Questions and Answers of Introduction To Probability Statistics

Let the chi square variables $\chi_{1}^{2}$, with $v_{1}$ degrees of freedom, and $\chi_{2}^{2}$, with $v_{2}$ degrees of freedom, be independent. Establish the result on page 211, that their
Let $X_{1}, X_{2}, \ldots, X_{8}$ be 8 independent random variables. Find the moment generating function\[M_{\sum X_{i}}(t)=E\left(e^{t\left(X_{1}+X_{2}+\cdots+X_{8}\right)}\right)\]of the sum when
Let $X_{1}, X_{2}, \ldots, X_{5}$ be 5 independent random variables. Find the moment generating function\[M_{\sum X_{i}}(t)=E\left(e^{t\left(X_{1}+X_{2}+\cdots+X_{5}\right)}\right)\]of the sum when
Let $X_{1}, X_{2}$, and $X_{3}$ be independent normal variables with\[\begin{array}{lll}E\left(X_{1}\right)=5 & \text { and } & \sigma_{1}^{2}=9 \\E\left(X_{2}\right)=-2 & \text { and } &
Refer to Exercise 6.36.(a) Show that $2 X_{1}-X_{2}-4 X_{3}-12$ has a normal distribution.(b) Find the mean and variance of the random variable in part (a).Data From Exercise 6.36 6.36 Let X1, X2,
Let $X_{1}, X_{2}$, and $X_{3}$ be independent normal variables with\[\begin{array}{lll}E\left(X_{1}\right)=-4 & \text { and } & \sigma_{1}^{2}=1 \\E\left(X_{2}\right)=0 & \text { and } &
Refer to Exercise 6.38.(a) Show that $7 X_{1}+X_{2}-2 X_{3}+7$ has a normal distribution.(b) Find the mean and variance of the random variable in part (a).Data From Exercise 6.38 6.38 Let X1, X2,
Let $X_{1}, X_{2}, \ldots, X_{r}$ be $r$ independent random variables each having the same geometric distribution.(a) Show that the moment generating function \(M_{\sum
Refer to Exercise 6.40. Let $X_{1}, X_{2}, \ldots, X_{n}$ be $n$ independent random variables each having a negative binomial distribution with success probability $p$ but where $X_{i}$ has
Referring to Example 16, verify that\[g(y)=\frac{1}{\sqrt{2 \pi}} y^{-1 / 2} e^{-y / 2}\]Data From Example 16 EXAMPLE 16 Not all samples will lead to a correct assessment of water quality Refer to
Use the distribution function method to obtain the density of $Z^{3}$ when $Z$ has a standard normal distribution.
Use the distribution function method to obtain the density of $1-e^{-X}$ when $X$ has the exponential distribution with $\beta=1$.
Use the distribution function method to obtain the density of $\ln (X)$ when $X$ has the exponential distribution with $\beta=1$.
Use the transformation method to obtain the density of $X^{3}$ when $X$ has density $f(x)=1.5 X$ for \(0
Use the transformation method to obtain the distribution of $-\ln (X)$ when $X$ has the uniform distribution on $(0,1)$.
Use the convolution formula, Theorem 6.9, to obtain the density of $X+Y$ when $X$ and $Y$ are independent and each has the exponential distribution with $\beta=1$.Data From Theorem 6.9
Use the transformation method, Theorem 6.9, to obtain the distribution of the ratio $Y / X$ when when $X$ and $Y$ are independent and each has the same gamma distribution.Data From Theorem 6.9
Use the discrete convolution formula, Theorem 6.10, to obtain the probability distribution of $X+Y$ when $X$ and $Y$ are independent and each has the uniform distribution on $\{0,1,2\}$.Data
The panel for a national science fair wishes to select 10 states from which a student representative will be chosen at random from the students participating in the state science fair.(a) Use Table
How many different samples of size $n=2$ can be chosen from a finite population of size(a) $N=12$;(b) $N=20$ ?
