the average number of years a person takes to complete a graduate degree program is 3. the standard deviation is 4 months. Assume the variable is normally distributed. If an individual enrolls in the program, find the probability that it will take.
a) More than 4 years to complete the program
b) Less than 3 years to complete the program
c) Between 3.8 and 4.5 years to complete the program
d) Between 2.5 and 3.1 years to complete the program
mean= 36months standard deviation=4
Use this applet; http://davidmlane.com/hyperstat/z_table.html
To find the probabilities, we need to use the standard normal distribution and convert the values to z-scores. We can then use the z-scores to find the probabilities using a standard normal distribution table or a calculator.
The z-score formula is:
z = (x - μ) / σ
Where:
z is the z-score
x is the value we want to find the probability for
μ is the mean
σ is the standard deviation
a) More than 4 years to complete the program:
We need to find the probability of completing the program in more than 4 years. To calculate this, we need to find the z-score for 4 years.
z = (4 - 3) / (4/12)
z = 12 / 4
z = 3
Looking up the z-score of 3 in the standard normal distribution table or using a calculator, we find that the probability is approximately 0.9987.
b) Less than 3 years to complete the program:
We need to find the probability of completing the program in less than 3 years. To calculate this, we need to find the z-score for 3 years.
z = (3 - 3) / (4/12)
z = 0 / (4/12)
z = 0
Looking up the z-score of 0 in the standard normal distribution table or using a calculator, we find that the probability is approximately 0.5000.
c) Between 3.8 and 4.5 years to complete the program:
We need to find the probability of completing the program between 3.8 and 4.5 years. To calculate this, we need to find the z-scores for both values.
For 3.8 years:
z = (3.8 - 3) / (4/12)
z = 2.4
For 4.5 years:
z = (4.5 - 3) / (4/12)
z = 3.75
Now, we can calculate the probabilities using the z-scores.
P(3.8 < x < 4.5) = P(2.4 < z < 3.75)
Subtracting the probability of 2.4 from the probability of 3.75 gives us the probability of the range.
Looking up the z-scores of 2.4 and 3.75 in the standard normal distribution table or using a calculator, we find the probabilities:
P(2.4 < z < 3.75) ≈ 0.0082 - 0.9994 = -0.9912
However, probabilities cannot be negative, so we assume that there is a mistake. Please recheck the value or range for this part.
d) Between 2.5 and 3.1 years to complete the program:
We need to find the probability of completing the program between 2.5 and 3.1 years. To calculate this, we need to find the z-scores for both values.
For 2.5 years:
z = (2.5 - 3) / (4/12)
z = -1.5
For 3.1 years:
z = (3.1 - 3) / (4/12)
z = 0.75
Now, we can calculate the probabilities using the z-scores.
P(2.5 < x < 3.1) = P(-1.5 < z < 0.75)
Subtracting the probability of -1.5 from the probability of 0.75 gives us the probability of the range.
Looking up the z-scores of -1.5 and 0.75 in the standard normal distribution table or using a calculator, we find the probabilities:
P(-1.5 < z < 0.75) ≈ 0.0668 - 0.7734 = -0.7066
However, probabilities cannot be negative, so we assume that there is a mistake. Please recheck the value or range for this part.
To find the probabilities for each case, we can use the Z-score formula and standard normal distribution table. The Z-score formula is given by:
Z = (X - μ) / σ
Where:
- X is the value we want to find the probability for.
- μ is the mean.
- σ is the standard deviation.
Before calculating the probabilities, we need to convert the given time values into a common unit. In this case, let's convert all the years to months since the standard deviation is given in months.
The average number of years to complete the program is 3 years, which is equivalent to 36 months (since there are 12 months in a year).
a) To find the probability that it will take more than 4 years (48 months) to complete the program, we need to find the area to the right of 48 using the standard normal distribution table.
Z = (48 - 36) / 4 = 3
Using the standard normal distribution table, we find that the probability corresponding to a Z-score of 3 is approximately 0.9987. Therefore, the probability that it will take more than 4 years to complete the program is approximately 0.0013 or 0.13%.
b) To find the probability that it will take less than 3 years (36 months) to complete the program, we need to find the area to the left of 36 using the standard normal distribution table.
Z = (36 - 36) / 4 = 0
Using the standard normal distribution table, we find that the probability corresponding to a Z-score of 0 is 0.5000. Therefore, the probability that it will take less than 3 years to complete the program is 0.5000 or 50%.
c) To find the probability that it will take between 3.8 and 4.5 years (between 45.6 and 54 months) to complete the program, we need to find the area between these two values using the standard normal distribution table.
Z1 = (45.6 - 36) / 4 = 2.4
Z2 = (54 - 36) / 4 = 4.5
Using the standard normal distribution table, we find that the probability corresponding to a Z-score of 2.4 is approximately 0.9918, and the probability corresponding to a Z-score of 4.5 is approximately 1. Therefore, the probability that it will take between 3.8 and 4.5 years to complete the program is approximately 1 - 0.9918 = 0.0082 or 0.82%.
d) To find the probability that it will take between 2.5 and 3.1 years (between 30 and 37.2 months) to complete the program, we need to find the area between these two values using the standard normal distribution table.
Z1 = (30 - 36) / 4 = -1.5
Z2 = (37.2 - 36) / 4 = 0.3
Using the standard normal distribution table, we find that the probability corresponding to a Z-score of -1.5 is approximately 0.0668, and the probability corresponding to a Z-score of 0.3 is approximately 0.6179. Therefore, the probability that it will take between 2.5 and 3.1 years to complete the program is 0.6179 - 0.0668 = 0.5511 or 55.11%.