The company states that 90% of the light bulbs have a lifetime of at least k hours. Find the value of k. Give your answer correct to the nearest hundred.

a) Mean = 5800 hours

b) P(5000 ≤ L ≤ 6000) = 0.420

c) k = 4700 hours

Step-by-step explanation:

The full correct question is attached to this solution.

From the graph of the distribution of lifetime of bulbs presented, it is evident that this is a normal distribution.

(a) Write down the mean lifetime of the light bulbs.

From the normal distribution graph, the mean is at the very centre of the graph

Mean = μ = 5800 hours

(b) The standard deviation of the lifetime of the light bulbs is 850 hours.

Find the probability that 5000 ≤ L ≤ 6000, for a randomly chosen light bulb.

Mean = μ = 5800 hours

Standard deviation = σ = 850 hours.

The required probability = P(5000 ≤ L ≤ 6000)

To find this, we first normalize/standardize 5000 and 6000.

The standardized score for any value is the value minus the mean then divided by the standard deviation.

For 5000

z = (L- μ)/σ = (5000 - 5800)/850 = -0.941

For 6000

z = (L- μ)/σ = (6000 - 5800)/850 = 0.235

The required probability

P(5000 ≤ L ≤ 6000) = P(-0.941 ≤ z ≤ 0.235)

We'll use data from the normal probability table for these probabilities

P(5000 ≤ L ≤ 6000) = P(-0.941 ≤ z ≤ 0.235)

= P(z ≤ 0.235) - P(z ≤ -0.941)

= 0.593 - 0.173 = 0.420

c) The company states that 90% of the light bulbs have a lifetime of at least k hours. Find the value of k. Give your answer correct to the nearest hundred.

P(L ≥ k) = 90% = 0.9

assuming that the z-score of k is z'

P(L ≥ k) = P(z ≥ z') = 0.90

P(z ≥ z') = 1 - P(z < z') = 0.9

P(z < z') = 0.1

Using the normal distribution table

z' = -1.282

z' = (k - μ)/σ

-1.282 = (k - 5800)/850

k = (-1.282 × 850) + 5800 = 4710.3 = 4700 hours to the nearest hundred.

Hope this Helps!!!

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1