# probability statistics due today

Probability and Statistics
Due Date: 31/October/2013

Please include all formulas and calculations, as well as a verbal explanation of the results.

1.       Collected data reveals that the average time for students to complete a college admission test is 75 minutes, with a standard deviation of 12 minutes. Also, it was determined that the time that takes the students to complete the test follows a normal distribution.  What is the maximum time that should be given to the students to complete the test if we want that 90% of the students have enough time to complete it?

2.       Assume that the average yearly salary of recently graduated chemical engineers is \$52,000 with a standard deviation of \$4,000. To confirm that average salary, a random sample of 60 recently graduated chemical engineers was taken and the arithmetic mean of their salaries was calculated.

a.       Completely describe the probability distribution of the sample mean .

b.      Between which limits would you expect the sample mean to be with a 95% probability?

c.       Calculate the probability that the sample mean will be above \$52,000.

3.       A study was made between two neighborhoods to compare the amount of emergency calls made during a period of 24 hours. To do this, a random sample of one hundred periods, 24-hours each, were selected from the record books with the following results:

Neighborhood A              Neighborhood B

Sample size                                        100                                         100
Sample mean                                    7.6                                          9.8
Sample variance                               1.45                                        2.65

Find an interval with a 90% confidence for the difference in the amount of emergency calls made between these two neighborhoods and interpret the result.

4.       The records for automotive accidents for a given year in a specific road section were classified for the amount of money in damages caused (under \$2,000 or \$2,000 or more), and if there were physical injuries to passengers involved. The results summarize as follows:

Under \$2,000                     \$2,000 or more
# of accidents                                                    34                                           45
# of accidents with injured pass.                               13                                           25

a.       Make an estimate of the true proportion of accidents in which people result injured when the damages are \$2,000 or more, and find the margin of error for said estimate.

b.      Obtain an estimate of the difference between the proportion of accidents in which people result injured when the damages are under \$2,000 and those in which the damages are \$2,000 or more.

5.       Samples of 600 plastic containers were taken from two injection molding machines. It was found that in the sample taken from “Machine A” there were 60 defective containers, while in the sample taken from “Machine B” there were 90 defective containers. Obtain an estimate of the difference in the fraction of defective containers produced by these two machines with a confidence coefficient of 90%.

6.       It is claimed that people who take a daily dose of 4 mg of vitamin C recover from a common flu in less time than those who don’t. To verify this claim an experiment was conducted with two groups of 35 randomly selected adults. One group was administered a placebo drug, while the other received the recommended vitamin C dose. When the subjects contracted the common flu the researchers observed their recovery time. The results are as follow:

Placebo                                                Vitamin C
Sample size                                        35                                           35
Sample mean                                    6.8 days                                5.8 days
Sample standard deviation          2.9 days                                1.2 days

a.       Identify the null and alternative hypotheses if we want to demonstrate that the consumption of vitamin C indeed reduces the recovering time after a common flu.

b.      Conduct a statistical hypothesis test with the observed data, using a value of (refer to document) and report your conclusion.