A random sample of 200 lightbulbs has a mean life of 600 hours and a standard deviation of 53 hours.(a) A histogram of the data indicates the sample data follow a bell-shaped distribution. According to theEmpirical Rule, 99.7% of lightbulbs have lifetimes between and hours.(b) Assuming the data are bell shaped, determine the percentage of lightbulbs that will have a life between494 and 706 hours.(c) Assuming the data are bell shaped, what percentage of lightbulbs will last between 547 and 706 hours?(d) If the company that manufactures the lightbulbs guarantees to replace any bulb that does not last at least 441 hours, what percentage of lightbulbs can the firm expect to have to replace, according to the Empirical Rule?

Given Information:

The average bulb life = μ = 600 hours

standard deviation = σ = 53 hours

Sample follows Bell-Shaped Distribution or Normal Distribution

Step-by-step explanation:

According to the Empirical Rule, 99.7% of the lightbulbs have lifetimes between μ - 3σ  and μ + 3σ

600 - 3(53) and 600 + 3(53)

441 and 759 hours

(b) assuming the data are bell shaped, determine the percentage of lightbulbs that will have a life between 494 and 706 hours.

494 = 600 - 106 = 600 - 2(53) =  μ - 2σ

706 = 600 + 106 = 600 + 2(53) = μ + 2σ

Therefore, according to the Empirical Rule, 95% of the lightbulbs have lifetimes between μ - 2σ  and μ + 2σ

(c) Assuming the data are bell shaped, what percentage of lightbulbs will last between 547 and 706 hours?

547 = 600 - 53 = 600 - 1(53) =  μ - σ

653 = 600 + 53 = 600 + 1(53) = μ + σ

Therefore, according to the Empirical Rule, 68% of the lightbulbs have lifetimes between μ - σ  and μ + σ

(d) If the company that manufactures the lightbulbs guarantees to replace any bulb that does not last at least 441 hours, what percentage of lightbulbs can the firm expect to have to replace, according to the Empirical Rule?

441 = 600 - 159 = 600 - 3(53) = μ - 3σ  

99.7% of the lightbulbs have lifetimes between  μ - 3σ  and  μ + 3σ  

100 - 99.7 = 0.3/2 = 0.15 %

According to the Empirical rule, 0.15% lightbulbs will last less than 441 hours.

step-by-step explanation:

the triangles are similar. therefore the corresponding sides are in proportion:

              cross multiply

            divide both sides by 32