The time required to complete a project is known to be normally distributed with a mean of 44 weeks and a standard deviation of 8 weeks.
(a) What is the probability that the project is finished in 40 weeks or fewer?
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40
To find the probability that the project is finished in 40 weeks or fewer, we need to find the area under the normal distribution curve to the left of 40 weeks.
We can use the standard normal distribution table or a calculator to find this probability.
First, we calculate the z-score for 40 weeks using the formula:
z = (x - μ) / σ
where x is the value we're interested in (40 weeks), μ is the mean (44 weeks), and σ is the standard deviation (8 weeks).
z = (40 - 44) / 8
z = -4 / 8
z = -0.5
Now, we can use the standard normal distribution table or a calculator to find the probability associated with the z-score of -0.5. Looking up the z-score in the table or using a calculator, we find that the probability is approximately 0.3085.
Therefore, the probability that the project is finished in 40 weeks or fewer is approximately 0.3085, or 30.85%.
To find the probability that the project is finished in 40 weeks or fewer, we need to use the normal distribution.
Step 1: Standardize the value
To work with the normal distribution, we need to convert the given value (40 weeks) to a standardized value using the z-score formula:
z = (x - mean) / standard deviation
where x is the value we want to standardize, mean is the mean of the distribution, and standard deviation is the standard deviation.
In this case:
x = 40 weeks,
mean = 44 weeks,
standard deviation = 8 weeks.
Plugging in these values, we get:
z = (40 - 44) / 8 = -0.5
Step 2: Find the corresponding cumulative probability
Next, we need to find the cumulative probability associated with the standardized value (-0.5). This can be done by looking up the value in the standard normal distribution table or using statistical software.
The cumulative probability for a z-score of -0.5 is approximately 0.3085.
Step 3: Interpret the result
The probability that the project is finished in 40 weeks or fewer is approximately 0.3085, or 30.85%.
Therefore, there is about a 30.85% chance that the project will be finished in 40 weeks or fewer.