Question (1) According to a recent study, 1 in every 10 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled 25 women and asked each whether she has been a victim of domestic abuse at some point in her life.
(a) Verify that the 4 assumptions required for an experiment to be a Binomial experiment are satisfied.
(b) Find the probabilitythat more than 22 of the women sampled HAVE NOT been the victim of domestic abuse.
Question (2) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 450 seconds and a standard deviation of 40 seconds. Find the probability that a randomly selected boy in secondary school can run the mile in less than 358 seconds.
Question (3) You are interested in purchasing a new car. One of the many points you wish to consider is the resale value of the car after 5 years. Since you are particularly interested in a certain foreign sedan, you decide to estimate the resale value of this car with a 95% confidence interval. (You have determined that the population of this 5-year-old foreign sedan is normal, but you do not know the population standard deviation). You manage to obtain data on 17 recently resold 5-year-old foreign sedans of this sedan. These 17 cars were resold at an average price of $12,610 with a standard deviation of $700.
What is the 95% confidence interval for the true mean resale value of a 5- year-old car of this model?
Question (4) A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will choose a random sample from the 100,000 items in the store's inventory in order to determine the proportion of merchandise that is outdated. The current owners have never determined the percentage of outdated merchandise and cannot help the buyers (that is, there is no prior estimate of outdated merchandise).
How large a sample do the buyers need in order to be 95% confident that the margin of error of their estimate is 5%? (Hint: See Example 8.14 in the Illowsky text).
Question (5) A local bakery has determined a probability distribution for the number of cheesecakes it sells in a given day. The distribution is as follows:
Determine the expected number of cheesecakes (that is, the mean of the numbers sold on a day) that this local bakery expects to sell in a day.
Question (6) The number of traffic accidents that occur on a particular stretch of road during a month follows a Poisson distributionwith a mean of 8.8. Find the probability that fewer than three accidents will occur next month on this stretch of road.
Question (7) In a box of 50 markers, 30 markers are either red or black, 20 are missing their caps, and 12 markers are either red or black AND are missing their caps.
Determine whether the events "red or black" and "missing cap" are dependent or independent events. Support your answer with an appropriate calculation. (Hint: What formula involving p(A) and p(B) must by true if the events A and B are independent?)
Question (8) Each manager of a corporation was rated as being either a good, fair, or poor manager by his/her boss. The manager's educational background was also noted. The data appear below:
Using the data in the above table, compute the probability that a randomly chosen manager is either a good manager OR has an advanced degree?
(Hint: Which rule of probability governs the OR of two events?)
Question (9) A typical woman in the 35-45 year-old age group has an IQ score of 106 with a standard deviation of 24. Assume the distribution of these IQ scores is normal. If 36 randomly selected women in this age group are selected (assume all have taken the IQ test), compute the probability that the sample mean IQ score of these 36 is between 96 and 116.
Question (10) The scores on a college entrance exam have an approximate normal distribution with a mean, μ = 52 points and a standard deviation, σ = 11 points. By means of the Empirical Rule, determine the following:
(a) About 68% of the data in the distribution are between what 2 values?
(b) About 95% of the data in the distribution are between what 2 values?
(c) Approximately what percentage of the data in the distribution is between the scores of 30 and 63?