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Statistics : Probability

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Question 1
Consider a random variable, z, that has a standardized normal distribution. Determine the following probabilities:
a. P(0 < z < 1.96)
b. P(z > 1.645)
c. P(1.28 < z < 2.33)
d. P(-2 < z < 3)
e. P(z > -1)

Question 2
Bowser Bites Industries (BBI) sells large bags of dog food to warehouse clubs. BBI uses an automatic filling process to fill the bags. Weights of the filled bags are approximately normally distributed with a mean of 50 kilograms and a standard deviation of 1.25 kilograms.

a. What is the probability that a filled bag will weigh less than 49.5 kilograms?
b. What is the probability that a randomly sampled filled bag will weigh between 48.5 and 51 kilograms?
c. What is the minimum weight a bag of dog food could be and remain in the top 15% of all bags filled?
d. BBI is unable to adjust the mean of the filling process. However, it is able to adjust the standard deviation of the filling process. What would the standard deviation need to be so that no more than 2% of all filled bags weigh more than 52 kilograms?

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