Calculating Binomial Probabilities Coursework Example

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Calculating Binomial Probabilities
Open a new Excel worksheet.
Open spreadsheet
In cell A1 type “success” as the label
Under that in column A, type 0 through 10 (these will be in rows 2 through 12)
In cell B1, type “one fourth”
In cell B2, type “=BINOM.DIST(A2,10,0.25,FALSE)” [NOTE: if you have Excel 2007, then the formula is BINOMDIST without the period]
Then copy and paste this formula in cells B3 through B12
In cell C1, type “one half”
In cell C2, type “=BINOM.DIST(A2,10,0.5,FALSE)”
Copy and paste this formula in cells C3 through C12
In cell D1 type “three fourths”
In cell D2, type “=BINOM.DIST(A2,10,0.75,FALSE)”
Copy and paste this formula in cells D3 through D12
Plotting the Binomial Probabilities
Create plots for the three binomial distributions above. You can create the scatter plots in Excel by selecting the data you want plotted, clicking on INSERT, CHARTS, SCATTER, then selecting the first chart shown which is dots with no connecting lines. 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 in the grey area below. (9 points)
Graph 1: One Fourth

Graph 2: One Half

Graph 3: Three Fourths

Calculating Descriptive Statistics
You will use the same class survey results that were entered into the worksheet for the Week 2 iLab Assignment for question 2.
Calculate descriptive statistics for the variable (Coin) where each of the thirty-five students flipped a coin 10 times.

Round your answers to three decimal places and type the mean and the standard deviation in the grey area below. (5 points)
Mean: 4.520
Standard deviation:1.499
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. Round all numeric answers to three decimal places.
List the probability value for each possibility in the binomial experiment calculated at the beginning of this lab, which was calculated with the probability of a success being ½. (Complete sentence not necessary; round your answers to three decimal places) (8 points)
P(x=0) 0.001 P(x=6) 0.205
P(x=1) 0.010
P(x=7) 0.117
P(x=2) 0.044 P(x=8) 0.044
P(x=3) 0.117 P(x=9) 0.010
P(x=4) 0.205 P(x=10) 0.001
P(x=5) 0.246 Give the probability for the following based on the calculations in question 3 above, with the probability of a success being ½. (Complete sentence not necessary; round your answers to three decimal places) (8 points)
P(x≥1) 0.999 P(x<0) 1
P(x>1) 0.989 P(x≤4) 0.377
P(4<x ≤7) 0.568 P(x<4 or x≥7) 0.227

Calculate (by hand) the mean and standard deviation for the binomial distribution with the probability of a success being ½ and n = 10. Either show work or explain how your answer was calculated. Use these formulas to do the hand calculations: Mean = np, Standard Deviation = (4 points)
Given that,
Mean = np and From the question n=10, P=12mean=12×10=5Standard Deviation =
=10×0.51-0.5=2.5Calculate (by hand) the mean and standard deviation for the binomial distribution with the probability of a success being ¼ and n = 10. Write a comparison of these statistics to those from question 5 in a short paragraph of several complete sentences. Use these formulas to do the hand calculations: Mean = np, Standard Deviation = (4 points)
Mean = np and From the Question p=14, n=10
mean=14×10 = 2.5
Standard Deviation =
=2.51-0.25=1.875
Comparison: Statistics in question 5 show a higher mean and standard deviation than those in question 6. The reason lies on the dispersion and volume of data in used in each case. While question 6 only used a quarter of the data set, question 5 used a half of the data set which was highly dispersed.
Calculate (by hand) the mean and standard deviation for the binomial distribution with the probability of a success being ¾ and n = 10. Write a comparison of these statistics to those from question 6 in a short paragraph of several complete sentences. Use these formulas to do the hand calculations: Mean = np, Standard Deviation = (4 points)
Mean = np: , n=10, p=0.75
Mean=34×10 =7.5Standard Deviation =
=7.51-0.75=1.369Comparison: Statistics in question 7 show a higher mean, but lower standard deviation than statistics in question 6.
Using all four of the properties of a Binomial experiment (see page 201 in the textbook) explain in a short paragraph of several complete sentences why the Coin variable from the class survey represents a binomial distribution from a binomial experiment. (4 points)
The coin variable in the class survey is a true representation of a binomial distribution because of four reasons. (1) The coin flipping experiment consisted of differently repeated trials. Each of the 35 students flipped the coin 10 times. (2) Each trial only resulted in two likely outcomes – heads or tails. (3) The probability of obtaining either head or tail in the experiment were similar (0.5). (4) The outcomes of all trials were independent; that is, probability getting tails on one trial did not affect whether the surveyor got tails on other trials.
Compare the mean and standard deviation for the Coin variable (question 2) with those of the mean and standard deviation for the binomial distribution that was calculated by hand in question 5. Explain how they are related in a short paragraph of several complete sentences. (4 points)
Mean from question #2: 4.520, if rounded would 5
Standard deviation from question #2:1.499Mean from question #5:5Standard deviation from question #5:2.5Comparison and explanation: The values of standard deviation and mean in both questions 2 and 5 are close indicating that the values can be similar and if some variables are to change.