5.9 Continuous random variables: lab i

Continuous Distribution Lab

Name:

I - student learning outcomes:

• The student will compare and contrast empirical data from a random number generator with the Uniform Distribution.

Ii - theoretical distribution

The theoretical distribution of X is X~U (0, 1). Use it for this part. In theory,

μ = _________ σ = _________ 1 ^st quartile = _________

40th percentile = _________ 3rd quartile = _________Median = _________

Iii collect the data

Use Minitab to generate 100 values between 0 and 1 (inclusive). (Calc → Random Data → Uniform) Using Minitab, calculate the following (include the session window):

$\overline{x}$ = _________ s = _________ 1 ^st quartile = _________

40th percentile = _________ (justify) 3rd quartile = _________ median = _________

Iv - comparing the data

1. For each part below, use a complete sentence to comment on how the value obtained from the experimental data (see part III) compares to the theoretical value you expected from the distribution in section II. ( How it is reflected in the corresponding data. Be specific! )

a. minimum value:

b. first quartile:

c. median:

d. third quartile

e. maximum value:

f. width of IQR:

V - Plotting the Data and Interpreting the Graphs.

1. What does the probability graph for the theoretical distribution look like? Draw it here and label the axis.

2. Use Minitab to construct a histogram a using 5 bars and density as the y-axis value. Be sure to attach the graphs to this lab.

a. Describe the shape of the histogram. Use 2 - 3 complete sentences. (Keep it simple. Does the graph go straight across, does it have a V shape, does it have a hump in the middle or at either end, etc.? One way to help you determine the shape is to roughly draw a smooth curve through the top of the bars.)

b. How does this histogram compare to the graph of the theoretical uniform distribution? Draw the horizontal line which represents the theoretical distribution on the histogram for ease of comparison. Be sure to use 2 – 3 complete sentences.

3. Draw the box plot for the theoretical distribution and label the axis.

4. Construct a box plot of the experimental data using Minitab and attach the graph.

a. Do you notice any potential outliers? _________

If so, which values are they? _________

b. Numerically justify your answer using the appropriate formulas.

c. How does this plot compare to the box plot of the theoretical uniform distribution? Be sure to use 2 – 3 complete sentences.

Vi - increasing the sample size. repeat the simulation with 500 data values.

1. Using Minitab, calculate the following (include the session window):

$\overline{x}$ = _________ s = _________ 1 ^st quartile = _________

40th percentile = _________ (justify) 3rd quartile = _________ median = _________

2. Does this data appear to reflect the theoretical data more closely than the original? Be sure to use 2 – 3 complete sentences. ( Be specific. )

3. Create a histogram with 5 bars and using density for the y-axis and box plot for this data. (attach to this lab)

4. How do these compare to the theoretical distribution? Be sure to use 2 – 3 complete sentences. ( Be specific. )