Download Lab 7. Confidence intervals (C.I.`s) for the mean (known σ) 1

Lab 7. Confidence intervals (C.I.’s) for the mean
(known σ)
www.nmt.edu/~olegm/283labs/Lab7stat.pdf
Note: the menus and other things you will read or type on the computer are in italics. Attach the
printouts whenever needed.
This week, we will look at how Minitab computes confidence intervals for the mean,
and conduct a simulation study to better understand their properties.
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Confidence intervals
The C% confidence interval (C.I.) for the unknown population mean µ is based on
the sample mean X:
σ
X ± z∗ √
n
Somewhat artificially, we will assume that σ (the population standard deviation) is
known. On the other hand, when n is large, then the sample standard deviation s is
a fairly good estimate for σ.
As an example of Minitab computing the C.I., let’s consider the data set of the times
(in minutes) it takes a certain professor to walk to work (see walkto.csv). We would
like to estimate the average time µ it takes him to walk to work. Let’s assume that
σ = 0.32.
Use Stat → Basic Stat → 1-sample Z and then
The default confidence level C is 95%. In order to choose a different level, click on
Options.
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Problem 1
(a) Construct an interval that contains 90% of all “walk to work” times. [Hint: To
compute, say, 80-th percentile of a sample, use Calculator, under Expression, enter
percentile(C1,0.8)]
(b) Construct a 90% C.I. for the “true” average time to walk to work; assume
σ = 0.32. Compare this with (a).
(c) Construct a 99% C.I. for the “true” average time to walk to work. Compare
this with (b).
Problem 2
We are studying precipitation in Southwest, which is really important for the water
resources. The file Silverton3.csv contains the total precipitation values for the
month of March at Silverton, CO, for the period 1950-1999. We would like to estimate
the mean monthly precipitation.
(a) Obtain a histogram of March totals.
(b) Explain why, even though the distribution is skewed, the normal distribution
used to construct the C.I. is still applicable.
(c) Construct the 95% C.I. for the “true” average March total precipitation1 (assume that σ = 1.2). Try to explain your result to a non-specialist, in plain
English.
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Behavior of C.I.’s
To better understand how C.I.’s work and give some substance to the “confidence”
concept, let’s consider a simulation example.
Suppose we are repeatedly drawing samples of n = 5 from a population of stock
returns, having the normal distribution, with the mean (somewhat optimistically)
µ = 13 and standard deviation σ = 17.
If we construct a 90% C.I. for each sample, what fraction of them will contain the
population mean µ?
We will simulate 100 such C.I.’s and observe the results.
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It’s a bit awkward to say “average total” but you are asked to estimate the value of the monthly
total precipitation, averaged over month of March of all years.
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Problem 3
Step 1. In rows 1-100 and columns C4-C8, put 100 samples of size 5 from the population
of stock returns. In column C1, put the sample averages (see Lab 5 for detailed
instructions).
Step 2. In column C2, put the lower ends of the 90% C.I.’s using Calculator and the
σ
formula X ± z ∗ √ . Assume z ∗ = 1.645 for 90% confidence.
n
Similarly, put the upper end into the column C3.
Graph the first five C.I.’s you obtained, with the vertical line showing the true
µ = 13:
Example
●
●
●
−10
0
10
20
30
40
What is the probability that all five of them will cover the “true” mean?
Step 3. Browsing through the columns C2 and C3, count how many of your C.I.’s did
not capture the “true” mean of 13. Then find the number of those that did.
Step 4. Using the same set of samples compute 95% C.I.’s (z* = _____?)
(Repeat steps 2, 3.) Fill in the table below.
Theoretical number of C.I.’s
Actual number of C.I.’s that
that should capture the
captured the “true” mean
“true” mean
90% C.I.
95% C.I.
What have you learned about the behavior of C.I.’s?
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