## B.9 Exercises

The files referenced in the exercises are available in the files associated with this chapter.

**Exercise B.1**The observations available in the text file,

*problem1.txt*, represent the count of the number of failures on a windmill turbine farm per year. Using the techniques discussed in the chapter recommend an input distribution model for this situation.

**Exercise B.2**The observations available in the text file,

*problem2.txt*, represent the time that it takes to repair a windmill turbine on each occurrence in minutes. Using the techniques discussed in the chapter recommend an input distribution model for this situation.

**Exercise B.3**The observations available in the text file,

*problem3.txt*, represent the time in minutes that it takes a repair person to drive to the windmill farm to repair a failed turbine. Using the techniques discussed in the chapter recommend an input distribution model for this situation.

**Exercise B.4**The observations available in the text file,

*problem4.txt*, represent the time in seconds that it takes to service a customer at a movie theater counter. Using the techniques discussed in the chapter recommend an input distribution model for this situation.

**Exercise B.5**The observations available in the text file,

*problem5.txt*, represent the time in hours between failures of a critical piece of computer testing equipment. Using the techniques discussed in the chapter recommend an input distribution model for this situation.

**Exercise B.6**The observations available in the text file,

*problem6.txt*, represent the time in minutes associated with performing a lube, oil and maintenance check at the local Quick Oil Change Shop. Using the techniques discussed in the chapter recommend an input distribution model for this situation.

**Exercise B.7**If \(Z \sim N(0,1)\), and \(Y = \sum_{i=1}^k Z_i^2\) then \(Y \sim \chi_k^2\), where \(\chi_k^2\) is a chi-squared random variable with \(k\) degrees of freedom. Setup an model to generate \(n\) = 32, 64, 128, 256, 1024 observations of \(Y\) with \(k = 5\). For each sample, fit a distribution to the sample.

**Exercise B.8**Consider the following sample (also found in

*problem8.txt*) that represents the time (in seconds) that it takes a hematology cell counter to complete a test on a blood sample.

23.79 | 75.51 | 29.89 | 2.47 | 32.37 |

29.72 | 84.69 | 45.66 | 61.46 | 67.23 |

94.96 | 22.68 | 86.99 | 90.84 | 56.49 |

30.45 | 69.64 | 17.09 | 33.87 | 98.04 |

12.46 | 8.42 | 65.57 | 96.72 | 33.56 |

35.25 | 80.75 | 94.62 | 95.83 | 38.07 |

14.89 | 54.80 | 95.37 | 93.76 | 83.64 |

50.95 | 40.47 | 90.58 | 37.95 | 62.42 |

51.95 | 65.45 | 11.17 | 32.58 | 85.89 |

65.36 | 34.27 | 66.53 | 78.64 | 58.24 |

Test the hypothesis that these data are drawn from a uniform distribution at a 95% confidence level assuming that the interval is between 0 and 100.

The interval of the distribution is between

*a*and*b*, where*a*and*b*are unknown parameters estimated from the data.Consider the following output for fitting a uniform distribution to a data set with the Arena Input Analyzer. Would you reject or not reject the hypothesis that the data is uniformly distributed.

```
Distribution Summary
Distribution: Uniform
Expression: UNIF(36, 783)
Square Error: 0.156400
Chi Square Test
Number of intervals = 7
Degrees of freedom = 6
Test Statistic = 164
Corresponding p-value < 0.005
Kolmogorov-Smirnov Test
Test Statistic = 0.495
Corresponding p-value < 0.01
Data Summary
Number of Data Points = 100
Min Data Value = 36.8
Max Data Value = 782
Sample Mean = 183
Sample Std Dev = 142
Histogram Summary
Histogram Range = 36 to 783
Number of Intervals = 10
```

**Exercise B.9**Consider the following frequency data on the number of orders received per day, \(x_j\), by a warehouse where \(c_j\) is the observed count.

