1 of 4 ID: MST.FET.PD.ND.01.0010A Let X be a normal random variable with a mean of -0.62 and a standard deviation of 4.69. a)Calculate the corresponding z-score (z) for the point x = 4.4. Give your answer to 2 decimal places. z= b)The area under the standard normal probability density function from negative infinity to z is interpreted as the probability that X is: less than or equal to 4.4 equal to 4.4 greater than or equal to 4.4 [2 marks]- 2 of 4 ID: MST.FET.PD.ND.02.0040A 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 23. 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.' x= [3 marks]- 3 of4 ID: MST.FET.PD.ND.03.0010A Sam 'Vandelay' Johnson plays basketball for his college team. You've observed that the probability of Sam making a given shot is 0.55 and that the success of a given shot is independent of other shots. Over the course of many games, Sam takes 90 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 53 successful shots from the 90 attempts. P(W = 53) = b)Find the probability that Sam makes at most 53 successful shots from the 90 attempts. P(W 53) = c)Find the probability that Sam makes between 40 and 50 successful shots from the 90 attempts. P(40 W 50) = [1 marks]- 3 of 4 ID: MST.FET.PD.ND.03.0010A A student provides four conditions that a function must satisfy in order to be a probability density function. However, only one of the conditions is actually correct. Of the following four statements, the one that accurately describes a property that a function must possess in order to be a probability density function is: The area between the function and the horizontal axis must be less than or equal to 1. The set of values along the horizontal axis that the function assigns values to must be the set of values between 0 and 1. The function must not go below the horizontal axis at any point. The function must not take values greater than 1