With reference to Exercise 6.52, what is the probability of choosing each sample in part(a) and the probability of choosing each sample in part (b), if the samples are to be random?Data From Exercise
Referring to Exercise 6.52, find the value of the finite population correction factor in the formula for $\sigma_{\bar{X}}^{2}$ for part(a) and part (b).Data From Exercise 6.52 6.52 How many
The time to check out and process payment information at an office supplies Web site can be modeled as a random variable with mean $\mu=63$ seconds and variance $\sigma^{2}=81$. If the sample
The number of pieces of mail that a department receives each day can be modeled by a distribution having mean 44 and standard deviation 8 . For a random sample of 35 days, what can be said about the
If measurements of the elasticity of a fabric yarn can be looked upon as a sample from a normal population having a standard deviation of 1.8 , what is the probability that the mean of a random
Adding graphite to iron can improve its ductile qualities. If measurements of the diameter of graphite spheres within an iron matrix can be modeled as a normal distribution having standard deviation
If 2 independent random samples of size $n_{1}=31$ and $n_{2}=11$ are taken from a normal population, what is the probability that the variance of the first sample will be at least 2.7 times as
If 2 independent samples of sizes $n_{1}=26$ and $n_{2}=8$ are taken from a normal population, what is the probability that the variance of the second sample will be at least 2.4 times the
When we sample from an infinite population, what happens to the standard error of the mean if the sample size is(a) increased from 100 to 200 ;(b) increased from 200 to 300 ;(c) decreased from 360 to
A traffic engineer collects data on traffic flow at a busy intersection during the rush hour by recording the number of westbound cars that are waiting for a green light. The observations are made
Explain why the following may not lead to random samples from the desired population:(a) To determine the mix of animals in a forest, a forest officer records the animals observed after each interval
Several pickers are each asked to gather 30 ripe apples and put them in a bag.(a) Would you expect all of the bags to weigh the same? For one bag, let $X_{1}$ be the weight of the first apple,
A construction engineer collected data from some construction sites on the quantity of gravel (in metric tons) used in mixing concrete. The quantity of gravel for n=24 sites4861 5158 8642
An industrial engineer collected data on the labor time required to produce an order of automobile mufflers using a heavy stamping machine. The data on times (hours) for $n=52$ orders of different
The manufacture of large liquid crystal displays (LCD's) is difficult. Some defects are minor and can be removed; others are unremovable. The number of unremovable defects, for each of $n=45$
In a study of automobile collision insurance costs, a random sample of 80 body repair costs for a particular kind of damage had a mean of $\$ 472.36$ and a standard deviation of $\$ 62.35$. If
Refer to Example 8. How large a sample will we need in order to assert with probability 0.95 that the sample mean will not differ from the true mean by more than 1.5. (replacing $\sigma$ by $s$
The dean of a college wants to use the mean of a random sample to estimate the average amount of time students take to get from one class to the next, and she wants to be able to assert with \(99
An effective way to tap rubber is to cut a panel in the rubber tree's bark in vertical spirals. In a pilot process, an engineer measures the output of latex from such cuts. Eight cuts on different
To monitor complex chemical processes, chemical engineers will consider key process indicators, which may be just yield but most often depend on several quantities. Before trying to improve a
With reference to the previous exercise, assume that the key performance indicator has a normal distribution and obtain a $95 \%$ confidence interval for the true value of the indicator.
Refer to Exercise 2.34, page 46, concerning the number for board failures for $n=32$ integrated circuits (IC). A computer calculates $\bar{x}=7.6563$ and $s=5.2216$. Obtain a 95\% confidence
Refer to the $2 \times 4$ lumber strength data in Exercise 2.58, page 48. According to the computer output, a sample of $n=30$ specimens had $\bar{x}=1908.8$ and $s=$ 327.1. Find a $95 \%$
Refer to the data on page 50, on the number of defects per board for Product B. Obtain a 95% confidence interval for the population mean number of defects per board. 5 Mean of defects 2 3 + 1 19 6 0
With reference to the thickness measurements in Exercise 2.41 , page 47 , obtain a $95 \%$ confidence interval for the mean thickness.Data From Exercise 2.41 2.41 The Aerokopter AK1-3 is an
Ten bearings made by a certain process have a mean diameter of $0.5060 \mathrm{~cm}$ and a standard deviation of $0.0040 \mathrm{~cm}$. Assuming that the data may be looked upon as a random
