals with a six-sided
die)

5.  In other words, my die would look like a pyramid.
You could roll a 1,2,3 or 4 and they are all equally
likely.

6.  To get the mean
Mean = ƩxP(x) = 1(1/4)+2(1/4)+3(1/4)+4(1/4)
= 10/4 or 2.5
Die Value times the probability
There are four equally likely sides, so
the probability for each would be ¼.

7.  To get the standard deviation
St. Dev. = sqrt((1-2.5)2(1/4)+ (2-2.5)2(1/4)+
(3-2.5)2(1/4)+ (4-2.5)2(1/4) = sqrt(1.250) =
1.118
REMEMBER WE WERE DEALING WITH A FOUR
SIDED DIE

8. 3. Give the mean for the mean column of the
Worksheet. Is this estimate centered about
the parameter of interest (the parameter of
interest is the answer for the mean in
question 2)?
I CHANGED THE DATA

10.  Type the word “Mean” to the right of Die10

11.  Pullup Calc > Row Statistics and select the
radio-button corresponding to Mean. For Input
variables: enter all 10 rows of the die data. Go
to the Store result in: and select the Mean
column. Click OK and the mean for each
observation will show up in the Worksheet.

13.  NowI have the means calculated in the “Mean”
column

14.  We also want to calculate the median for the
10 rolls of the die. Label the next column in
the Worksheet with the word Median. Repeat
the above steps but select the radio-button
that corresponds to Median and in the Store
results in: text area, place the median
column.

16.  Same process as Mean, except choose Median
radial button and change Store result to
Median by double clicking on Median in your
list on the left.

19.  Calculate descriptive statistics for the mean
and median columns that where created
above. Pull up Stat > Basic Statistics >
Display Descriptive Statistics and set
Variables: to mean and median. The output
will show up in your Session Window. Print
this information.

22. So number 3 wants the “Mean of Means”
(From previous chart)
Mean of means = 2.64 yes, this is generally
centered around the parameter of interest
(the 2.5 I calculated in number 2)
Honestly I would have liked for it to be a tad
closer (but remember I changed data at a
whim and probably put too many 3’s and
4’s in rather than actually rolling a 4-sided
die)

23. 4. Give the mean for the median column of the
Worksheet. Is this estimate centered about
the parameter of interest (the parameter of
interest is the answer for the mean in
question 2)?
Mean of medians = 2.775, this is definitely
farther away from the parameter of
interest (the 2.5 I calculated
mathematically in number 2)

24. 5. Give the standard deviation for the mean
and median column. Compare these and be
sure to identify which has the least
variability?
StDev of means = 0.3202
StDev of medians = 0.472
The standard deviation of the means is
smaller, thus it has less variability than the
medians. This would mean the data for the
means is grouped closer together.

25. 6. Based on questions 3, 4, and 5 is the mean
or median a better estimate for the
parameter of interest? Explain your
reasoning.
In my case, the mean seems to be a better
estimate because it is closer to the
mathematically calculated mean and the
standard deviation is less than that of the
medians meaning the means are grouped
closer together.

26. 7. Give and interpret the 95% confidence
interval for the hours of sleep a student gets.
I changed the data! So these are not the
answers to your lab!

27.  We are interested in calculating a 95%
confidence interval for the hours of sleep a
student gets. Pull up Stat > Basic Statistics >
1-Sample t and set Samples in columns: to
Sleep. Click the OK button and the results
will appear in your Session Window.

30. Therefore, the 95% confidence interval would be (6.232, 8.168). I
would be 95% confident that the true mean number of hours a student
sleeps would be between those two values.
I changed the data! So these are not the answers to your lab!

31. 8. Give and interpret the 99% confidence
interval for the hours of sleep a student gets.
(Same approach as number 7, but MAKE SURE
you click options and change confidence to
99%)

32. 9. Compare the 95% and 99% confidence
intervals for the hours of sleep a student
gets. Explain the difference between these
intervals and why this difference occurs.
The 99% confidence interval is wider than
the 95%, which is always the case. To get
more confidence, the bounds widened (i.e.
it’s the only way you can get more
certainty).

33. Iwill post a link to these charts in the
“Statcave” at www.facebook.com/statcave
 See you next time!