An electrical firm manufactures light bulbs that have a usuable life that is normally distributed with a mean of 1000 hours and a standard deviation of 80 hours. Find the probability that a package of 4 bulbs would last at least 4400 hours.
To find the probability that a package of 4 bulbs would last at least 4400 hours, we need to first calculate the probability that a single bulb will last at least 4400 hours, and then raise this probability to the power of 4 (since there are 4 bulbs in the package).
Given that the usable life of the bulbs is normally distributed with a mean (μ) of 1000 hours and a standard deviation (σ) of 80 hours, we can use the standard deviation formula to standardize the value.
Z = (X - μ) / σ
Where:
Z is the z-score representing the number of standard deviations from the mean
X is the value we want to calculate the probability for
In this case, we want to find the probability that a bulb lasts at least 4400 hours, so X = 4400.
Now, we can calculate the z-score:
Z = (4400 - 1000) / 80
Z = 3400 / 80
Z = 42.5
The next step is to find the cumulative probability (area under the curve) to the right of this z-score using a standard normal distribution table or a calculator. This area represents the probability of a single bulb lasting at least 4400 hours.
P(X ≥ 4400) = 1 - P(Z < 42.5)
Since the z-score is very large, the probability is extremely close to 1 (or 100%). Therefore, for a single bulb, the probability that it lasts at least 4400 hours is essentially 1.
Finally, we need to find the probability for all 4 bulbs in the package. Since the bulbs are independent, we can raise the probability of a single bulb to the power of 4.
P(package of 4 bulbs lasting at least 4400 hours) = P(single bulb lasting at least 4400 hours) ^ 4
P(package of 4 bulbs lasting at least 4400 hours) = 1^4
Hence, the probability that a package of 4 bulbs would last at least 4400 hours is 1 or 100%.
To find the probability that a package of 4 bulbs would last at least 4400 hours, we can use the concept of the Central Limit Theorem and the properties of the normal distribution.
First, let's calculate the mean and standard deviation of the distribution of the sum of the lifetimes of the 4 bulbs:
Mean of the sum = Mean of one bulb x Number of bulbs = 1000 hours x 4 = 4000 hours
Standard deviation of the sum = Standard deviation of one bulb x Square root of the number of bulbs = 80 hours x sqrt(4) = 80 hours x 2 = 160 hours
Now, we need to standardize the value of 4400 hours to calculate the probability using the z-score formula:
z = (x - mean) / standard deviation
z = (4400 - 4000) / 160 = 400 / 160 = 2.5
Next, we need to find the area under the standard normal distribution curve to the right of z = 2.5. This represents the probability that the package of 4 bulbs will last at least 4400 hours.
Using a standard normal distribution table or a calculator, we find that the area to the right of z = 2.5 is approximately 0.0062.
Therefore, the probability that a package of 4 bulbs would last at least 4400 hours is approximately 0.0062, or 0.62%.