Ivo Dinov
UCLA Statistics, Neurology, LONI

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Statistics 157
Spring  2010

Probability and Statistics Data Modeling and Analysis using SOCR

Department of Statistics

## Instructor: Ivo Dinov

Homework 1
Due Date: Friday, April 09, 2010

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 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
 Relative Frequency Cumulative Relative Frequency Model Probabilities (model used) ___________

• 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: This SOCR Histogram Activity and SOCR Charts; copy the second column frequencies and  paste them in the Data tab. Then Go to Graph tab and plot the relative frequencies. Save the graph and include in your HW document.)
• Visually choose a model distribution for these data using  SOCR Distributions, 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). Try to make the support and the shape of the model distribution similar to their corresponding sample-histogram (relative frequency plot).
• 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.
 Time 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 and sample-median?
• By how much should the largest time be increased so that the sample median is half the sample mean?
• Use SOCR Charts to interactively compute/validate your answers (for example you can use the Index Chart).

• (HW_1_3) The accompanying data (millions of revolutions) for the load-life of bearings of certain type 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.
• Validate your manually-computed results against the summaries reported in SOCR Charts.

• (HW_1_4) Run the Virtual Roulette game 10 times (you can also use the Spinner Experiment from the SOCR Experiments, 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?

 Ivo Dinov's Home http://www.stat.ucla.edu/~dinov Visitor number, since Jan. 01, 2002 I.D.Dinov  2010