# 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 ]

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