Statistics and Probability Answers

A computer manager needs to know how efficiency of her new computer program depends
on the size of incoming data. Efficiency will be measured by the number of processed
requests per hour. Applying the program to data sets of different sizes, she obtains the
following results,
Data size (gigabytes) 6 7 7 8 10 10 15
Processed requests 40 55 50 41 17 26 16
i. Draw the scatterplot for the data. Be sure to label your axes.
ii. Is there any correlation between the processing request and the size of incoming data?
What is the correlation coefficient?
iii. By what percentage is the processing time dependent on the size of incoming data?

Problem:
The standard deviation of the breaking strengths of certain cables produced by a company is given as 240 lbs. After a change was introduced in the process of manufacture of these cables, the breaking strengths of a sample of 8 cables showed a standard deviation of 300 lbs. investigate the significance of the apparent increase in variability, using a significance level of (a) 0.05 and (b)0.01.

A random variable X has the cumulative distribution function given as
F (x) =
8><>:
0; for x < 1
x2 − 2x + 2
2 ; for 1 ≤ x < 2
1; for x ≥ 2
Calculate the variance of X

A random variable X has the cumulative distribution function given as
F (x) =
8><>:
0; for x < 1
x2 − 2x + 2
2 ; for 1 ≤ x < 2
1; for x ≥ 2
Calculate the variance of X

Problem:
Two streams, A and B, suspected of being contaminated, were tested for their degree of acidity. Analysis of the 6 water samples taken from stream A showed that the mean acidity level is 7.52 with a standard deviation of 0.024 while the 5 samples taken at Stream B showed an acidity level of 7.49 with a standard deviation of 0.032. Using a 0.05 significance level, determine whether the streams have different acidity levels.

The loss due to an earthquake in a commercial building is modelled by a random
variable X with density function
f(x) =  0:005(20 0; − x); for 0 elsewhere < x < 20
Given that the fire loss exceeds 10, what is the probability that it exceeds 18

A random variable X has the cumulative distribution function given as
F (x) =
8><>:
0; for x < 1
x2 − 2x + 2
2 ; for 1 ≤ x < 2
1; for x ≥ 2
Calculate the variance of X

A package of 6 fuses are tested where the probability an individual fuse is defective is
0.05. (That is, 5% of all fuses manufactured are defective).
(a) What is the probability one fuse will be defective?
(b) What is the probability at least one fuse will be defective?
(c) What is the probability that more than one fuse will be defective, given that at
least one is defective?

The lifetime of a machine is continuous on the interval (0,40) with probability density function f, where f(t) is proportional to (t + 10)−2, and t is the lifetime in years. Calculate the probability that the lifetime of the machine part is less than 10 years. Hint: Show that f(t) is legitimate and find the proportionality constant.

A store is having a contest. Gold tickets worth $100 are hidden in 5 of 500 coffee cups. When you buy a coffee, a cup is selected and given to you to take home.
If you buy two coffees per day over the course of two weeks, what is the probability that you will win at least one ticket?