STATS HARD QUESTIONS NEED HELP

Question

If the lifetime length of light bulbs from a certain factory follows a normal distribution with mean 2500 hours and standard deviation 50 hours.

1. What is the percentage of light bulbs that can last for more than 2598 hours?

2. A light bulb is identified as defective if it lasts for less than 2418 hours. What is the percentage of defective light bulbs from this factory?

3. How many hours does a light bulb can last for, if it is identified as top 5% of all the light bulbs from the factory

 

Please show help with steps

Answer 1

NOTE: I used the Z-score table for all the probability calculations.

 

(a) Calculate the Z-score for 2,598 hours

 

Z = (x - μ)/σ = (2598-2500)/50 = 1.96

 

P(x>2,598)

= P(Z>1.96)

= 1 - P(Z≤1.96)

= 1 - 0.9750

= 0.025

= 2.5%

 

(b) Calculate the Z-score for 2,418 hours

 

Z = (x - μ)/σ = (2418-2500)/50 = -1.64

 

P(x<2418)

= P(Z<-1.64)

= 0.0505

= 5.1%

 

(c) Top 5% means that P(Z>Z*) = 0.05

 

Let Z* be value of X that is the cut off for the top 5%.

 

Using the Z-score table, a probability of 5% translates to Z*=1.645

 

Z = (x - μ)/σ

 

Rearrange the equation to solve for x.

 

x* = μ + Zσ

    = 2500 + 1.645(50)

x* = 2,582.25 hours ≈ 2,582 hours