Question

1	For problems 1a through 1.c, assume that the length of a population of fish is normally distributed with population mean μ = 63 cm and population standard deviation σ = 9 cm.
1.a	What proportion of the individual fish are longer than 76 cm?
1.b	What proportion of the fish are between 42 and 84 cm long?
2	For problem 2.a through 2.c, assume that a population of automobile engines has a population mean useful life μ = 120,000 miles and population standard deviation σ  = 8,000 miles.
2.a	What proportion of the engines last more than 140,000 miles?
2.b	What proportion of the engines last between 128,400 to 151,600 miles?
2.c	The manufacturer wants to write a warranty so that only 0.8%  (0.008) of the engines fail while under warranty.  For how long should the warranty be written?
3	A sociology professor finds that his student’s scores on an exam are normally distributed with population mean μ = 80 and population standard deviation σ = 6.  Find the 40thpercentile.
4	Use the following data for problems 6.a and 6.b.   A community college instructor finds that his students score on an exam is normally distributed with a population mean µ = 83 and population standard deviation  σ = 5.
4.a	The instructor wants to pass 95% of the class.  What should be the minimum passing grade?
4.b	The instructor wants to give A’s to 30% of his students.  What should be the minimum grade for an A?
5	A manufacturer of high intensity lamps finds that the useful life of the lamps is normally distributed with population mean μ = 70 months and population standard deviation s = 12 months.
	The manufacturer wants to write a warranty so that only 1.5% (0.015) of the lamps fail while still under warranty.  For how long should the warranty be written?
6	The time required for laboratory rats to complete a maze is normally distributed with  population mean              µ = 45 minutes with population standard deviation σ = 5.4 minutes.   What proportion of the rats complete the maze with time between 37 to 53 minutes?

Answer 1

1. Let X denotes the length of randomly selected fish.

X ~ Normal(63, 9²)

a) The proportion of the individual fish that are longer than 76 cm

b) The proportion of fish that are between 42 and 84 cm long

2. Let X denotes the lifetime of a randomly selected automobile engine.

X ~ Normal(120000, 8000²)

The proportion of engines last more than 140,000 miles

The proportion of engines last between 128,400 to 151,600 miles