Questions and Answers of Introduction To Probability Statistics

With reference to Exercise 10.42, verify that the mean of the observed distribution is 1.6 , corresponding to \(40 \%\) of the cars requiring repairs. Then look up the probabilities for \(n=5\) and
The following is the distribution of the hourly number of trucks arriving at a company's warehouse:Find the mean of this distribution, and using it (rounded to one decimal place) as the parameter
Among 100 purification filters used in an experiment, 46 had a service life of less than 20 hours, 19 had a service life of 20 or more but less than 40 hours, 17 had a service life of 40 or more but
A chi square test is easily implemented on a computer. With the countsfrom Example 8 in columns 1-4, the MINITAB commandsproduce the output Expected counts are printed below observed countsRepeat the
The procedure in Exercise 10.46 also calculates the chi square test for independence. Do Exercise 10.40 using the computer.Data From Exercise 10.46Data From Exercise 10.40 10.46 A chi square test is
In a sample of 100 ceramic pistons made for an experimental diesel engine, 18 were cracked. Construct a \(95 \%\) confidence interval for the true proportion of cracked pistons using the large sample
With reference to Exercise 10.48, test the null hypothesis \(p=0.20\) versus the alternative hypothesis \(pData From Exercise 10.48 10.48 In a sample of 100 ceramic pistons made for an ex- perimental
In a random sample of 160 workers exposed to a certain amount of radiation, 24 experienced some ill effects. Construct a \(99 \%\) confidence interval for the corresponding true percentage using the
With reference to Exercise 10.50, test the null hypothesis \(p=0.18\) versus the alternative hypothesis \(p eq 0.18\) at the 0.01 level.Data From Exercise 10.50 10.50 In a random sample of 160
In a random sample of 150 trainees at a factory, 12 did not complete the training. Construct a 99% confidence interval for the true proportion of trainees not completing their training using the
With reference to Exercise 10.52, test the hypothesis \(p=0.05\) versus the alternative hypothesis \(p>0.05\) at the 0.05 level.Data From Exercise 10.52 10.52 In a random sample of 150 trainees at
Refer to Example 5 but suppose there are two additional design plans \(\mathrm{B}\) and \(\mathrm{C}\) for making miniature drones. Under B, 10 of 40 drones failed the initial test and under C 15 of
As a check on the quality of eye glasses purchased over the internet, glasses were individually ordered from several different online vendors. Among the 92 lenses with antireflection coating, 61
With reference to Exercise 10.55, find a large sample 95% confidence interval for the true difference of probabilities.Data From Exercise 10.55 10.55 As a check on the quality of eye glasses
Two bonding agents, \(A\) and \(B\), are available for making a laminated beam. Of 50 beams made with Agent \(A, 11\) failed a stress test, whereas 19 of the 50 beams made with Agent \(B\) failed. At
With reference to Exercise 10.57, find a large sample 95% confidence interval for the true difference of the probabilities of failure.Data From Exercise 10.57 10.57 Two bonding agents, A and B, are
Cooling pipes at three nuclear power plants are investigated for deposits that would inhibit the flow of water. From 30 randomly selected spots at each plant, 13 from the first plant, 8 from the
Two hundred tires of each of four brands are individually placed in a testing apparatus and run until failure. The results are obtained the results shown in the following table:(a) Use the 0.01 level
The following is the distribution of the daily number of power failures reported in a western city on 300 days:Test at the 0.05 level of significance whether the daily number of power failures in
With reference to Example 13, repeat the analysis after combining the categories below average and average in the training program and the categories poor and average in success. Comment on the form
Mechanical engineers, testing a new arc-welding technique, classified welds both with respect to appearance and an X-ray inspection.Test for independence using \(\alpha=0.05\) and find the individual
With reference to the previous exercise, construct a 90% confidence interval for the true population mean quantity of gravel in concrete mixes.Data From Previous Exercise 7.1 A construction engineer
With reference to the previous exercise, construct a 95% confidence interval for the true population mean labor time.Data From Previous Exercise 7.3 An industrial engineer collected data on the labor
With reference to the previous exercise, construct a 98% confidence interval for the true population mean number of unremovable defects per display.Data From Previous Exercise 7.5 The manufacture of
