Suppose the number of hours a type of light bulb will burn is normally distributed with a mean 84 with a standard deviation of 9 hours. what is the probability one of these bulbs will burn for more than 90 hours? If a sample of 24 of this type of bulb is taken, what is the probability their mean number burning hours is less than 86 hours? can we apply the CLT and why?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z
score.
Z = (score-mean)/SEm
SEm = SD/√n
Use same table.
What is CLT?
CLT is the central limit theorem
To find the probability that one of these bulbs will burn for more than 90 hours, we can use the properties of a normal distribution.
Step 1: Standardize the value of 90 hours using the formula z = (x - mean) / standard deviation.
z = (90 - 84) / 9 = 0.67
Step 2: Find the probability corresponding to the z-value using a standard normal distribution table or calculator. For z = 0.67, the table or calculator will give you the probability of approximately 0.7486.
So, the probability that one of these bulbs will burn for more than 90 hours is approximately 0.7486 or 74.86%.
Now, let's move on to the second part of the question regarding the mean number of burning hours for a sample of 24 bulbs.
To determine the probability that the mean number of burning hours is less than 86 hours, we need to apply the Central Limit Theorem (CLT). The CLT states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
In this case, we have a sample size of 24 bulbs. Since the sample size is reasonably large, we can apply the CLT.
Step 1: Calculate the standard deviation of the sample mean using the formula standard deviation of the sample mean = standard deviation / square root of sample size.
standard deviation of the sample mean = 9 / sqrt(24) ≈ 1.8368
Step 2: Standardize the value of 86 hours using the formula z = (x - mean) / standard deviation of the sample mean.
z = (86 - 84) / 1.8368 ≈ 1.0876
Step 3: Find the probability corresponding to the z-value using a standard normal distribution table or calculator. For z = 1.0876, the table or calculator will give you the probability of approximately 0.8619.
Therefore, the probability that the mean number of burning hours for a sample of 24 bulbs is less than 86 hours is approximately 0.8619 or 86.19%.
In summary, we can apply the Central Limit Theorem (CLT) in this case because the sample size (24 bulbs) is reasonably large (usually considered greater than 30). This allows us to approximate the distribution of the sample means as a normal distribution, regardless of the underlying distribution of the population.