Assume that a random variable under consideration has a density curve. Answer the following questions.

Suppose that the density curve has an area of 0.361 to the left of 13 and an area of 0.428 to the right of 15. What is the size of the area between 13 and 15? (2 marks)

Determine the z-score that marks the 80th percentile of the standard normal distribution. Also determine the z-score that marks the upper 0.6% of the standard normal distribution. (4 marks)

A pharmaceutical company manufactures bottles of a certain medicine. The ml in the bottles follows a normal distribution with a mean of 100 ml and a standard deviation of 2 ml. Find the percentage of medicine bottles that contain more than 104.5 ml of the medicine. (3 marks) Find the percentage of medicine bottles containing between 98 ml and 102 ml. (3 marks) Determine the 1% percentile of the distribution of ml for the medicine bottles. (3 marks) Above what ml are the top 2% of mls among the medicine bottles? (3 marks) If a medicine bottle contains no more than 94 ml of the medicine, it is rejected by the quality control board. That is, a bottle is rejected by the quality control department if it contains less than 94 ml. What percentage of bottles are rejected? (3 marks) A medicine bottle is rejected by the quality control department if it contains less than 94 ml. Compute the probability of at least one rejection among a random sample of 10 bottles. (5 marks)

Suppose we are studying a population with mean =50 and standard deviation =10. Answer the following questions.

What are the parameters (the mean and standard deviation) of the sampling distribution of the same mean for samples of size n = 100? (2 marks) Suppose that the distribution of the population is right-skewed. What is the shape of the sampling distribution of the sample mean in a)? Explain. (2 marks) If the sample size n changes, do the parameters of the sampling distribution of the sample mean also change? Explain your answer fully, considering how they change if they do. (3 marks) Suppose we collect a sample of size n = 20 from this population. Under what condition is the sampling distribution of the sample mean normally distributed? (2 marks)

Let X denote the amount of time a randomly selected customer spends on hold with some insurance company. Suppose X follows a distribution with a mean of 8 hours and a variance of 16 hours. The population density curve for X is shown below. Is the population approximately normally distributed? Explain. (2 marks) If you randomly pick sixteen customers, describe the sampling distribution of the average time spent on hold; provide the mean, standard deviation, and shape. (3 marks) If you randomly pick 121 customers, describe the sampling distribution X of the average time spent in on hold; provide the mean, standard deviation, and shape. (3 marks) If you randomly pick 121 customers, find the probability that the average time spent on hold is within 1 hour of the population mean. HINT: We want P(-1