STAT 30100: Elementary Statistical Methods I Project # 3 FALL 2015

QUESTION

Problem # 1

The manager of a
local soft-drink bottling company believes that when a new beverage-dispensing
machine is set to dispense 7 ounces, it in fact dispenses an amount xat
random anywhere between 6.5 and 7.5 ounces, inclusive. Suppose xhas a
uniform probability distribution.

Now we will explore the following questions or ideas.

(a) Using calculator, find the mean and
standard deviation of the variable x. [½
+ ½ = 1 pt]

m =

s =

(b) Generate 5000 samples of 50 observations
each from the uniform distribution.
NO RESULTS TO BE REPORTED.

[ StatCrunch: Data -> Simulate data -> Uniform (Rows: 5000, Columns: 50, Uniform parameters
a: 6.5, b: 7.5) -> Simulate ]

Note that each row indicates a sample
consisting of 50 data values.

(c) Calculate
the 5000 sample means. NO RESULTS TO BE REPORTED

[ StatCrunch: Stat-> Summary Stats ->Rows (select all
variables from left to right)
Select next
Keep
only mean on the right side of the box by clicking each on the left
side except mean
Check mark at “Store output in data
table”
Calculate

You will see a new column has been added named “Row Mean” in the
dataset. This Row
mean is the mean of each sample.

(d) By the Central Limit Theorem, we know the
variable “Row Mean” has an approximate Normal distribution.

What are the parameter
values? [½ + ½ = 1 pt]
[ StatCrunch: Go to Stat -> Summary Stats -> Column
(select Row Mean) -> Calculate
Report
mean and standard deviation only ]

(ii) Draw a histogram of “Row Mean” with the
appropriate Normal PDF overlaid. Does this graph look as you would have
expected? [½ + ½ = 1 pt]

[ StatCrunch:
GO TO Graphics -> Histogram -> Choose the variable Row Mean -> Next ->
Next->
Choose Overlay density Normal -> Next

X axis label: Means of 5000 samples
Y
axis label: Frequency
Title:
Histogram of Normal Distribution Exploration by the CLT ]

Click
on Create Graph.

(e) Now
pretend that ? is unknown, buts is still known. Calculate the margin of error(ME)
) for
a 95% confidence interval for ? based
on a sample of size n = 50. [1 pt]

(f) Imagine calculating a 95% CI for ? using
each of the 5000 samples. Of the 5000 CIs, how many would you expect to contain
the true value of ?? Explain. [ ½ pt]

(g) Calculate the lower and upper limits of
the 95% CIs.NO
RESULTS TO BE REPORTED
[ StatCrunch: GO
TO Data->Compute expression -> “Row Mean”-ME [ ½ pt]
GIVE
New column name: lower
SIMILARLY
DO “Row Mean”+ME
GIVE New column name: upper ]

(h) How many of the 5000 CIs covered ?? Is
this consistent with your expectations in part (f)?
[½ + ½ = 1 pt]

[ StatCrunch: GO
TO Data -> Compute expression
TYPE:
between(m, lower, upper)
NOTE the value
ofm comes from part (a)
GIVE
new column name: tallycount

Again,
GO TO Data -> Compute expression
ifelse(tallycount=”true”,
1,0)
GIVE
new column name: tallycount1
GO TO Stat -> Tables-> Frequency (select
variable tallycount1) and see the percentage for 1 ]

ANSWER:

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