Question

Accrotime is a manufacturer of quartz crystal watches. Accrotime researchers have shown that the watches have an average life of 26 months before certain electronic components deteriorate, causing the watch to become unreliable. The standard deviation of watch lifetimes is 4 months, and the distribution of lifetimes is normal.

(a) If Accrotime guarantees a full refund on any defective watch for 2 years after purchase, what percentage of total production will the company expect to replace? (Round your answer to two decimal places.)

(b) If Accrotime does not want to make refunds on more than 6% of the watches it makes, how long should the guarantee period be (to the nearest month)?

Suppose the heights of 18-year-old men are approximately normally distributed, with mean 67 inches and standard deviation 6 inches.

(a) What is the probability that an 18-year-old man selected at random is between 66 and 68 inches tall? (Round your answer to four decimal places.)


(b) If a random sample of twenty 18-year-old men is selected, what is the probability that the mean height x is between 66 and 68 inches? (Round your answer to four decimal places.)


(c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?

The probability in part (b) is much higher because the mean is larger for the x distribution.

The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.    

The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.

The probability in part (b) is much higher because the standard deviation is larger for the x distribution.

The probability in part (b) is much higher because the mean is smaller for the x distribution.

Answer 1

Solution :

Given that ,

1) mean = = 26 months

standard deviation = = 4 months

a) P(x < 24) = P[(x - ) / < (24 - 26) / 4]

= P(z < -0.50)

Using z table,

= 0.3085

The percentage is = 30.85%

b) Using standard normal table,

P(Z < z) = 6%

= P(Z < z) = 0.06  

= P(Z < -1.55) = 0.06

z = -1.55

Using z-score formula,

x = z * +

x = -1.55 * 4 + 26

x = 19.8

x = 20 months.

2) Given that ,

mean = = 67 inches.

standard deviation = = 6 inches.

a) P( 66 < x < 68) = P[(66 - 67 )/ 6 ) < (x - ) /  < (68 - 67 ) / 6) ]

= P(-0.17 < z < 0.17)

= P(z < 0.17) - P(z < -0.17)

Using z table,

= 0.5675 - 0.4325

= 0.1350

b) n = 20

=   = 67

= / n = 6 / 20 = 1.34

P(66 < < 68)  

= P[(66 - 67) /1.34 < ( - ) / < (68 - 67) / 1.34)]

= P( -0.75 < Z < 0.75)

= P(Z < 0.75) - P(Z < -0.75)

Using z table,  

= 0.7734 - 0.2266

= 0.5468

c) The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.