A population forms a normal distribution with a mean of sigma=75 and a standard deviation of 20.
- What proportion of the sample means for samples of n=25 have vales less than 80? In other words, find p(M>79) for n=25
- What proportion of the sample means for n=100 have values greater than 79? In other words, find p(m>79) for n=100.
mean of sample means same as population mean. I assume you have a typo and mean mu = 75
sample s = s/sqrt(n)
if n = 25, then sample sigma = 20/5 = 4
Our 79 is 4 more than the means which is 4/4 = 1.0 sigma above the mean
I assume you have a normal distribution table that says that for various values of z, F(z) = integral from - infinity to z of p(z)
for z = 1 F(z) = .8413. That is the probability of mean of sample being less than 1 sigma greater than mean of population
I think you can repeat that for n = 100. However now they are asking for how many are MORE than 4 from the mean. find the table integral from - infinity to your new z and your answer is 1 - F(z)
To find the proportion of sample means that have values less than or greater than a certain value, we need to calculate the z-score for that value and then use the Z-table or Z-score calculator to find the corresponding proportion.
The formula to calculate the z-score for a sample mean is:
Z = (X - μ) / (σ / sqrt(n))
Where:
Z is the z-score
X is the value we want to find the proportion for (in this case, 80 or 79)
μ is the population mean (75 in this case)
σ is the population standard deviation (20 in this case)
n is the sample size
For the first question, we want to find p(M > 79) for n = 25. Let's calculate the z-score:
Z = (80 - 75) / (20 / sqrt(25))
Z = 5 / 4
Z = 1.25
Now, we can look up the proportion corresponding to a z-score of 1.25 in the Z-table or use a Z-score calculator. Let's assume the proportion is P1.
So, p(M > 79) = 1 - P1
For the second question, we want to find p(m > 79) for n = 100. Let's calculate the z-score:
Z = (79 - 75) / (20 / sqrt(100))
Z = 4 / 2
Z = 2
Again, we can look up the proportion corresponding to a z-score of 2 in the Z-table or use a Z-score calculator. Let's assume the proportion is P2.
So, p(m > 79) = 1 - P2