Statistics: Lab #3

 

Chapter 6: Continuous Random Variables and Probability and Chapter 7: Sampling Distributions

You will have two files for this lab, they are separated below as parts A and B.

 

Part A

Population Data

  1. Your first step should be to enter Excel. Begin a file called "Stat Lab 3 Exercise, yourname" and save it. Also open Winter 2007 Student Database. Throughout this assignment you will be asked several questions. When asked to do so, please respond on a separate document or within your Excel file. You may wish to consult the Winter 2007 survey to remind yourself of how the questions were phrased.
  2. Type your name in A1 of Stat Lab 3 and the date in cell A2.
  3. Copy the EXERCISE column of the database to your Stat Lab 3 file in column B. Recall that the students were asked how many hours of exercise they typically received in a week. For the time being, consider this to be our population.
  4. Scroll down, looking for missing data. If you see any, delete the entire row.
  5. Create a table of descriptive statistics for the data and place the descriptive statistics in a new worksheet, separate from your column. Tools, Data Analysis, Descriptive Statistics. See earlier labs for more details. Interpret the mean and standard deviation and standard error. Why is there such a difference between the standard deviation and standard error? (4 points)

Question: Can we consider this to be a continuous random variable? Which probability distribution would you expect this random variable to represent? Explain your responses. Keep in mind our assumption of what is serving as the population for now. (2 points)

1. To investigate the shape of this distribution, create a histogram with bins 1 hour wide. Put the histogram in a separate worksheet so you can see it in all its glory. Again, see earlier labs for more details. Adjust the chart so it looks professional with appropriate titles and axes. (2 points)

Question: Describe the probability distribution of your population. Does your histogram approximate a normal distribution? If not, which continuous probability distribution does it most represent? Explain. (3 points)

 

Population Probability Problem

In your separate document, set up and solve each of the following 3 probability problems, including a diagram showing the area you wish to find. (3 points each)

1. QUESTION: What is the probability that one randomly selected Hanover student will exercise less than 12 hours? By now you probably have realized that you need the exponential function to solve.

2. Do the above problem using the Function Wizard. Insert the formula in C2 at the top of your Lab 3 worksheet. Clearly label C1 "Exercise1".

 

The exponential function in Excel requires a little explaining. There are three boxes to complete in the Function Wizard. The box for x is the value of the random variable. The box for lambda is actually the inverse of the population mean. For example, if mu was equal to 4, you would enter .25 for lambda. The cumulative box requires a �true� for our purposes. By entering "TRUE", you are telling Excel to compute the probability of x taking that value or less. You may need to adjust the result given in the Function Wizard to answer your specific question.

 

1. QUESTION: What is the probability that one randomly selected Hanover student will exercise between 10 hours and 5 hours in a typical week?

2. Do the above problem using the Function Wizard. Insert the formula near the previous function at the top of your Lab 3 worksheet. Clearly label an adjoining cell "Exercise2".

3. QUESTION: What is the probability that a randomly selected Hanover student would exercise more than 15 hours per week?

4. Do the above problem using the Function Wizard. Insert the formula near the previous function at the top of your Lab 3 worksheet. Clearly label an adjoining cell "Exercise3".

 

Save your file. Remember you only have to submit hard copies of your written solutions and responses to questions above. Show your work. The Excel file must be emailed to me as an attachment.

 

 

PART B

 

Sampling from a Population.

  1. Begin an Excel file called "Stat Lab 3 Age, yourname" and save it. Throughout this assignment you will be asked several questions.
  2. Type your name in A1 and the date in A2.
  3. Copy the AGE column from the database to your "Stat Lab 3" file in column B. You may recall that this question asked a student for his/her age in years and months. For now, these students will serve as our population.
  4. Delete any rows that include missing values.
  5. After deleting any missing values, in cells near the top of the page, use formulas to calculate the mean (=average(data range)) and standard deviation (=stdevp(data range)) of the population. (2 points)

Drawing a random sample.

  1. Excel can compute a random sample from a population. Enter in cell E1, "Samp1".
  2. From the menu, choose Tools, Data Analysis, Sampling. Your input range is your population data. Number of samples is n=30, output range is E2..E31. Make sure you choose a random sample. Hit OK, and you should have 30 values in cells E2 through E31.
  3. In cells D33 and D34, type "Mean" and "StdDev" respectively.
  4. In cells E33 and E34, write formulas that calculate the sample mean and sample standard deviation. The formula for the mean is the same as above, but the sample standard deviation is =Stdev(data range). NOTE, make sure you use this formula for the standard deviation and not STDEVP that assumes this is a population. You are not dealing with a population anymore.

Draw another random sample of n=30, call it "Samp2" and place it in the next column. Repeat step 4 for your second sample, placing your formulas in F33 and F34. Repeat one more time so you have 3 samples of size n=30. (3 points for three random samples and correct formulas)

1. How many different samples of 30 could be drawn from this population? Should the values in cells E33 and E34 differ from values in F33, F34, and G33 and G34, and from your population mean and standard deviation? Why or why not? (3 points)

2. If you sampled many times (n=30) from this population, you would produce a sampling distribution for the mean age for students ().

Question: Should you consider this population finite or infinite? Explain. What is the expected value (mean) and standard error of the sampling distribution?

For the next 3 questions, you MUST set up and calculate the same question using Z-scores and the standard normal table. Include these solutions, including a diagram, in your written/typed responses. (3 points each)

3. Question: Using the expected value and standard error of the sampling distribution, calculate the probability that a random sample of n=30 will yield a sample mean that has a sampling error of less than 6 months. Again, do this problem first with Z-scores and the standard normal probability table, then with Excel and the NORMDIST formula. Put output in a cell near the top of the sheet, labeled clearly as Age1. There is an example of how to use this formula in the appendix to chapter 6.

 

4. Question: What is the probability that a random sample of n=30 is drawn from this population and the sample mean age is greater than 21? Again, do this problem first with Z-scores and the standard normal probability table, then with Excel and the NORMDIST formula. Put output in a cell near the top of the sheet, labeled clearly as Age2.

 

5. Question: How low would a sample mean need to be such that only 3% of sample means lie below it? Again, do this problem first with Z-scores and the standard normal probability table, then with Excel and the NORMINV formula. Put output in a cell near the top of the sheet, labeled clearly as Age3.

 

 

Save your file. Remember you only have to submit hard copies of your written solutions and responses to questions above. The Excel file must be emailed to me as an attachment.

 

OK, you have carefully completed both parts A and B to this lab. You have double-checked that you have all of your typed/written responses to questions. Bring responses and typed comments to class on the due date. Your emailed files must also be sent by the beginning of class on the due date. When you email your file, make sure your name is in the filename. Late assignments or those sent incorrectly will be penalized. Check the syllabus or ask for help if you are uncertain of how to do any of this.