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# Probability and Statistics : Confidence Intervals and Random Variables

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1) A die with three sides {1,2,3} is tossed two times. Let X equal the maximum of two observations and let Y equal the minimum. Find the coefficient of X and Y.

2) Let X1, X2, ...Xn be random variables denoting n independent bids for an item that is for sale. Suppose each Xi is uniformly distributed on the interval [100,200]. If the seller sells to the highest bidder, how much can he expect to earn of the sale?

3) The average zinc concentration from a sample of zinc measurements in 36 different locations is found to be 2.6 grams per millimeter. Find the 95% and the 97% confidence intervals for the mean zinc concentration in the river. Assume that the population standard deviation is .5.

4) The following random sample was selected from a normal population: 7,4,10,5,6,4. Construct a 80%, 98%, and 99% confidence interval for the population mean.

5) A new design on the braking system on a certain type of car has been proposed. For the current system, the true average braking distance at 40 mph under specified conditions is known to be 120ft. It is proposed that the new design be implemented only if sample data strongly indicates a reduction in true average braking distance for the new design.

a) Define the parameter of interest and state the relevant hypothesis
b) Suppose braking distance for the new system is normally distributed with sigma=10. Let X bar denote the sample average braking distance for a random sample of 36 observations. Which of the following rejection regions is appropriate: R1={Xbar: is greater than or equal to 124.80}, R2={xbar: is less than or equal to 115.20}, R3={Xbar: is either greater than or equal to 125.13 or less than or equal to 114.87}
c) What is the significance level for the appropriate region of part (b)? How would you change the region to obtain a test with alpha=.001?
d) What is the probability that the new design is not implemented when its true average braking distance is actually 115 ft and the appropriate region from part (b) is used?

6) Because of variability in the manufacturing process, the actual yielding point of a sample of mild steel subjected to increasing stress will usually differ from the theoretical yielding point. Let p denote the true proportion of samples that yield before their theoretical yielding point. If on the basis of a sample it can be concluded that more than 20% of all specimens yield before the theoretical point, the production process will have to be modified.
a) If 15 of 60 specimens yield before the theoretical point, what is the P-value when the appropriate test is used, and what would you advise the company to do?
b) If the true percentage of "early yields" is actually 50% ( so that the theoretical point is the median of the yield distribution) and a level .01 test is used, what is the probability that the company concludes a modification of the process is necessary?

7) People with such initials as ACE, JOY, and GOD are likely to live longer than those whose name spell out words like APE, MAD, and RAT, a study suggests. The study, done by researchers from the university of California at San Diego, looked at thousands of death certificates of people who died in California from 1969 through 1997...The study found 11 "good" sets of initials and 19 "bad" ones...All in all, men with WINing initials lived 4.48 years longer than a control group of people with neutral or ambiguous monograms, while DUDs, BUGs, and such died an average of 2.8 years earlier that a control group. What do you think...

a) This was a wrong application of statistics
b) You agree with the psychologist: the argument is that there's some psychological symbolic factor that can exert its impact cumulatively over the years
c) There is a better explanation of these data, then give it...

##### Solution Summary

A variety of probability and statistics problems involving confidence intervals and random variables are solved. The solution is detailed and well presented.

Solution provided by:
###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
• "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
• "Thank you"
• "Thank you very much for your valuable time and assistance!"

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