Question

In: Statistics and Probability

An expensive watch is powered by a​ 3-volt lithium battery expected to last three years. Suppose...

An expensive watch is powered by a​ 3-volt lithium battery expected to last three years. Suppose the life of the battery has a standard deviation of 0.3 year and is normally distributed.a. Determine the probability that the​ watch's battery will last longer than 3.8 years.b. Calculate the probability that the​ watch's battery will last more than 2.15 years.c. Compute the​ length-of-life value for which 15​% of the​ watch's batteries last longer.

a. The probability that the battery will last longer than 3.8 years is​

b. The probability that the battery will last more than 2.15 years is

c. The​ length-of-life value for which 15​% of the batteries last longer is years.

Solutions

Expert Solution

Answer:

Given that,

An expensive watch is powered by a​ 3-volt lithium battery expected to last three years.

Suppose the life of the battery has a standard deviation of 0.3 year and is normally distributed.

i.e,

The mean= 3 years

The standard deviation= 0.3 years

(a).

Determine the probability that the​ watch's battery will last longer than 3.8 years:

P(X>3.8) = 1 - P(X<3.8)

= 1 - P(Z < (X - mean)/standard deviation)

= 1 - P(Z < (3.8 - 3)/0.3)

= 1 - P(Z < 2.67)

= 1 - 0.9962

= 0.0038

Therefore, the probability that the watch's battery will last longer than 3.8 years = 0.0038.

(b).

Calculate the probability that the​ watch's battery will last more than 2.15 years:

P(X > 2.15) = 1 - P(X<2.15)

= 1 - P(Z < (2.15-3)/0.3)

= 1 - P(Z < -2.83)

= 1 - 0.0023

= 0.9977

Therefore, the probability that the watch's battery will last longer than 2.15 years = 0.9977.

(c).

Compute the​ length-of-life value for which 15​% of the​ watch's batteries last longer:

Let the value for which 15% of the watch's batteries last longer be A

P(X > A) = 0.15

P(X < A) = 1-0.15 = 0.85

P(Z < (A - mean)/standard deviation) = 0.85

From standard normal distribution table,

(A - 3)/0.3 = 1.0364

A-3=1.0364 (0.3)

A-3=0.3109

A=0.3109+3

A = 3.3109


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