STAT 30100: Elementary Statistical Methods I
Department of Mathematical Sciences
School of Science, IUPUI
Project # 2
Spring 2016, Total 30 points
Name: __________________________________________________________
Due Date: 02/26/2016
1
PROBLEM # 1
(Answer this question by typing directly, or by writing by hand and then scanning it into a pdf. A
calculator is allowed. But do not use StatCrunch to do this problem.)
A fast food restaurant just leased a new freezer and food fryer for three years. The service contract for
the freezer offers unlimited repairs for a fee of $125 a year plus a $35 service charge for each repair
needed. The restaurant’s research suggested that during a given year 80% of these freezers need no
repairs, 11% need to be serviced once, 5% twice, 4% three times, and none required more than three
repairs.
Let x be the number of repairs this kind of freezer will need each year.
(a) Write the probability distribution of x.
0
P(x)
1
2
80/100=80%
11/100=11%
5/100=5%
(b) Find the expected value of X and interpret the result. Show your work.
E(x)= 0(0.80)+1(0.11)+2(0.05)+3(0.04)=0.33 repairs
(c) Find the standard deviation of X. Show your work.
Var(x)= (0-0.33)^2(0.80) +(2-0.33)^2(0.05) + (3-0.33)^2(0.04)= 0.561
Standard Deviation= 0.561sqrt= 0.749
2
3
4/100=4%
PROBLEM # 2
An unprincipled used-car dealer sells a car to an unsuspecting buyer, even though the dealer
knows that the car will have a major breakdown within the next 8 months. The dealer provides
a warranty of 45 days on all cars sold. Let x represent the length of time until the breakdown
occurs. Assume that x is a uniform random variable with values between 0 and 8 months.
Now we will explore the following questions or concepts.
(a)
Find P(x ≥ 2)? (The answer for this part can be typed or written by hand and scanned.
But do not use Statcrunch.)
(b)
Using StatCrunch, generate 5000 samples of 50 observations each from the uniform
distribution. NO RESULTS TO BE REPORTED for this part.
[ StatCrunch: Data -> Simulate data -> Uniform (Rows: 5000, Columns: 1, Uniform
parameters a: 0, b: 8) -> Simulate ]
You will see a column named Uniform1 has been created on the StatCrunch spreadsheet.
(c) Using StatCrunch, find P(x>2)?
[ StatCrunch: Data -> Compute Expression
Under Expression: type “Uniform1”>=2
And under Column Label: type tally and then hit Compute!
Go to STAT->Tables->Frequency
Under Select Column(s): click on tally
And then hit Compute!
Now report the Relative Frequency corresponds to True ]
(d) Does your answer found in part (c) resonate with part (a)?
3
PROBLEM # 3
The height of adult women in the USA is normally distributed with mean 64.5 inches and standard
deviation 2.4 inches. Find the probability that a randomly chosen woman is between 63 and 70 inches
tall.
(Answer this question in the following two ways: Part A and Part B)
Part A: Answer the question by typing or by writing by hand and scanning.
i)
Formulate the problem. (Label the random variable x, state the parameters, and write an
appropriate probability statement)
ii) Sketch an appropriate normal distribution curve and shade the area under the curve that
corresponds to the problem in (i)
iii) Convert the x variable into z value.
iv) Put z value on the sketch in (ii), (writing z below x )
v) Use Standard Normal Table to find the desired probability.
4
Part B: Answer the question by Computer Simulation
Perform the following steps to answer this part and then complete the provided table below.
Step 1.
Simulate data from normal distribution with mean 64.5 inches and standard deviation
2.4 inches.
Open STATCRUNCH
Go to DATA->SIMULATE DATA-> NORMAL
Enter Rows:2000
Column: 1
Mean: 64.5
Std. Dev.: 2.4
You will see a column named Normal1 has been created on the StatCrunch spreadsheet.
Step 2.
Calculate summary statistics of the column "Normal1" column to check that it has mean
of approximately 64.5 and standard deviation of approximately 2.4. Also, create a
histogram of the data in "normal1".
GO TO Stat->Summary Stats ->Columns
CLICK ON Normal1 and then Compute!
(include the summary table and histogram in your project write up)
Step 3.
Calculate
x x 64.5
z
2 .4
for each row
GO TO Data->Compute expression,
TYPE UNDER EXPRESSION (Normal1-64.5)/2.4
UNDER NEW COLUMN NAME zscore
You will see a column named zscore has been created on the StatCrunch spreadsheet.
Step 4.
Calculate summary statistics of the column z-score to check that it has mean 0 and
standard deviation of 1 (as we have learnt in class that z-score has standard normal
distribution with mean 0 and standard deviation 1.
GO TO Stat->Summary Stats ->Columns
DOUBLE CLICK ON zscore and then Compute!
(include the summary table and histogram in your project write up)
Step 5.
Calculate the probability in Part A that P(* < z< ** )=?
Do NOT put * and **, instead use the values of * and ** found in
PART A.
GO TO Data->Compute expression
TYPE UNDER EXPRESSION ifelse(zscore >=* and zscore <= **, 1,0)
Again, * and ** should be replaced with z-score values found in part (A)
UNDER NEW COLUMN NAME prob and then Compute!
5
Note that prob=1 indicates (
* zscore * *
, that is, z scores between * and **);
and prob=0 indicates the opposite.
Step 6. To see answer of P(
* zscore * *
)=?
GO TO Stats->Tables->Frequency
CLICK ON prob and then Compute!
Now report the Relative Frequency corresponds to 1]
The answer you have now should be the same or pretty close to the answer you obtained
in PART A.
(include the frequency table in your project)
PART B Answer Sheet
Using StatCrunch, display the summary statistics and histogram you created in Step 2 (based
on the data stored in the "Normal1" column created in Step 1)
(Use as much space as you need to insert the items.)
Using StatCrunch, display the summary statistics and histogram you created in Step 4 (based
on the data stored in the Z-score column created in Step 3)
(Use as much space as you need to insert the items.)
6
Report the results that you obtained in Step 6 on the StatCrunch output.
How close are the results from Part A and Part B? Briefly explain why there may be some small differences.
What might you do differently to improve your result from Part B (to make it closer to the value found in
Part A)?
