28. Let ^ be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate.
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1 Exercises Section 4.3 [page 154] 28. Let ^ be a standard normal random variable and calculate the following probabilities, drawing pictures wherever appropriate. a. T Ð! Ÿ ^ Ÿ #Þ"(Ñ b. TÐ! Ÿ ^ Ÿ "Ñ c. TÐ #Þ&! Ÿ ^ Ÿ!Ñ d. T Ð #Þ&! Ÿ ^ Ÿ #Þ&!Ñ e. T Ð^ Ÿ "Þ$(Ñ f. TÐ "Þ(& Ÿ ^Ñ g. T Ð "Þ&! Ÿ ^ Ÿ #Þ!!Ñ h. T Ð"Þ$( Ÿ ^ Ÿ #Þ&!Ñ i. T Ð"Þ&! Ÿ ^Ñ j. T Ðl^l Ÿ #Þ&!Ñ
2 29. In each case, determine the value of the constant - that makes the probability statement correct. a. FÐ-Ñ œ Þ*)$) b. T Ð! Ÿ ^ Ÿ -Ñ œ Þ#*" c. T Ð- Ÿ ^Ñ œ Þ"#" d. TÐ -Ÿ^Ÿ-ÑœÞ'') e. T Ð- Ÿ l^lñ œ Þ!"' 31. Determine D α for the following: a. α œ Þ!!&& b. α œþ!* c. α œ Þ''$
3 32. If \ is a normal rv with mean )! and standard deviation "!, compute the following probabilities by standardizing. a. T Ð\ Ÿ "!!Ñ b. T Ð\ Ÿ )!Ñ c. T Ð'& Ÿ \ Ÿ "!!Ñ d. TÐ(! Ÿ \Ñ e. T Ð)& Ÿ \ Ÿ *&Ñ f. T Ðl\ )!l Ÿ "!Ñ
4 33. Suppose the force acting on a column that helps to support a building is normally distributed with mean "&Þ! kips and standard deviation "Þ#& kips. What is the probability that the force is a. At most ") kips? b. Is between "! and "# kips? c. Differs from "&Þ! kips by at most "Þ& standard deviations? 34. The article Reliability of Domestic-Waste Biofilm Reactors ( J. of Envir. Engr., 1995: ) suggests that substrate $ concentration (mg/cm ) of influent to a reactor is normally distributed with. œþ$! and 5 œþ!'. a. What is the probability that the concentration exceeds Þ#&? b. What is the probability that the concentration is at most Þ"!? c. How would you characterize the largest % of all concentration & values?
5 35. Suppose the diameter at breast height (in.) of trees of a certain type is normally distributed with. œ)þ) and 5 œ#þ), as suggested in the article Simulating a Harvester-Forwarder Softwood Thinning ( Forest Products J., May 1997: 36-41). a. What is the probability that the diameter of a randomly selected tree will be at least "! in.? Will exceed "! in.? b. What is the probability that the diameter of a randomly selected tree will exceed #! in.? c. What is the probability that the diameter of a randomly selected tree will be between & and "! in.? d. What value - is such that the interval Ð)Þ) -ß )Þ) -Ñ includes *)% of all diameter values? e. If four trees are independently selected, what is the probability that at least one has a diameter exceeding "! in.? 38. There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean $ cm and standard deviation!þ" cm. The second machine produces corks with diameters that have a normal distribution with mean $Þ!% cm and standard deviation Þ!# cm. Acceptable corks have diameters between #Þ* cm and $Þ" cm. Which machine is more likely to produce an acceptable cork?
6 39. If a normal distribution has. œ$! and 5 œ&, what is the *" st percentile of the distribution? b. What is the ' th percentile of the distribution? c. The width of a line etched on an integrated circuit chip is normally distributed with mean $Þ!!!. m and standard deviation Þ"%!.m. What width value separates the widest "!% of all such lines from the other *!%? 40. The article Monte Carlo Simulation Tool for Better Understanding of LRFD ( J. Structural Engr., 1993: ) suggests that yield strength (ksi) for A36 grade steel is normally distributed with. œ%$ and 5 œ%þ&. a. What is the probability that yield strength is at most %!? b. What yield strength value separates the strongest % from the (& others?
7 44. If bolt thread is normally distributed, what is the probability that the thread length of a randomly selected bolt is a. Within "Þ& SDs of its mean? b. Farther than #Þ& SDs from its mean value? c. Between " and # SDs from its mean value? 45. A machine that produces ball bearings has initially been set so that the true average diameter of the bearings it produces is Þ&!! in. A bearing is acceptable if its diameter is within Þ!!% in. of this target value. Suppose, however, that the setting has changed during the course of production, so that the bearings have normally distributed diameters with mean value Þ%** in. and standard deviation Þ!!# in.. What percentage of the bearings produced will not be acceptable? 47. The weight distribution of parcels sent in a certain manner is normal with mean value "# lb and standard deviation $Þ& lb. The parcel service wishes to establish a weight value - beyond which there will be no surcharge. What value of - is such that **% of all parcels are at least " lb under the surcharge weight?
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