The freshness of produce at a mega-store is rated a scale of 1 to 5 , with 5 being very fresh. From a random sample of 36 customers, the average score was 3.5 with a standard deviation of 0.8.(a)
A café records that in $n=81$ cases, the coffee beans for the coffee machine lasted an average of 225 cups with a standard deviation of 22 cups.(a) Obtain a 90\% confidence interval for $\mu$,
In an air-pollution study performed at an experiment station, the following amount of suspended benzenesoluble organic matter (in micrograms per cubic meter) was obtained for eight different samples
Modify the formula for $E$ on page 216 so that it applies to large samples which constitute substantial portions of finite populations, and use the resulting formula for the following problems:(a)
Instead of the large sample confidence interval formula for $\mu$ on page 230, we could have given the alternative formula\[\bar{x}-z_{\alpha / 3} \cdot \frac{\sigma}{\sqrt{n}}Explain why the one
Suppose that we observe a random variable having the binomial distribution. Let $X$ be the number of successes in $n$ trials.(a) Show that $\frac{X}{n}$ is an unbiased estimate of the binomial
The statistical program MINITAB will calculate the small sample confidence interval for $\mu$. With the nanopillar height data in $\mathrm{C} 1$,produces the output(a) Obtain a $90 \%$
You can simulate the coverage of the small sample confidence intervals for $\mu$ by generating 20 samples of size 10 from a normal distribution with $\mu=20$ and $\sigma=5$ and computing the
Refer to Example 13, Chapter 3, where 294 out of 300 ceramic insulators were able to survive a thermal shock.(a) Obtain the maximum likelihood estimate of the probability that a ceramic insulator
Refer to Example 7, Chapter 10, where 48 of 60 transceivers passed inspection.(a) Obtain the maximum likelihood estimate of the probability that a transceiver will pass inspection.(b) Obtain the
The daily number of accidental disconnects with a server follows a Poisson distribution. On five days\[\begin{array}{lllll}2 & 5 & 3 & 3 & 7\end{array}\]accidental disconnects are observed.(a) Obtain
In one area along the interstate, the number of dropped wireless phone connections per call follows a Poisson distribution. From four calls, the number of dropped connections is\[\begin{array}{llll}2
Refer to Exercise 7.12.(a) Obtain the maximum likelihood estimates of $\mu$ and $\sigma$.(b) Find the maximum likelihood of the probability that the next run will have a production greater than
Refer to Exercise 7.14.(a) Obtain the maximum likelihood estimates of $\mu$ and $\sigma$.(b) Find the maximum likelihood of the coefficient of variation $\sigma / \mu$.Data From Exercises
Find the maximum likelihood estimator of $p$ when\[f(x ; p)=p^{x}(1-p)^{1-x} \quad \text { for } \quad x=0,1\]
Let $x_{1}, \ldots, x_{n}$ be the observed values of a random sample of size $n$ from the exponential distribution $f(x ; \beta)=\beta^{-1} e^{-x / \beta}$ for $x>0$.(a) Find the maximum
Let $X$ have the negative binomial distribution$f(x)=\left(\begin{array}{l}x-1 \\ r-1\end{array}\right) p^{r}(1-p)^{x-r}$ for $x=r, r+1, \ldots$(a) Obtain the maximum likelihood estimator of
A civil engineer wants to establish that the average time to construct a new two-storey building is less than 6 months.(a) Formulate the null and alternative hypotheses.(b) What error could be made
A manufacturer of four-speed clutches for automobiles claims that the clutch will not fail until after 50,000 miles.(a) Interpreting this as a statement about the mean, formulate a null and
An airline claims that the typical flying time between two cities is 56 minutes.(a) Formulate a test of hypotheses with the intent of establishing that the population mean flying time is different
A manufacturer wants to establish that the mean life of a gear when used in a crusher is over 55 days. The data will consist of how long gears in 80 different crushers have lasted.(a) Formulate the
A statistical test of hypotheses includes the step of setting a maximum for the probability of falsely rejecting the null hypothesis. Engineers make many measurements on critical bridge components to
Suppose you are scheduled to ride a space vehicle that will orbit the earth and return. A statistical test of hypotheses includes the step of setting a maximum for the probability of falsely
Suppose that an engineering firm is asked to check the safety of a dam. What type of error would it commit if it erroneously rejects the null hypothesis that the dam is safe? What type of error would
Suppose that we want to test the null hypothesis that an antipollution device for cars is effective. Explain under what conditions we would commit a Type I error and under what conditions we would
If the criterion on page 242 is modified so that the manufacturer's claim is accepted for $\bar{X}>1640$ cycles, find(a) the probability of a Type I error;(b) the probability of a Type II error
Suppose that in the electric car battery example on page 242, $n$ is changed from 36 to 50 while everything else remains the same. Find(a) the probability of a Type I error;(b) the probability of a