With reference to the \(n=50\) interrequest time observations in Example 6, Chapter 2, which have mean 11,795 and standard deviation 14,056, what can one assert with \(95 \%\) confidence about the
With reference to the previous exercise, construct a 95% confidence interval for the true mean inter-request time.Data From Previous Exercise 7.7 With reference to the n = 50 interrequest time ob-
With reference to the previous exercise, assume that production has a normal distribution and obtain a \(99 \%\) confidence interval for the true mean production of the pilot process.Data From
Refer to Example 1 and the data on the resiliency modulus of recycled concrete.(a) Obtain a 95% confidence interval for the population mean resiliency modulus \(\mu\).(b) Is the population mean
Suppose that in the preceding exercise the first measurement is recorded incorrectly as 16.0 instead of 14.5. Show that, even though the mean of the sample increases to \(\bar{x}=14.7\), the null
With reference to the preceding exercise, construct a \(95 \%\) confidence interval for the true average increase in the pulse rate of astronaut trainees performing the given task.Data From Preceding
Suppose that in the lithium car battery example on page \(242, n\) is changed from 36 to 50 while the other quantities remain \(\mu_{0}=1600, \sigma=192\), and \(\alpha=\) 0.03. Find(a) the new
With reference to the previous exercise, find a \(90 \%\) confidence interval for the difference of the two means.Data From Previous Exercise 8.26 With reference to Exercise 2.64, test that the mean
With reference to the preceding exercise, find the corresponding distribution function and use it to determine the probabilities that a random variable having this distribution function will take on
With reference to the preceding exercise, find the corresponding distribution function, and use it to determine the probabilities that a random variable having the distribution function will take on
With reference to the preceding exercise, for which temperature is the probability 0.05 that it will be exceeded during one day?Data From Preceding Exercise Determining a joint cumulative
With reference to the preceding exercise, find the probabilities that the random variable will take on a value(a) less than 8.0;(b) between 4.5 and 6.5 .Data From Preceding Exercise Determining a
With reference to the preceding exercise, find the marginal densities of the two random variables.Data From Preceding Exercise Determining a joint cumulative distribution function Find the joint
With reference to the preceding exercise, find the joint cumulative distribution function of the two random variables and use it to verify the value obtained for the probability.Data From Preceding
With reference to the preceding exercise, check whether(a) the three random variables are independent;(b) any two of the three random variables are pairwise independent.Data From preceding Exercise
A pair of random variables has the circular normal distribution if their joint density is given by\[\begin{aligned}& f\left(x_{1}, x_{2}\right) \\& \quad=\frac{1}{2 \pi \sigma^{2}}
The MINITAB commandswill create a normal scores plot from observations that were set in C1. (MINITAB uses a variant of the normal scores, \(m_{i}\), that we defined.) Construct a normal scores plot
With reference to the preceding exercise, find the corresponding distribution function and use it to determine the probabilities that a random variable having this distribution function will take on
Use the computing formula for \(\sigma^{2}\) to rework part (b) of the preceding exercise.Data From Preceding Exercise Determining a joint cumulative distribution function Find the joint cumulative
Refer to the example on page 84 but suppose the manufacturer has difficulty getting enough LED screens. Because of the shortage, the manufacturer had to obtain \(40 \%\) of the screens from the
Use the data of Exercise 7.14 to estimate \(\sigma\) for the key performance indicator in terms of(a) the sample standard deviation;(b) the sample range.Compare the two estimates by expressing their
With reference to Example 7, Chapter 8, use the range of the second sample to estimate \(\sigma\) for the resiliency modulus of recycled materials from the second location. Compare the result with
Use the data of part(a) of Exercise 8.13 to estimate \(\sigma\) for the Brinell hardness of Alloy 1 in terms of (a) the sample standard deviation;(b) the sample range.Compare the two estimates by
With reference to Exercise 7.56, construct a 95% confidence interval for the variance of the yield.Data From Exercise 7.56 7.65 The statistical program MINITAB will calculate t tests. With the
