Suppose that the lifetimes of light bulbs are approximately normally distributed, with a mean of 56 hours and a standard deviation of 3.3 hours. with this information, answer the following questions. (a) what proportion of light bulbs will last more than 61 hours? (b) what proportion of light bulbs will last 53 hours or less? (c) what proportion of light bulbs will last between 57 and 62 hours? (d) what is the probability that a randomly selected light bulb lasts less than 46 hours?

a. 0.07  or 7%

b. 0.1841  or 18%

c. 0.35 or 35%

d. 0%

Step-by-step explanation:

Given

Mean=μ=56 hours

and

Standard Deviation=σ=3.3 hours

In order to find the proportions of light bulbs we will have to find the z-scores.

The formula for z-score is:

z-score=(x-μ)/σ

So,

a) proportion of light bulbs that will last more than 61 hours

z-score=(61-56)/3.3

=5/3.3

=1.5151

The area to the left of z-score 1.5151 is 0.9344. As we have to caclculate the proportion of light bulbs greater than 61 hours.

Area to the right of z-score=1-0.9344

=0.0656

Rounding off to the nearest 10 will give us 0.07

7% of the bulbs will last more than 61 hours.  

b) What proportion of light bulbs will last 53 hours or less?

z-score=(53-56)/3.3

=(-3)/3.3

=-0.9090

Area to the left of z-score is 0.18406. Rounding off will give us:

=0.1841

18% of the bulbs will last 53 hours or less.  

c) What proportion of light bulbs will last between 57 and 62 hours?  

z-score for 57=(57-56)/3.3  

=1/3.3

=0.3030

The area to the left of z-score 0.3030 is 0.61791

z-score for 57=(62-56)/3.3  

=6/3.3

=1.8181

The area to the left of z-score 1.8181 is 0.96485

Area between z-scores of 57 and 62=0.96485-0.61791

=0.34694

Rounding off gives 0.35

35% of the bulbs will last between 57 to 62 hours.

d) What is the probability that a randomly selected light bulb lasts less than 46 hours?

z-score for 46=(46-56)/3.3  

=(-10)/3.3

=-3.0303

Area to the left of z-score is 0.00122. Rounding off gives 0.00. So almost zero percent of the bulbs will last less than 46 hours ..