Assignment 5
1. If all possible samples of size 16 are drawn from a normal population with mean equal to 50 and standard deviation equal to 5, what is the probability that a sample mean will fall in the interval from to ? Assume that the sample means can be measured to any degree of accuracy.
2. Given the discrete uniform population
find the probability that a random sample of size 54, selected with replacement, will yield a sample mean greater than 4.1 but less than 4.4. Assume the means to be measured to the nearest tenth.
3. A random sample of size 25 is taken from a normal population having a mean of 80 and a standard deviation of 5. A second random sample of size 36 is taken from a different normal population having a mean of 75 and a standard deviation of 3. Find the probability that the sample mean computed from the 25 measurements will exceed the sample mean computed from the 36 measurements by at least 3.4 but less than 5.9. Assume the means to be measured to the nearest tenth.
4. The mean score for freshmen on an aptitude test, at a certain college, is 540, with the standard deviation of 50. What is the probability that two groups of students selected at random, consisting of 32 and 50 students, respectively, will differ in their mean scores by
(a) more than 20 points?
(b) an amount between 5 to 10 points?
Assume the means to be measured to any degree of accuracy.
5. Find the probability that a random sample of 25 observations, from a normal population with variance s2 = 6, will have a variance s2
(a) greater than 9.1;
(b) between 3.462 and 10.745.
Assume the sample variance to be continuous measurements.
6. The scores on a placement test given to college freshman for the past five years are approximately normally distributed with a mean m = 74 and a variance s2 = 8. Would you consider s2 = 8 to be a valid value of the variance if a random sample of 20 students who take this placement test this year obtain a value of s2 = 20?
7. A manufacturing firm claims that the batteries used in their electronic games will last an average of 30 hours. To maintain this average, 16 batteries are tested each month. If the computed t-value falls between – t0.025 and t0.025, the firm is satisfied with its claim. What conclusion should the firm draw from a sample that has a mean = 27.5 hours and a standard deviation s = 5 hours? Assume the distribution of battery lives to be approximately normal.
8. A random sample of 100 automobile owners shows that, in the state of Virginia, an automobile is driven on the average 23,5000 kilometers per year with a standard deviation of 3900 kilometers.
(a) Construct 99% confidence interval for the average number of kilometers an automobile is driven annually in Virginia.
(b) What can we assert with 99% confidence about the possible size of our error if we estimate the average number of kilometers driven by car owners in Virginia to be 23,500 kilometers per year?
9. A machine is producing metal pieces that are cylindrical in shape. A sample of pieces is taken and the diameters are 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01 and 1.03 centimeters. Find a 99% confidence interval for the mean diameter of pieces from this machine, assuming an approximate normal distribution.
10. An experiment was conducted to determine whether surface finish has an effect on the endurance limit of steel. An existing theory says that polishing increases the average endurance limit (reverse bending). An experiment was performed on 0.4% carbon steel using both unpolished and polished smooth-turned specimens was obtained by polishing with No. 0 and No. 00 emery cloth. The data are as follows:
Endurance limit (psi) for
Polished 0.4% carbon unpolished 0.4% carbon
85,500 82,600
91,900 82,400
89,400 81,700
84,000 79,500
89,900 74,400
78,700 69,800
87,500 79,900
83,100 83,400
From a practical point of view, polishing should not have any effect on the standard deviation of the endurance limit, which as known from the performance of numerous endurance limit experiments, to be 4000 psi. Find a 95% confidence interval for the difference between the population means for the two methods, assuming that the populations are approximately normally distributed.