With reference to Exercise 7.63, construct a \(99 \%\) confidence interval for the variance of the population sampled.Data From Exercise 7.63 7.63 A manufacturer claims that the average tar content
Use the value \(s\) obtained in Exercise 9.3 to construct a \(98 \%\) confidence interval for \(\sigma\), measuring the actual variability in the hardness of Alloy 1.Data From Exercise 9.3Data From
With reference to Exercise 7.62, test the null hypothesis \(\sigma=600\) psi for the compressive strength of the given kind of steel against the alternative hypothesis \(\sigma>600\) psi. Use the
If 15 determinations of the purity of gold have a standard deviation of 0.0015 , test the null hypothesis that \(\sigma=0.002\) for such determinations. Use the alternative hypothesis \(\sigma eq
With reference to Exercise 8.5, test the null hypothesis that \(\sigma=0.75\) hours for the time that is required for repairs of the second type of bulldozer against the alternative hypothesis that
Use the 0.01 level of significance to test the null hypothesis that \(\sigma=0.015\) inch for the diameters of certain bolts against the alternative hypothesis that\(\sigma eq 0.015\) inch, given
Playing 10 rounds of golf on his home course, a golf professional averaged 71.3 with a standard deviation of 2.64 .(a) Test the null hypothesis that the consistency of his game on his home course is
The fire department of a city wants to test the null hypothesis that \(\sigma=10\) minutes for the time it takes a fire truck to reach a fire site against the alternative hypothesis \(\sigma eq 10\)
Explore the use of the two sample \(t\) test in Exercise 8.9 by testing the null hypothesis that the two populations have equal variances. Use the 0.02 level of significance.Data From Exercise 8.9
With reference to Exercise 8.10, use the 0.10 level of significance to test the assumption that the two populations have equal variances.Data From Exercise 8.10 8.10 We know that silk fibers are very
Two different computer processors are compared by measuring the processing speed for different operations performed by computers using the two processors. If 12 measurements with the first processor
With reference to Exercise 8.6, where we had \(n_{1}=\) \(40, n_{2}=30, s_{1}=15.2\), and \(s_{2}=18.7\), use the 0.05 level of significance to test the claim that there is a greater variability in
With reference to Example 20, Chapter 7, construct a 95% confidence interval for the true standard deviation of the lead content.Data From Example 20 EXAMPLE 20 At test of a normal population mean
If 44 measurements of the refractive index of a diamond have a standard deviation of 2.419 , construct a 95% confidence interval for the true standard deviation of such measurements. What assumptions
Past data indicate that the variance of measurements made on sheet metal stampings by experienced quality-control inspectors is 0.18 (inch) \({ }^{2}\). Such measurements made by an inexperienced
Thermal resistance tests on 13 samples of Enterococcus hirae, present in milk, yield the following results in degrees Celsius:Another set of seven samples of milk was tested after pasteurization to
With reference to the Example 8, Chapter 8, test the equality of the variances for the two aluminum alloys. Use the 0.02 level of significance.Data From Example 8 EXAMPLE 8 Graphics to accompany a
With reference to the Example 8, Chapter 8, find a 98% confidence interval for the ratio of variances of the two aluminum alloys.Data From Example 8 EXAMPLE 8 Graphics to accompany a two sample /
MINITAB calculation of \(t_{\alpha}, \chi_{v}^{2}\), and \(F_{\alpha}\)The software finds percentiles, so to obtain \(F_{\alpha}\), we first convert from \(\alpha\) to \(1-\alpha\). We illustrate
A bioengineering company manufactures a device for externally measuring blood flow. Measurements of the electrical output (milliwatts) on a sample of 16 units yields the data plotted in Figure
An inspector examines every twentieth piece coming off an assembly line. List some of the conditions under which this method of sampling might not yield a random sample.
Large maps are printed on a plotter and rolled up. The supervisor randomly selects 12 printed maps and unfolds a part of each map to verify the quality of the printing. List one condition under which
Explain why the following will not lead to random samples from the desired populations.(a) To determine what the average person spends on a vacation, a market researcher interviews passengers on a
A market research organization wants to try a new product in 8 of 50 states. Use Table 7W or software to make this selection.Data From Table 7W Table 1.3 Random digits (portion of Table 7W) 1306 1189
How many different samples of size \(n=4\) can be chosen from a finite population of size(a) \(N=15\) ?(b) \(N=35\) ?
