Did you ever buy an incandescent light bulb that failed (either burned out or did not work) the first time you turned the switch on? When you put a new light bulb into a light fixture, you expect it to light and most of the time it does. Consider 8-packs of 60-watt bulbs and let x be the number of bulbs in a pack that 'fail' the first time they are used. If 0.02 of all the bulbs of this type fail on their first use and each 8-pack is considered a random sample.
A. List the probability distribution and draw the histogram of x.
B. What is the probability that any one 8-pack has no bulbs that fail on first use?
C. What if the probability that any one 8-pack has no more than one bulb that fails on the first use?
D. Find the mean and standard deviation of x.
E. What is the proportion of the distribution is between m - sigma and m + sigma?
What is the proportion of the distribution is m- 2(sigma) and m + 2(sigma) ?
A. The probability distribution for x can be represented as follows:
x | P(x)
--------------
0 | 0.98
1 | 0.02
2 | 0
3 | 0
...
The histogram of x will have bars representing each value of x, with the height of the bar corresponding to the probability P(x).
B. The probability that any one 8-pack has no bulbs that fail on first use is the probability of x being 0. From the probability distribution, we know that P(x=0) = 0.98.
C. The probability that any one 8-pack has no more than one bulb that fails on the first use is the probability of x being 0 or 1. So, P(x≤1) = P(x=0) + P(x=1) = 0.98 + 0.02 = 1.
D. To find the mean of x, we multiply each value of x by its corresponding probability and sum the products. In this case, the mean (µ) can be calculated as:
µ = (0 * 0.98) + (1 * 0.02) + (2 * 0) + (3 * 0) + ...
Since all the probabilities for x = 2, 3, ... are 0, these terms will not contribute to the sum. Therefore, the mean is simply:
µ = (0 * 0.98) + (1 * 0.02) = 0.02
To find the standard deviation (σ) of x, we need to calculate the variance first. The variance (σ^2) is given by:
σ^2 = [(0 - µ)^2 * 0.98] + [(1 - µ)^2 * 0.02] + [(2 - µ)^2 * 0] + [(3 - µ)^2 * 0] + ...
Again, since the probabilities for x = 2, 3, ... are 0, these terms will not contribute to the sum. Hence, the variance simplifies to:
σ^2 = [(0 - 0.02)^2 * 0.98] + [(1 - 0.02)^2 * 0.02] = 0.0196
Finally, the standard deviation is calculated as the square root of the variance:
σ = √0.0196 ≈ 0.14
E. The proportion of the distribution between m - sigma and m + sigma is the area under the probability distribution curve between µ - σ and µ + σ. Since we have a discrete probability distribution, this can be calculated by summing the probabilities for x within this range.
Proportion between m - sigma and m + sigma = P(x=0) + P(x=1) ≈ 0.98 + 0.02 = 1
The proportion of the distribution between m - 2(sigma) and m + 2(sigma) is the area under the probability distribution curve between µ - 2σ and µ + 2σ. Again, for a discrete probability distribution, this can be calculated by summing the probabilities for x within this range.
Proportion between m - 2(sigma) and m + 2(sigma) = P(x=0) + P(x=1) ≈ 0.98 + 0.02 = 1
To answer the given questions, we need to understand the probability distribution and then calculate the probabilities and statistics based on that distribution. Let's go step by step:
A. Probability Distribution and Histogram:
Since the number of bulbs that fail on their first use follows a probability distribution, we can assume that it follows a binomial distribution. In this case, we have a random sample of 8 bulbs, and the probability of failure for each bulb is 0.02.
Using the binomial distribution formula, we can calculate the probabilities for different values of x:
P(x) = (n choose x) * p^x * (1-p)^(n-x)
Where n is the total number of trials (8 bulbs in this case), x is the number of failures (ranging from 0 to 8), and p is the probability of failure for each trial (0.02).
We can calculate these probabilities for x = 0, 1, 2, ..., 8, and then create a histogram to visualize the distribution.
B. Probability of no failures in any one 8-pack:
To find the probability that an 8-pack has no bulbs that fail on first use, we need to calculate the probability of x = 0 using the binomial distribution formula. Plug in the values n = 8 and p = 0.02, and calculate P(x = 0).
C. Probability of no more than one failure in any one 8-pack:
To find the probability that an 8-pack has no more than one bulb that fails on first use, we need to calculate the probabilities P(x = 0) and P(x = 1) using the binomial distribution formula. Add these probabilities together to get the desired probability.
D. Mean and Standard Deviation:
To find the mean and standard deviation of the distribution, we can use the formulas for the mean and variance of a binomial distribution.
Mean (mu) = n * p
Standard Deviation (sigma) = sqrt(n * p * (1 - p))
Substitute the values n = 8 and p = 0.02 into these formulas to find the mean and standard deviation.
E. Proportion within m - sigma and m + sigma:
To find the proportion of the distribution within one standard deviation of the mean, we need to calculate the probabilities for x values in the range m - sigma to m + sigma. Sum up these probabilities to get the desired proportion.
Proportion within m - 2(sigma) and m + 2(sigma):
Similarly, to find the proportion within two standard deviations of the mean, calculate the probabilities for x values in the range m - 2(sigma) to m + 2(sigma), and then sum them up.
By following these steps, you should be able to answer all the given questions related to the probability distribution and statistical measures of the number of failed light bulbs in an 8-pack.