Ivo Dinov
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STAT 35
Winter 2005

Interactive and Computational Probability

Department of Statistics

## Instructor: Ivo Dinov

Homework 1
Due Date: Wednesday, Jan. 26, 2005

Please, submit your homework before lecture on the due date. Correct solutions to any 4 out of the 4 problems carry full credit. See the HW submission rules. On the front page include the following header

• (HW_1_1)  The clearness index was determined for the skies over Baghdad for each of the 365 days during a particular year. It was a contribution to the study of the solar radiation of the Baghdad environment. The following table summarized the resulting data.
 Class interval for the clearness index Number of days Relative Frequency Cumulative Relative Frequency Model Probabilities (model used) ___________ 0.16 - 0.20 3 0.21 - 0.25 5 0.26 - 0.30 6 0.31 - 0.35 8 0.36 - 0.40 12 0.41 - 0.45 16 0.46 - 0.50 24 0.51 - 0.55 39 0.56 - 0.60 51 0.61 - 0.65 106 0.66 - 0.70 84 0.71 - 0.75 11

• Determine the Relative Frequency and the Cumulative Relative Frequency (fill in the two last columns of the table)
• Sketch the Relative Frequency histogram and comment on it (use: http://dostat.stat.sc.edu/webstat/3.0/oldindex.html, copy the second column frequencies and  paste them in the Data link. Then Go to Graphs and plot the relative frequencies. Save the graph and include in your HW document.)
• Visually choose a model distribution from http://socr.stat.ucla.edu/test/SOCR_Distributions.html, compute and enter the corresponding model probabilities for each range in the last column (by varying the left & right limit of the regions using mouse clicks).
• Cloudy days are those with the clearness index < 0.35. What proportion of the days were cloudy? How different are the data and model probabilities?
• Clear days are those for which the clearness index is at least 0.66. What proportion of the days were clear?

• (HW_1_2) A sample of 26 offshore oil workers took part in a simulated escape exercise, resulting in the accompanying data on time (sec) to complete the escape ("Oxygen Consumption and Ventilation During Escape from an Offshore Platform." Ergonomics, 1997: 281-292). The escape times are shown below. The operators of the platform have requested your assistance in improving the platform. Specifically, the operators want to know what escape time corresponds with a 1% chance of being exceeded.
 389 356 359 363 375 424 325 394 402 373 373 370 364 366 364 325 339 393 392 369 374 359 356 403 334 397

• How different are the sample mean and median?
• By how much should the largest time be increased so that the sample median is half the sample mean?

• (HW_1_3) The accompanying data on bearing load-life (million revolutions) for bearings of certain type when subjected to a 9.56 kN load are (Lubric. Eng., 1984, 153-159)
 14.5 25.6 52.4 66.3 69.3 69.8 76.2
• What is the five-number summary for this data?
• Calculate the following sample measures of spread: variance, standard deviation and the mean-absolute-deviation.

• (HW_1_4) Run the virtual Roulette game 10 times (you can use the Spinner Experiment under the SOCR Experiments page http://socr.stat.ucla.edu/test/SOCR_Experiments.html, you must set the number of experiments, n=38). A Roulette wheel consists of 18 red, 18 black and 2 green spaces. The wheel is spun and a marble falls at random into one of the 38 spaces. Each sector in the Roulette wheel is numbered (0, 00, 1, 2, ..., 36).
• Should all 38 possible outcomes occur the same number of times? Why?
• Does it appear as if some outcomes are just too frequent and some are too rare?
• How large is the difference between the even and odd outcomes in your 10 experiments? Is this expacted to vary for different students?
• Would the answers to the above questions change if we did 1,000 experiments, instead of just 10?

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