\(j\) | \(x_j\) | \(c_j\) | \(np_j\) | \(\frac{(c_j - np_j)^2}{np_j}\) |
---|---|---|---|---|

1 | 0 | 10 | ||

2 | 1 | 42 | ||

3 | 2 | 27 | ||

4 | 3 | 12 | ||

5 | 4 | 6 | ||

6 | 5 or more | 3 | ||

Totals | 100 |

- Compute the sample mean for this data.
- Perform a \(\chi^2\) goodness of fit test to test the hypothesis (use a 95% confidence level) that the data is Poisson distributed. Complete the provided table to show your calculations.

**Exercise B.10**The number of electrical outlets in a prefabricated house varies between 10 and 22 outlets. Since the time to perform the install depends on the number of outlets, data was collected to develop a probability distribution for this variable. The data set is given below and also found in file

*problem10.txt*:

14 | 16 | 14 | 16 | 12 | 14 | 12 | 13 | 12 | 12 |

12 | 12 | 14 | 12 | 13 | 16 | 15 | 15 | 15 | 21 |

18 | 12 | 17 | 14 | 13 | 11 | 12 | 14 | 10 | 10 |

16 | 12 | 13 | 11 | 13 | 11 | 11 | 12 | 13 | 16 |

15 | 12 | 11 | 14 | 11 | 12 | 11 | 11 | 13 | 17 |

Fit a probability model to the number of electrical outlets per prefabricated house.

**Exercise B.11**Test the following fact by generating instances of \(Y = \sum_{i=1}^{r} X_i\), where \(X_i \sim \mathit{expo}(\beta)\) and \(\beta = E[X_i]\). Using \(r=5\) and \(\beta =2\). Be careful to specify the correct parameters for the exponential distribution function. The exponential distribution takes in the mean of the distribution as its parameter. Generate 10, 100, 1000 instances of \(Y\). Perform hypothesis tests to check \(H_0: \mu = \mu_0\) versus \(H_1: \mu \neq \mu_0\) where \(\mu\) is the true mean of \(Y\) for each of the sample sizes generated. Use the same samples to fit a distribution using the Input Analyzer. Properly interpret the statistical results supplied by the Input Analyzer.

**Exercise B.12**Suppose that we are interested in modeling the arrivals to Sly’s BBQ Food Truck during 11:30 am to 1:30 pm in the downtown square. During this time a team of undergraduate students has collected the number of customers arriving to the truck for 10 different periods of the 2 hour time frame for each day of the week. The data is given below and also found in file

*problem12.csv*

Obs# | M | T | W | R | F | S | SU |
---|---|---|---|---|---|---|---|

1 | 13 | 4 | 4 | 3 | 8 | 8 | 9 |

2 | 6 | 5 | 7 | 7 | 5 | 6 | 8 |

3 | 7 | 14 | 10 | 5 | 5 | 5 | 10 |

4 | 12 | 6 | 10 | 5 | 12 | 7 | 4 |

5 | 6 | 8 | 8 | 5 | 4 | 11 | 9 |

6 | 10 | 6 | 9 | 3 | 3 | 6 | 4 |

7 | 9 | 5 | 5 | 5 | 7 | 5 | 4 |

8 | 7 | 10 | 11 | 9 | 7 | 10 | 13 |

9 | 8 | 4 | 2 | 7 | 6 | 5 | 7 |

10 | 9 | 2 | 6 | 8 | 7 | 4 | 9 |

- Visualize the data.
- Check if the day of the week influences the statistical properties of the count data.
- Tabulate the frequency of the count data.
- Estimate the mean rate parameter of the hypothesized Poisson distribution.
- Perform goodness of fit tests to test the hypothesis that the number of arrivals for the interval 11:30 am to 1:30 pm has a Poisson distribution versus the alternative that it does not have a Poisson distribution.

**Exercise B.13**A copy center has one fast copier and one slow copier. The copy time per page for the fast copier is thought to be lognormally distributed with a mean of 1.6 seconds and a standard deviation of 0.3 seconds. A co-op Industrial Engineering student has collected some time study data on the time to copy a page for the slow copier. The times, in seconds, are given in the data file

*problem13.txt*. Recommend a probability distribution for the time to copy on the slow copier. Justify your recommendation using the statistical methods described in the chapter.