Final Exam

1.

You’ve started recording how long your cell phone battery lasts between charges. You find the length of time between charges, x, is normally distributed with a mean of 10 hours and a standard deviation of 1.5 hours. What is the probability that your cell phone battery will last between 8 and 12 hours between charges?

2.  A university asked 10 graduate students how many hours of homework they were planning to do that week. Here are their responses:

14       13       15      21      24      25      28      25      31

Answer the following questions about this data:

1. What’s the mean number of hours of homework done by these students per week? What’s the median? Why are these figures different?
2. What’s the range of the data?
3. Is the data skewed? If so in what direction? How do you know?

3.

Your economics professor is analyzing exam grades to determine whether her exams are too hard or too easy. The stem-and-leaf plot below was made of the final exam grades of student in all her courses. Answer the following questions about the data.

Stem-and-leaf plot:

1. Create a box plot from the data and draw a line at the median grade. What is the median grade?
2. At your university, grades in the 90s get an A, those in the 80s get a B, those in the 70s get a C, those in the 60s get a D, and those below 60 get an F. What is the mean and mode of the data set? What letter grades do they represent?
3. If your economics professor is aiming for a B average, are her final exams too difficult? How do you know?

4. A local charity is holding a prize raffle to raise money for their cause. The raffle is being run by a lottery, in which each participant picks a set of 3 numbers ranging from 1 to 21, The three winning numbers are selected at random, and the prize drawing is done only once. If you decide to buy a raffle ticket what are your chances winning?

5.  The mean time for a I00 meter race at a college track meet is 13.2 seconds, with a standard deviation of 0.9 seconds. To win, the next sprinter needs to run the race in 12.5 seconds or less. Assuming this random variable is normally distributed, what is the probability of the sprinter running the race in short enough time to take the lead?

6. A denim company sells its jeans both online and at a retail store. Assume that 80% of the company’s  sales are retail, and of 20% sales are online.

1. What’s the probability that all of the next four pairs of Jeans are sold online?
2. What’s the probability that three out of the next four pairs of Jeans are sold online?
3. Use your answers from parts (a) and (b) to derive a formula for p(x), the probability distribution of the binomial random variable x, the number of the next four pairs of jeans sold online.

7.  Suppose you purchase shares of stock in five different companies. Assume that 70% of each of these investments will increase in value, and the performance of each company’s  stock is  independent of one another, and that the probability disPtribution for x, the number of  successful investments  out of the five is as follows:

Find the mean of the discrete random variable, x; then use the mean to find the variance σ² , and the standard deviation, σ.

8. A toy manufacturer needs to test its new product to make sure it doesn’t present defect to small children.  It needs to estimate the fraction of toys that could potentially be defective, less than .01 with a 95% confidence interval.  A sample fraction of defective from other ṕ is 0.04. How many toys would the manufacturer have to randomly sample to make sure toys aren’t defective?

9.  An environmental scientist is catching fish from the Great Lakes and testing them for the presence of an invasive parasite. After a sample analysis that included 175 rounds testing, the scientist recorded that the sample mean infection rate, µ is 54.37 fish per round and the sample standard deviation is 7.07 fish per round.

1. Construct a 95% confidence interval for the mean infection rate in the total fish population. Explain what this figure means.
2. Explain how your work in part would have been different if the sample size had been 12 rounds testing instead of 175.

10.  Up to 20% of people in the United States will catch the flu this year. Use this figure as the probability mean and assume you work in an office space with 30 coworkers:

1. How many of your coworkers would you expect to get the flu this year?
2. What’s the standard deviation?
3. Using the appropriate approximation, determine P(x > 1), that is, the probability that one of your coworker will get the flu this year.