Statistical Concepts:

• Probability

• Binomial Probability Distribution

Calculating Binomial Probabilities

Open a new MINITAB worksheet.

We are interested in a binomial experiment with 10 trials. First, we will make the probability of a success ¼. Use MINITAB to calculate the probabilities for this distribution. In column C1 enter the word ‘success’ as the variable name (in the shaded cell above row 1. Now in that same column, enter the numbers zero through ten to represent all possibilities for the number of successes. These numbers will end up in rows 1 through 11 in that first column. In column C2 enter the words ‘one fourth’ as the variable name. Pull up Calc > Probability Distributions > Binomial and select the radio button that corresponds to Probability. Enter 10 for the Number of trials: and enter 0.25 for the Event probability:. For the Input column: select ‘success’ and for the Optional storage: select ‘one fourth’. Click the button OK and the probabilities will be displayed in the Worksheet.

Now we will change the probability of a success to ½. In column C3 enter the words ‘one half’ as the variable name. Use similar steps to that given above in order to calculate the probabilities for this column. The only difference is in Event probability: use 0.5.

Finally, we will change the probability of a success to ¾. In column C4 enter the words ‘three fourths’ as the variable name. Again, use similar steps to that given above in order to calculate the probabilities for this column. The only difference is in Event probability: use 0.75.

Plotting the Binomial Probabilities

1. Create plots for the three binomial distributions above. Select Graph > Scatter Plot and Simple then for graph 1 set Y equal to ‘one fourth’ and X to ‘success’ by clicking on the variable name and using the “select” button below the list of variables. Do this two more times and for graph 2 set Y equal to ‘one half’ and X to ‘success’, and for graph 3 set Y equal to ‘three fourths’ and X to ‘success’. Paste those three scatter plots below.

Calculating Descriptive Statistics

Open the class survey results that were entered into the MINITAB worksheet.

2. Calculate descriptive statistics for the variable where students flipped a coin 10 times. Pull up Stat > Basic Statistics > Display Descriptive Statistics and set Variables: to the coin. The output will show up in your Session Window. Type the mean and the standard deviation here.

Mean:

Standard deviation:

Short Answer Writing Assignment – Both the calculated binomial probabilities and the descriptive statistics from the class database will be used to answer the following questions.

3. List the probability value for each possibility in the binomial experiment that was calculated in MINITAB with the probability of a success being ½. (Complete sentence not necessary; type with three decimal places.)

P(x=0) P(x=6)

P(x=1) P(x=7)

P(x=2) P(x=8)

P(x=3) P(x=9)

P(x=4) P(x=10)

P(x=5)

4. Give the probability for the following based on the MINITAB calculations with the probability of a success being ½. (Complete sentence not necessary; type with three decimal places.)

P(x≥1) P(x<0)

P(x>1) P(x≤4)

P(4<x ≤7) P(x<4 or x≥7)

5. Calculate the mean and standard deviation (by hand) for the MINITAB created binomial distribution with the probability of a success being ½. Either show work or explain how your answer was calculated. Mean = np, Standard Deviation =

Mean =

Standard deviation =

6. Calculate the mean and standard deviation (by hand) for the MINITAB created binomial distribution with the probability of a success being ¼ and compare to the results from question 5. Mean = np, Standard Deviation =

Mean =

Standard deviation =

Comparison:

7. Calculate the mean and standard deviation (by hand) for the MINITAB created binomial distribution with the probability of a success being ¾ and compare to the results from question 6. Mean = np, Standard Deviation =

Mean =

Standard deviation =

Comparison:

8. Explain why the coin variable from the class survey represents a binomial distribution.

9. Give the mean and standard deviation for the coin variable and compare these to the mean and standard deviation for the binomial distribution that was calculated in question 5. Explain how they are related. Mean = np, Standard Deviation =

Mean =

Standard deviation =

Comparison: