Questions and Answers of Introduction To Probability Statistics

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
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

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