Commuters were asked how many times a week they stop for coffee on their way to work. A survey found a mean of 3 times a week with a standard deviation of 0.55. Find the probability that the sum of 100 values is more than 290.
I think the same as the probability that one is more than 2.90
mean = 3
we are at .1 below mean
Z = -.1/.55 = -.18
from normal table if Z = -.18 then F(z) about .421
so about 42% are below so about 58 % are above 2.9
For accurate table use
http://davidmlane.com/hyperstat/z_table.html
57.21% or .5721
To find the probability that the sum of 100 values is more than 290, we need to use the Central Limit Theorem.
The Central Limit Theorem states that if we have a large enough sample size, the distribution of the sample mean will be approximately normal, regardless of the shape of the original population distribution.
In this case, we are given the mean of the original population, which is 3 times a week, and the standard deviation, which is 0.55. We also know that the sample size is 100.
To transform the problem, we need to find the mean and standard deviation of the sample mean. The mean of the sample mean is the same as the mean of the original population, which is 3 times a week.
The standard deviation of the sample mean, also known as the standard error, is calculated by dividing the standard deviation of the original population by the square root of the sample size.
So, the standard error = standard deviation / √n
= 0.55 / √100
= 0.055
Now that we have the mean and standard deviation of the sample mean, we can use the z-score formula to find the probability. The z-score is calculated by subtracting the hypothesized value from the mean and dividing by the standard deviation.
z = (x - μ) / σ
= (290 - 3) / 0.055
= 5145.45
Since we are interested in finding the probability that the sum of 100 values is more than 290, we need to find the area under the normal curve to the right of the z-score.
Using a standard normal table or a calculator, we find that the probability corresponding to a z-score of 5145.45 is essentially 1.
Therefore, the probability that the sum of 100 values is more than 290 is approximately 1.