suppose the replacement times for washing machines are normally distributed with mean 8.6 years and standard devistion of 1.6years.
a. what percentage of washing machines have replacement times less than 7 years?
b. find the probability that a washing machine will have a replacement time of more than 6.5 years?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
To solve these questions, we need to use the concept of Z-scores and the standard normal distribution. A Z-score tells us how many standard deviations a value is from the mean.
a. To find the percentage of washing machines with replacement times less than 7 years, we need to calculate the Z-score for 7 years using the formula:
Z = (X - μ) / σ
where X is the value we're interested in, μ is the mean, and σ is the standard deviation.
Z = (7 - 8.6) / 1.6 = -1
We then look up the Z-score in the standard normal distribution table (or use a calculator) to find the corresponding percentage. A Z-score of -1 corresponds to approximately 0.1587 or 15.87%. Therefore, approximately 15.87% of washing machines have replacement times less than 7 years.
b. To find the probability that a washing machine will have a replacement time of more than 6.5 years, we can use the Z-score again.
Z = (6.5 - 8.6) / 1.6 = -1.31
Looking up the Z-score in the standard normal distribution table, we find that a Z-score of -1.31 corresponds to approximately 0.0948 or 9.48%. Therefore, the probability that a washing machine will have a replacement time of more than 6.5 years is approximately 9.48%.
To answer these questions, we need to use the concept of the standard normal distribution, also known as the Z-score. The Z-score allows us to calculate the probabilities associated with different values in a normally distributed dataset.
In order to calculate the Z-score, we need to know the mean and standard deviation of the dataset. In this case, the mean replacement time is 8.6 years and the standard deviation is 1.6 years.
a. To calculate the percentage of washing machines with replacement times less than 7 years, we first need to calculate the Z-score for the value 7. The formula for calculating the Z-score is:
Z = (X - μ) / σ
Where:
Z is the Z-score
X is the value we want to calculate the probability for
μ is the mean of the dataset (8.6 years in this case)
σ is the standard deviation of the dataset (1.6 years in this case)
Plugging in the values, we get:
Z = (7 - 8.6) / 1.6
Z = -1.0
Once we have the Z-score, we can use a Z-table or a calculator to find the corresponding percentage. Looking up the Z-score of -1.0 in the Z-table, we find that the corresponding percentage is 15.87%. Thus, approximately 15.87% of washing machines have replacement times less than 7 years.
b. To find the probability that a washing machine will have a replacement time of more than 6.5 years, we again need to calculate the Z-score for the value 6.5:
Z = (6.5 - 8.6) / 1.6
Z = -1.31
Using the Z-table or a calculator, we find that the percentage associated with a Z-score of -1.31 is approximately 9.15%. However, we're interested in the probability of a replacement time of more than 6.5 years, so we need to subtract this percentage from 100% to get the desired probability:
Probability = 100% - 9.15% = 90.85%
Therefore, the probability that a washing machine will have a replacement time of more than 6.5 years is approximately 90.85%.