Lab on Confidence Intervals
The story: Studies have shown that the random variable X, the processing time required to do a
multiplication on a new 3-D computer, is normally distributed with mean  and standard deviation 2
microseconds. A random sample of 16 observations is to be taken (i.e. 16 random multiplications will be
performed and the time that it takes to perform each one of them will be annotated). The sample mean
will be calculated.
Part 1- The very basics
1. Before we collect the random sample, what can we say about the sampling distribution of x ?
2. Before we collect the random sample, what is the probability that the sample mean x will have a
value that is
2
2
greater than or equal to   1.96
, and less than or equal to   1.96
16
16
In other words, what is the probability that the sample mean is within 0.98 microseconds
2
( 1.96
 0.98 ) of the true (unknown) mean time?
16
3. A random sample was taken (i.e.16 multiplications were done) and the processing times were as
follows:
42.65 45.15 39.32 44.44 41.63 41.54
41.59 45.68 46.50 41.35 44.37 40.27
43.87 43.79 43.28 40.70
Round the times to the nearest tenth (e.g. 42.65 ~
42.6) and do the stem-and-leave display
Do the stem and leaf display.
x
Calculate the sample mean
39
40
41
42
43
44
45
46
4. Now we will use our knowledge of the value of the sample mean for this particular sample and the
relationship between sample means (in general) and population mean (see questions 1 and 2 of
this lab) to make an intelligent guess (estimation) of the population mean. Use the formula
x  z*

to calculate a 95 % confidence interval for the population mean  (Show your work)
n
(
,
)
From the Minitab menu, use STAT>BASIC STATISTIC>ONE SAMPLE Z to check your calculations.
Calculate the width of the confidence interval _______________
5. Interpret the confidence interval you just found:
We are 95% confident that the true mean processing time required to do a multiplication on a new 3D computer lies between __________ and __________ microseconds.
(The meaning of this is the following: Think of all the possible samples (all the possible sets of 16 multiplications) that we could
have been done with this type of computer; 95% of those samples would have produced confidence intervals that contain the
true mean, the other 5% would have been off)
Part 2- Additional issues: how the width of the confidence interval is influenced
by sample size, confidence and variability. Finding the appropriate sample size.
6. Assume that the sample size was 100 instead of 16 but that the sample mean was still the same
sample mean you calculated in question 3
Calculate the 95% confidence interval in this case
x  z*

n
Calculate the width of the confidence interval ___________
Now that the sample size has increased, your confidence interval is narrower or wider than the
confidence interval in question 4? ______
Do you think this will always happen? YES NO Why?
7. Assume that you have the same sample that was described in question 3 but that the standard
deviation of the processing time is 4 instead of 2. Calculate the 95% confidence interval in this case
x  z*

n
Calculate the width of the confidence interval ___________
Now that the variability has increased, your confidence interval is narrower or wider wider than the
confidence interval in question 4? ______
Do you think this will always happen? YES NO Why?
8. Now we want to calculate the 90% confidence interval. Keep the same sample mean and sample size
(16) of the original problem and calculate the 90% confidence interval
x  z*

n
Calculate the width of the confidence interval ___________
Now that the confidence interval has decreased, your confidence interval is narrower or wider wider than
the confidence interval in question 4? ______
Do you think this will always happen? YES NO Why?
9. Summarize your findings in questions 6, 7 and 8 filling the blank cells on the right with either 'narrower'
or 'wider'.
When…..
The Confidence interval becomes…
….the sample size increases
….the variability in the population is larger
….the confidence decreases
10. A rather picky person is not satisfied with the precision of the confidence interval you found in
question 3. He says that he would like to see have a margin of error of only 1/2 second and still have 95%
confidence. How many multiplications we would need to perform in the computer in order to produce a
confidence interval that satisfies him.
2
 z * 
Use the formula n  
 . Show your work.
 m 