Let X be a normal random variable with a mean of -0.11 and a standard deviation of 4.83.
a)Calculate the corresponding z-score (z) for the point?x = 4.0. Give your answer to 2 decimal places.
b)The area under the standard normal probability density function from z to infinity is interpreted as the probability that X is:
|less than or equal to 4.0|
|equal to 4.0|
|greater than or equal to 4.0|
American Expresso sells coffee and assorted pastries. A manager at a particular coffee stand has noticed that muffins contribute a substantial amount to the costs of running that stand. She wants to make sure that the stand is selling enough muffins to justify their continued existence at the stand. Ideally, the manager would like to sell at least 120 muffins each day. However, she notices that on far too many days this condition is not being met.
Some research has shown that the average number of muffins sold each day is 96, and the standard deviation in the number of muffins sold is 25.
You may find this?standard normal table?useful throughout the following questions.
a)Calculate the percentage of days on which the coffee stand sells less than the ideal number of muffins. Give your answer as a percentage to 1 decimal place.
Percentage of 'failure' days =?%
b)The manager is not happy with this at all. She would like to have no more than 2.5% of days being failures. However, at the moment the manager can't do anything about the average number of muffins sold and the standard deviaiton in muffins sold. Her only option is to try and lower costs, to change the limit of what is considered a 'failure'. Calculate the number x such that the following statement is true:
'The coffee stand sells less than x muffins on 2.5% of days or less.'
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Sam 'Vandelay' Johnson plays basketball for his college team. You've observed that the probability of Sam making a given shot is 0.5 and that the success of a given shot is independent of other shots. Over the course of many games, Sam takes 100 attempted shots at the basket. Let W be the random variable that is the number of successful shots.
You should use the normal approximation to the binomial to calculate the probabilities in parts b) and c). Give your answers as decimals to 4 decimal places.
a)Find the probability that Sam makes exactly 52 successful shots from the 100 attempts.
P(W = 52) =?
b)Find the probability that Sam makes at most 52 successful shots from the 100 attempts.
P(W ? 52) =?
c)Find the probability that Sam makes between 50 and 50 successful shots from the 100 attempts.
P(50 ? W ? 50) =?
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