If X is a random variable with distribution X~N(350,110)
find (a)P(X<400) and b)P(250<X<400)
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 calculated.
To find the probabilities in this problem, you will need to use the cumulative distribution function (CDF) of the normal distribution. The CDF gives you the probability of a random variable being less than or equal to a specific value.
For a), we need to find P(X < 400).
Step 1: Standardize the value
To use the CDF, we need to standardize the value 400 using the mean and standard deviation of the distribution.
z = (x - mean) / standard deviation
z = (400 - 350) / 110
z = 0.4545
Step 2: Look up the probability
Using a standard normal distribution table or a calculator, you can find the probability P(Z < 0.4545). This represents the probability of the standardized value being less than 0.4545.
P(Z < 0.4545) ≈ 0.6736
Therefore, P(X < 400) ≈ 0.6736.
For b), we need to find P(250 < X < 400).
Step 1: Standardize the values
To use the CDF, we need to standardize both the lower and upper values using the mean and standard deviation of the distribution.
For the lower value:
z1 = (250 - 350) / 110
z1 = -0.9091
For the upper value:
z2 = (400 - 350) / 110
z2 = 0.4545
Step 2: Calculate the probability
We need to find P(z1 < Z < z2). This represents the probability of the standardized value being between -0.9091 and 0.4545.
Using a standard normal distribution table or a calculator, you can find the probability P(-0.9091 < Z < 0.4545).
P(-0.9091 < Z < 0.4545) ≈ 0.6726
Therefore, P(250 < X < 400) ≈ 0.6726.