Homework
Set #1
2.1. The probability distribution of x is 𝑓(𝑥) = 𝑘𝑒 −𝑥 , 0 ≤ 𝑥 < ∞. Find the appropriate value
of k. Find the mean and variance of x.
2.2. A manufacturer of electronic calculators offers a one-year warranty. If the calculator fails
for any reason during this period, it is replaced. The time to failure is well modelled by the
following probability distribution:
𝑓(𝑥) = 0.125𝑒 −0.125𝑥
𝑥>0
a. What percentage of the calculators will fail within the warranty period?
b. The manufacturing cost of a calculator is $50, and the profit per sale is $25. What is
the effect of warranty replacements on profit?
2.3. The net contents in ounces of a canned soft drink is a random variable with probability
distribution
𝑓(𝑥) = {
4(𝑥 − 11.75)
4(12.75 − 𝑥)
11.75 ≤ 𝑥 ≤ 12.25
}
12.25 ≤ 𝑥 ≤ 12.75
Find the probability that a can contains less than 12 oz of product.
2.4. A lot of size N = 30 contains five nonconforming units. What is the probability that a
sample of five units selected at random contains exactly one nonconforming units? What is
the probability that it contains one or more nonconformances?
2.5. A quality characteristic of a product is normally distributed with mean µ and standard
deviation one. Specifications on the characteristic are 6 ≤ 𝑥 ≤ 8. A unit that falls within
specification on this quality characteristic results in a profit of C0. However, if 𝑥 < 6, the
profit is −C1, while if 𝑥 > 8, the profit is −C2. Find the value of µ that maximizes the
expected profit.
Set #2
3.1 The inside diameters of bearings used in an aircraft landing gear assembly are known to
have a standard deviation of 𝜎 =0.002 cm. A random sample of 15 bearing has an average
inside diameter of 8.2535 cm. Construct a 95% two-sided confidence interval on the mean
bearing diameter.
3.2 Two machines are used for filling glass bottles with a soft drink beverage. The filling
processes have known standard deviations 𝜎1 = 0.010 litre and 𝜎2 = 0.015 litre, respectively.
A random sample of 𝑛1 = 25 bottles from machine 1 and 𝑛2 = 20 bottles from machine 2
results in average net contents of 𝑥
̅̅̅1 = 2.04 litres and ̅̅̅
𝑥2 = 2.07 litres. Construct a 95%
confidence interval on the difference in mean fill volume. Based on the results of this
calculation, would you conclude that both machines fill to the same volume?
3.3. A random sample of 200 printed circuit boards contains 18 defective or nonconforming
units. Estimate the process fraction nonconforming. Construct a 90% two-sided confidence
interval on the true fraction nonconforming in the production process.
3.4. Consider the data in Question 3.1 above. Test the hypothesis that the mean inside bearing
diameter equals 8.25 cm.
3.5. A new purification unit is installed in a chemical process. Before its installation, a random
sample yielded the following data about the percentage of impurity: ̅̅̅
𝑥1 = 9.85, 𝑆12 = 81.73,
and 𝑛1 = 10. After installation, a random sample resulted in ̅̅̅
𝑥2 = 8.08, 𝑆22 = 78.46, and 𝑛2 =
8.
a. Can you conclude that the two variances are equal?
b. Can you conclude that the new purification device has reduced the mean percentage of
impurity?
3.6. Suppose we wish to test the hypothesis
𝐻0 : µ = 15
𝐻1 : µ ≠ 15
where we know that 𝜎 2 = 9.0. If the true mean is really 20, what sample size must be used to
ensure that the probability of type II error is no greater than 0.10? Assume that 𝛼 = 0.5
Set #3
4.1. Discuss the concepts of chance and assignable causes of variability and the part they play
in statistical quality control.
4.2. A normally distributed quality characteristic is monitored by a control chart with 3-sigma
control limits. Develop a general expression for the probability that a point will plot outside
the control limits when the process is really in control.
Set #4
4.3. A process produces rubber belts in lots of size 2500. Inspection records on the last 20 lots
reveal the following data:
a. Compute trial control limits for a fraction nonconforming control chart.
b. If you wanted to set up a control chart for controlling future production, how would you
use the above data to obtain the center line and control limits for the chart?
4.4. A control chart indicates that the current process fraction nonconforming is 0.02. If the 50
items are inspected each day, what is the probability of detecting a shift in the fraction
nonconforming to 0.04 on the first day after the shift? By the end of the third day following the
shift?
4.5. A control chart is used to control the fraction nonconforming for a plastic part
manufactured in an injection molding process. Ten subgroups yield the following data:
a. Set up a control chart for the number nonconforming in samples of n = 100.
b. For the chart established in part (a), what is the probability of detecting a shift in the
process fraction nonconforming to 0.30 on the first sample after the shift has occurred?
4.6. A fraction nonconforming control chart with centre line 0.10, UCL = 0.19, and LCL = 0.01
is used to control a process.
a. If 3-sigma limits are used, to find the sample size for the control chart.
b. Use the Poisson approximation to the binomial to find the probability of type I error.
c. Use the Poisson approximation to the binomial to find the probability of type II
error if the process fraction defective is actually p = 0.20.
4.7. The data below are 𝑥̅ and R values for 24 samples of size n = 5 taken from a process
producing bearings. The measurements are made on the inside diameter of the bearing, with
only the last three decimals recorded (i.e., 34.5 should be 0.50345).
a. Set up 𝑥̅ and R charts on this process. Does the process seem to be in statistical
control? If necessary, revise the trial control limits.
b. If specifications on this diameter are 0.5030 ± 0.0010, fine the percentage of nonconformities produced by this process.
4.8. An 𝑥̅ chart is used to control the mean of a quality characteristic. It is known that σ = 6.0
and n = 4. The centre line = 200, UCL = 209, and LCL = 191. If the process mean shifts to 188,
find the probability that this shift is detected on the first subsequent sample.
Set #5
5.1 The data from an experiment to investigate the effect of glass type and phosphor type
on the brightness of a television tube is shown in the table below. The response measured
is the current necessary (in microamps) to obtain a specified brightness level. Analyse the
data and draw conclusions.
Glass Type
1
Phosphor Type
2
1
280
290
285
230
235
240
300
310
295
260
240
235
2
3
290
285
290
220
225
230
Set #6
6.1 A manufacturer claims that he can design and manufacture a radar set that will have a mean
time between failures of 240 hours based on the exponential distribution. Suppose that a
certain mission requires failure-free operation of the set for 24 hours. What is the chance that
the set will complete a mission without failure?
6.2 The average life of subassembly A is 2,000 hours. Data indicate that this life characteristic is
exponentially distributed.
a. What percent of the subassemblies in the population will last at least 200 hours?
b. The average life subassembly B is 1,000 hours and the life is exponentially distributed.
What percent of the subassemblies in the population will last at least 200 hours?
c. These subassemblies are independently manufactured and then connected in series to
form the total assembly. What percent of assemblies in the population will last at least
200 hours?