Download Lab 2 solutions

Lab Activity 2: Probability Plots and Sampling Distributions
Feel free to work together on lab activities! Before you raise your hand to ask me, try to
learn from one another!
Normal Probability Plots
In Sections 7.3 and 7.4, a central question is whether observations come from a normal
distribution. Although there are various ways to test for normality, the graphical method
of normal probability plots is the best method in my opinion. This method is covered in
Section 4.6, but we’ll hit the high points below.
1. The idea is this: If a sample is really drawn from a normal distribution, then the pth
percentile of the sample should match closely with the pth percentile of a truly normal
population. Plotting one against the other should result in a straight line. Furthermore, it
doesn’t matter which “truly normal population” we compare with, since any normal
population is just a linear combination of any other.
Start Minitab.
Construct a normal probability plot “by hand” as follows: First, obtain a sample of size
100 from a normal distribution with mean 400 and standard deviation 20 and store it in
column C1. Go to Calc > Random Data > Normal…. These data will go on the X-axis
after they are sorted. Sort by going to Calc > Sort… and entering C1 in three separate
boxes: “sort”, “store in”, and “sort by”.
Next, obtain the percentiles of the standard normal distribution that will go on the Y-axis
as explained in the box at the bottom of page 192. Generate a series of values (i-.5)/n for
n=100 and i=1, …, 100 by going to Calc > Make Patterned Data > Simple set of
numbers…. The patterned data should go from .5/100 to 99.5/100 in steps of 1/100 and
they should be stored in column C2. Next, obtain the corresponding standard normal
quantiles by going to Calc > Probability Distributions > Normal…. Obtain the inverse
cumulative probability values corresponding to column C2, then store the results in
column C3.
Finally, with the data in C1 and the true standard normal percentiles in C3, produce a
normal probability plot, which is simply a scatterplot with the data on one axis (often the
x-axis) and the percentiles on the other axis (often the y-axis). A scatterplot is produced
by going to Graph > Plot…. Attach your normal probability plot below, noting its
straight-line appearance.
"Handmade" normal probability plot
3
2
C3
1
0
-1
-2
-3
350
400
450
C1
2. Do #88 (from Chapter 4) in Minitab. Ordinarily it’s not necessary to do probability
plots by hand! Enter the values in the fourth column of the worksheet. Go to Graph >
Probability Plot…. Enter “C4” in the “Variables” box and make sure that the
distribution is set to “Normal”.
To make the necessary calculations (square root and cubed root) in Minitab, go to Calc >
Calculator. To obtain the square root of the data values, choose “Square root” from the
list of functions, enter “C4” inside the parentheses, and “Store result in” C5. There is no
function in the list for cubed root, but you can raise C4 to the 1/3 power instead and store
the result in C6. In Minitab, the double-star (**) button is the same as a ^ (raises a value
to a power).
Copy and paste your three normal probability plots, and comment on them below.
(a) The plot below shows an obvious curvature, indicating that normality is
probably not a safe assumption.
(b) The square-root-transformed plot below looks less curved than the original,
though there is still a slight curvature evident.
(c) The cube-root-transformed plot below is the straightest of the three. If we’re
trying to transform the data to normality, the cube root looks like the best
bet.
Sampling distributions
We have been learning about the distribution of X . The distributions of statistics such as
X are often referred to sampling distributions, because their randomness is solely the
result of the fact that they are based on the values found in a random sample.
3. IQ scores for PSU students are normally distributed with mean 100 and standard
deviation 15. Using the empirical rule (p. 167), sketch by hand a picture of this
distribution, giving the correct scale on the horizontal axis.
Here is a sketch:
55
70
85
100
115
130
145
What would the sampling distribution of the sample mean IQ look like for repeated
samples of 16 PSU students? Go to Calc > Random Data > Normal…. Generate
10,000 rows of data, enter the mean and standard deviation of IQ scores, and store the
values in columns C7-C22. (Note: You can actually type “C7-C22”.)
Each row (between C7-C22) is a sample of size 16 from the population. To calculate the
mean IQ for each sample, go to Calc > Row Statistics…. You want to calculate the
mean of C7-C22 (“Input variables” box) and store the result in column C23.
Now, create a histogram of the values in C23 (Graph > Histogram…). This histogram
gives you an approximation of the sampling distribution of the sample mean IQ ( X ) for
repeated samples of 16 PSU students! Print and attach this histogram. Below,
describe the shape, location, and spread of the histogram (in comparison to the picture of
the population distribution you drew above).
Here is the histogram:
600
Frequency
500
400
300
200
100
0
85
95
105
115
C23
Now, using the theory presented at the end of lecture on Wednesday, give by hand the
approximate shape, the mean, and the standard deviation of the sampling distribution of
X for repeated samples of size n = 16. (Hint: CENTRAL LIMIT THEOREM!)
According to the central limit theorem, X is approximately normally distributed (in
this case, it’s EXACTLY normally distributed because the population itself is
normally distributed). The mean and standard deviation of the distribution of
X are 100 and 15/sqrt(16)=3.75, respectively.