STAT 30100: Elementary Statistical Methods I

STAT 30100: Elementary Statistical Methods I

Department of Mathematical Sciences

School of Science, IUPUI

Project # 3

FALL 2015, Total 25 points

Due Date:11/18/2015

Name:________________________________________________________

Important Instructions:

You must work independently to answer all questions. Provide complete and clear explanations to all questions. Graphs must be labeled for clear communication. (Remember the importance of context throughout!) Your project must be typed, easy to read and have the answer to each question identified by question number. Copy and paste the specified StatCrunch output, including graphs and number summaries, into the specified table. The indicated StatCrunch output MUST appear with the discussion, not as separate pages. Also, consider at least three decimals for your calculation throughout the project.

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 x at random anywhere between 6.5 and 7.5 ounces, inclusive. Suppose x has 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]

 =

 =

(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.

1. 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 μ is unknown, but  is still known. Calculate the margin of error(ME) ) for a 95% confidence interval for μ based on a sample of size n = 50. [1 pt]

(f) Imagine calculating a 95% CI for μ using each of the 5000 samples. Of the 5000 CIs, how many would you expect to contain the true value of μ? 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 μ? Is this consistent with your expectations in part (f)?

[½ + ½ = 1 pt]

[ StatCrunch: GO TO Data -> Compute expression

TYPE: between( , lower, upper)

NOTE the value of  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 ]

Problem # 2

Studies were conducted in Los Angeles to determine the carbon monoxide (CO) concentration near freeways. The basic technique used was to capture air samples in special bags and to then determine the carbon monoxide concentration by using a spectrophotometer. The measurements in ppm (parts per million) over a sampled period during the year can be found in the Excel file CARBON.xlsx via Oncourse. Find a 90% confidence interval for µ, the mean CO concentration (in ppm) during the year.

Consider the following steps for your answer.

1. Label the parameter.[1 pt]

1. Verify all required conditions. (using StatCrunc, produce a boxplot to verify the normality condition) [3 pts]

1. Interval calculations. (Both StatCrunch output and manual calculation must be reported; note that answer should be same in both cases) [2+2 = 4 pts]

StatCrunch output:

Manual calculation:

1. Interpret your result in part (iii) in the context of the problem.[1 pt]

Problem # 3

In a number of previous Phase I and II studies of male, non-insulin-dependent diabetic (NIDDM) patients conducted by Mylitech Biosystems, Inc., the mean body mass index (BMI) was found to be 28.4. An investigator has 16 male NIDDM patients enrolled in a new study and wants to know if the BMI from this sample is consistent with previous findings. BMI is computed as the ratio of weight in kilograms to the square of the height in meters. The Excel data file BMI.xlsx shows the BMI for 16 males. Conduct a hypothesis test at 5% level of significance to answer the question.

Consider the following steps for your hypotheses test.

1. Formulate the hypotheses. (Level the parameter, null hypothesis, alternative hypothesis)[2 pts.]

1. Test statistic. (Name of the test you are going to use and verify all required conditions for your test using the data; using StatCrunch, produce a boxplot to verify the normality condition) [3 pts.]

1. Calculation of your proposed test statistic and p-value. (Both StatCrunch output and manual calculation must be reported; note that answer should be same in both cases) [3 pts.]

StatCrunch output:

Manual calculation:

1. Decision and conclusion in the context. [2 pts.]