It is desired to test the null hypothesis $\mu=30$ minutes against the alternative hypothesis \(\mu
Several square inches of gold leaf are required in the manufacture of a high-end component. Suppose that, the population of the amount of gold leaf has $\sigma=8.4$ square inches. We want to test
A producer of extruded plastic products finds that his mean daily inventory is 1,250 pieces. A new marketing policy has been put into effect and it is desired to test the null hypothesis that the
Specify the null and alternative hypotheses in each of the following cases.(a) A car manufacturer wants to establish the fact that in case of an accident, the installed safety gadgets saved the lives
Refer to Exercise 7.1 where a construction engineer recorded the quantity of gravel (in metric tons) used in concrete mixes. The quantity of gravel for $n=24$ sites has $\bar{x}=5,818$ tons and
Refer to data in Exercise 7.3 on the labor time required to produce an order of automobile mufflers using a heavy stamping machine. The times (hours) for $n=52$ orders of different parts has
Refer to Exercise 7.5, where the number of unremovable defects, for each of $n=45$ displays, has $\bar{x}=$ 2.467 and $s=3.057$ unremovable defects.(a) Conduct a test of hypotheses with the
Refer to Exercise 7.12, where, in a pilot process, vertical spirals were cut to produce latex from $n=8$ trees to yield (in liters) in a week.26.8 32.5 29.7 24.6 31.5
Refer to Exercise 7.14, where $n=9$ measurements were made on a key performance indicator.\[\begin{array}{lllllllll}123 & 106 & 114 & 128 & 113 & 109 & 120 & 102 &
Refer to Exercise 7.22, where, in $n=81$ cases, the coffee machine needed to be refilled with beans after 225 cups with a standard deviation of 22 cups.(a) Conduct a test of hypotheses with the
Refer to Exercise 2.34, page 46, concerning the number of board failures for $n=32$ integrated circuits. A computer calculation gives $\bar{x}=7.6563$ and $s=$ 5.2216. At the 0.01 level of
In 64 randomly selected hours of production, the mean and the standard deviation of the number of acceptable pieces produced by a automatic stamping machine are $\bar{x}=1,038$ and $s=146$. At
With reference to the thickness measurements in Exercise 2.41 , test the null hypothesis that $\mu=30.0$ versus a two-sided alternative. Take $\alpha=0.05$.Data From Exercise 2.41 2.41 The
A random sample of 6 steel beams has a mean compressive strength of 58,392 psi (pounds per square inch) with a standard deviation of 648 psi. Use this information and the level of significance
A manufacturer claims that the average tar content of a certain kind of cigarette is $\mu=14.0$. In an attempt to show that it differs from this value, five measurements are made of the tar content
The statistical program MINITAB will calculate $t$ tests. With the nanopillar height data in $\mathrm{C} 1$,You must compare your preselected $\alpha$ with the printed $P$-value in order to
Refer to the nanopillar height data on page 25. Using the $95 \%$ confidence interval, based on the $t$ distribution, for the mean nanopillar height(a) decide whether or not to reject \(H_{0}:
Repeat Exercise 7.66 but replace the $t$ test with the large sample $Z$ test.Data From Exercise 7.66 7.66 Refer to the nanopillar height data on page 25. Using the 95% confidence interval,
Refer to the green gas data on page 241 . Using the $95 \%$ confidence interval, based on the $t$ distribution for the mean yield(a) decide whether or not to reject \(H_{0}: \mu=5.5
Refer to the labor time data in Exercise 7.3. Using the $90 \%$ confidence interval, based on the $t$ distribution, for the mean labor time(a) decide whether or not to reject $H_{0}: \mu=1.6$
Repeat Exercise 7.69 but replace the $t$ test with the large sample $Z$ test.Data From Exercise 7.69 7.69 Refer to the labor time data in Exercise 7.3. Using the 90% confidence interval,
Refer to the example concerning average sound intensity on page 260 . Calculate the power at $\mu_{1}=77$ when(a) the level is changed to $\alpha=0.03$.(b) $\alpha=0.05$ but the alternative is
MINITAB calculation of power These calculations pertain to normal populations with known variance and provide an accurate approximation in the large sample case where $\sigma$ is unknown. To
Use computer software to repeat Exercise 7.71.Data From Exercise 7.71 77 7.71 Refer to the example concerning average sound inten- sity on page 260. Calculate the power at when (a) the level is
MINITAB calculation of sample size Refer to Exercise 7.72, but this time leave Sample size blank but Type 0.90 in power to obtain the partial output concerning sample sizeRefer to the example
MINITAB calculation of power or OC curve Refer to the steps in Exercise 7.72, but enter a range of values for the difference.Here 0:3/.1 goes in steps from 0 to 3 in steps of . 1 for Example 22.With
Specify the null hypothesis and the alternative hypothesis in each of the following cases.(a) An engineer hopes to establish that an additive will increase the viscosity of an oil.(b) An electrical

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