Question

In: Statistics and Probability

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

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

five 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 5.4 years.

b. Calculate the probability that the​ watch's battery will last more than 4.45 years.

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

2. Doggie Nuggets Inc.​ (DNI) sells large bags of dog food to warehouse clubs. DNI uses an automatic filling process to fill the bags. Weights of the filled bags are approximately normally distributed with a mean of 56 kilograms and a standard deviation of 0.86 kilograms. Complete parts a through d below.

a. What is the probability that a filled bag will weigh less than 55.2 kilograms?

Solutions

Expert Solution

1)

a)

Here, μ = 5, σ = 0.3 and x = 5.4. We need to compute P(X >= 5.4). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (5.4 - 5)/0.3 = 1.33

Therefore,
P(X >= 5.4) = P(z <= (5.4 - 5)/0.3)
= P(z >= 1.33)
= 1 - 0.9082 = 0.0918


b)

Here, μ = 5, σ = 0.3 and x = 4.45. We need to compute P(X >= 4.45). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (4.45 - 5)/0.3 = -1.83

Therefore,
P(X >= 4.45) = P(z <= (4.45 - 5)/0.3)
= P(z >= -1.83)
= 1 - 0.0336 = 0.9664


c)

z value at 15% = 1.04

z = (x - mean)/s
1.04 = (x - 5)/0.3
x = 0.3 *1.04 + 5

x = 5.31


2)'
Here, μ = 56, σ = 0.86 and x = 55.2. We need to compute P(X <= 55.2). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (55.2 - 56)/0.86 = -0.93

Therefore,
P(X <= 55.2) = P(z <= (55.2 - 56)/0.86)
= P(z <= -0.93)
= 0.1762


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