With reference to Exercise 6.5, what is the probability of each sample in part(a) and the probability of each sample in part(b) if the samples are to be random?Data From Exercise 6.5 6.5 How many
Take 30 slips of paper and label five each -4 and 4, four each -3 and 3 , three each -2 and 2 , and two each \(-1,0\) and 1 .(a) If each slip of paper has the same probability of being drawn, find
Repeat Exercise 6.7, but select each sample with replacement; that is, replace each slip of paper and reshuffle before the next one is drawn.Data From Exercise 6.7 6.7 Take 30 slips of paper and
Given an infinite population whose distribution is given bylist the 25 possible samples of size 2 and use this list to construct the distribution of \(\bar{X}\) for random samples of size 2 from the
Suppose that we convert the 50 samples referred to on page 197 into 25 samples of size \(n=20\) by combining the first two, the next two, and so on. Find the means of these samples and calculate
When we sample from an infinite population, what happens to the standard error of the mean if the sample size is(a) increased from 40 to 1,000 ?(b) decreased from 256 to 65 ?(c) increased from 225 to
What is the value of the finite population correction factor in the formula for \(\sigma_{\bar{X}}^{2}\) when(a) \(n=8\) and \(N=640\) ?(b) \(n=100\) and \(N=8,000\) ?(c) \(n=250\) and \(N=20,000\) ?
For large sample size \(n\), verify that there is a \(50-50\) chance that the mean of a random sample from an infinite population with the standard deviation \(\sigma\) will differ from \(\mu\) by
The mean of a random sample of size \(n=25\) is used to estimate the mean of an infinite population that has standard deviation \(\sigma=2.4\). What can we assert about the probability that the error
Engine bearings depend on a film of oil to keep shaft and bearing surfaces separated. Insufficient lubrication causes bearings to be overloaded. The insufficient lubrication can be modeled as a
A wire-bonding process is said to be in control if the mean pull strength is 10 pounds. It is known that the pull-strength measurements are normally distributed with a standard deviation of 1.5
If the distribution of scores of all students in an examination has a mean of 296 and a standard deviation of 14 , what is the probability that the combined gross score of 49 randomly selected
If \(X\) is a continuous random variable and \(Y=X-\mu\), show that \(\sigma_{Y}^{2}=\sigma_{X}^{2}\).
Prove that \(\mu_{\bar{X}}=\mu\) for random samples from discrete (finite or countably infinite) populations.
The tensile strength (1,000 psi) of a new composite can be modeled as a normal distribution. A random sample of size 25 specimens has mean \(\bar{x}=45.3\) and standard deviation s=7.9. Does this
The following is the time taken (in hours) for the delivery of 8 parcels within a city: 28,32,20,26, 42,40,28, and 30 . Use these figures to judge the reasonableness of delivery services when they
The process of making concrete in a mixer is under control if the rotations per minute of the mixer has a mean of 22 rpm. What can we say about this process if a sample of 20 of these mixers has a
Engine bearings depend on a film of oil to keep shaft and bearing surfaces separated. Samples are regularly taken from production lines and each bearing in a sample is tested to measure the thickness
A random sample of 15 observations is taken from a normal population having variance \(\sigma^{2}=90.25\). Find the approximate probability of obtaining a sample standard deviation between 7.25 and
If independent random samples of size \(n_{1}=n_{2}=8\) come from normal populations having the same variance, what is the probability that either sample variance will be at least 7 times as large as
Find the values of(a) \(F_{0.95}\) for 15 and 12 degrees of freedom;(b) \(F_{0.99}\) for 5 and 20 degrees of freedom.
The chi square distribution with 4 degrees of freedom is given by\[f(x)= \begin{cases}\frac{1}{4} \cdot x \cdot e^{-x / 2} & x>0 \\ 0 & x \leq 0\end{cases}\]Find the probability that the variance of
The \(t\) distribution with 1 degree of freedom is given by\[f(t)=\frac{1}{\pi}\left(1+t^{2}\right)^{-1} \quad-\inftyVerify the value given for \(t_{0.05}\) for \(v=1\) in Table 4 .Data From Table 4
The \(F\) distribution with 4 and 4 degrees of freedom is given by\[f(F)= \begin{cases}6 F(1+F)^{-4} & F>0 \\ 0 & F \leq 0\end{cases}\]If random samples of size 5 are taken from two normal
Let \(Z_{1}, \ldots, Z_{5}\) be independent and let each have a standard normal distribution.(a) Specify the distribution of \(Z_{2}^{2}+Z_{3}^{2}+Z_{4}^{2}+Z_{5}^{2}\).(b) Specify the distribution
Let \(Z_{1}, \ldots, Z_{6}\) be independent and let each have a standard normal distribution. Specify the distribution
Let \(Z_{1}, \ldots, Z_{7}\) be independent and let each have a standard normal distribution.(a) Specify the distribution of \(Z_{1}^{2}+Z_{2}^{2}+Z_{3}^{2}+Z_{4}^{2}\).(b) Specify the distribution

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