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Unformatted text preview: SIEO 3600 (Non-IEOR Majors) Introduction to Probability and Statistics Assignment #2 26th January 2010 Assignment #2 – due February 2nd, 2010 1. Use the data provided in the table below to do the following tasks (a) Draw a scatterplot of salaries in private …rms and in government; calculate their correlation coe¢ cient. (b) Draw a scatterplot of salaries in government and in universities; calculate their correlation coe¢ cient. Is there anything strange? (Note: do not include the “Total”row in the analysis because it is di¤erent from other rows.)
Median salaries for recent U.S. doctorate recipients (1-3 years after degree), by sector of employment: 1999 Private Four-year Ph.D …eld Total noneducational Government institution Total 48,400 68,000 55,000 43,400 Computer sciences 75,000 82,000 66,000 53,000 Engineering 66,700 70,000 65,000 56,300 Life sciences 35,000 61,000 48,000 42,500 Math sciences 45,000 60,500 55,200 39,500 Social sciences 45,000 53,000 52,400 40,000 Physical sciences 52,000 64,000 58,000 39,400
Source: N ational Science Foundation D ivision of Science Resources Statistics (N SD /SR S). Survey of D octorate Recipients, 1999. 2. Prove Property 3 of sample correlation coe¢ cient r, i.e., suppose we have a set of paired date (xi ; yi ), 1 i n. Show that if r is sample correlation of (xi ; yi ), then it is also the sample correlation of (a xi + b; c yi + d) for a c > 0. 3. Show that the sample correlation coe¢ cient jrj
n X i=1 1. Hint: start with y sy
2 xi sx x yi 0: 4. Let S = f1; 2; 3; 4; 5; 6; 7g, E = f1; 3; 5; 7g, F = f7; 4; 6g, G = f1; 4g. Find (b) E [ (F \ G) (d) (E \ F c ) [ G (c) E \ Gc (a) E \ F (e) E c \ (F [ G) (f) (E \ G) [ (F \ G) 5. A system is composed of four components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector (x1 ; x2 ; x3 ; x4 ) where xi is equal to 1 if component i is working and is equal to 0 if component i is failed. (a) How many outcomes are in the sample space of this experiment? 2 SIEO 3600, Assignment #2 (b) Suppose that the system will work if components 1 and 2 are both working or if components 3 and 4 are both working. Specify all the outcomes in the event that the system works. (c) Let E be the event that components 1 and 3 are both failed. How many outcomes are contained in event E ? 6. Find simple expressions for the events (a) E [ E c (c) (E [ F ) \ (E [ F c ) (e) (E [ F ) \ (F [ G) (b) E \ E c (d) (E [ F ) \ (E c [ F ) \ (E [ F c ) 7. Prove Bonferroni’ inequality, namely that s P(E \ F ) When does the equality hold? Hint: Combine the two known results introduced in class: (i) P(E [ F ) = P(E ) + P(E ) P(E \ F ); and (ii) P(G) P(H ) if E F . P(E ) + P(F ) 1: ...
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