A random sample of \(80\) apartments has a mean monthly rent of \(\$1450\). The goal is to estimate the mean monthly rent \(\mu\) of all apartments in the city. Identify the parameter, the estimator, and the numerical estimate.

A school district wants to know the true mean commute time \(\mu\) for all of its employees. A random sample of employees has sample mean \(\bar X\), and the observed sample gives \(\bar x=24.6\) minutes. Match each item with the correct term: parameter, estimator/statistic, or estimate. 1. \(\mu\) 2. \(\bar X\) 3. \(24.6\) minutes

Two estimators, \(A\) and \(B\), are used to estimate a parameter whose true value is \(50\). Their sampling distributions have the following summaries. <table> <tr><th>Estimator</th><th>Sampling mean</th><th>Sampling standard deviation</th></tr> <tr><td>\(A\)</td><td>\(50\)</td><td>\(5\)</td></tr> <tr><td>\(B\)</td><td>\(50\)</td><td>\(2\)</td></tr> </table> a) Which estimators are unbiased? b) Which estimator is more precise? Explain.

A city wants to know the average number of library books borrowed per resident each year. It surveys \(80\) residents, and their sample mean is \(6.2\) books. What population quantity is being estimated, and which number is the estimate?

A vertical reference line marks the true population proportion \(p\). The histogram shows estimates from many random samples. Do the estimates tend to be too high, too low, or centered correctly? Explain.

Two unbiased estimators of the same population mean produce the sampling distributions shown. Which estimator is more precise, a) or b)? Explain.

A population has true mean \(50\). Two estimators are used many times: <table> <tr><th>Estimator</th><th>Mean of estimates</th><th>SD of estimates</th></tr> <tr><td>A</td><td>\(50\)</td><td>\(4\)</td></tr> <tr><td>B</td><td>\(53\)</td><td>\(1\)</td></tr> </table> Which estimator is unbiased? Which is more precise?

A researcher estimates a city's average daily water use by taking a random sample of households. The researcher could use either \(20\) or \(200\) households. Which sample size would generally give a more precise sample-mean estimate, assuming the same sampling method? Explain.

In a simple random sample of \(240\) registered voters, \(78\) support a proposed transit plan. The city has \(12{,}000\) registered voters. a) Give the point estimate of the population proportion who support the plan. b) Use that estimate to estimate the number of registered voters who support the plan. c) Name the estimator used in part a).

An estimator \(T\) is used for a parameter whose true value is \(\theta=5.4\). Its sampling distribution is shown below. <table> <tr><th>\(T\)</th><th>\(4\)</th><th>\(5\)</th><th>\(6\)</th></tr> <tr><th>Probability</th><td>\(0.20\)</td><td>\(0.50\)</td><td>\(0.30\)</td></tr> </table> Find the expected value and bias of \(T\). State whether it tends to overestimate or underestimate \(\theta\).

An estimator \(T_n\) based on a sample of size \(n\) satisfies \(E(T_n)=\theta+\frac{10}{n}\). a) Write an unbiased estimator of \(\theta\). b) For \(n=25\), the observed value is \(T_{25}=18.7\). Find the corrected estimate. c) Compare the size of the correction at \(n=25\) and \(n=100\).

A measurement is uniformly distributed from \(0\) to an unknown upper limit \(\theta\). For a random sample, the sample mean is \(\bar X\). It is known that \(E(X)=\frac{\theta}{2}\). a) Construct an unbiased estimator of \(\theta\) using \(\bar X\). b) Calculate the estimate when \(\bar x=7.4\).

A parameter of interest is \(\psi=3\mu-2\), where \(\mu\) is a population mean. A sample mean \(\bar X\) is unbiased for \(\mu\), and its standard deviation is \(1.4\). a) Construct an unbiased estimator of \(\psi\). b) Find its standard deviation. c) Calculate the estimate when \(\bar x=11\).

Two estimators are used to estimate the population median \(\theta=100\): the sample mean and a trimmed mean. Simulations give these sampling-distribution summaries for two populations that both have median \(100\). <table> <tr><th>Population</th><th>Estimator</th><th>Sampling mean</th><th>Sampling standard deviation</th></tr> <tr><td>Clean, nearly symmetric</td><td>Sample mean</td><td>\(100\)</td><td>\(2\)</td></tr> <tr><td>Clean, nearly symmetric</td><td>Trimmed mean</td><td>\(99.5\)</td><td>\(2.4\)</td></tr> <tr><td>Right-skewed with extreme values</td><td>Sample mean</td><td>\(107\)</td><td>\(5\)</td></tr> <tr><td>Right-skewed with extreme values</td><td>Trimmed mean</td><td>\(101\)</td><td>\(3\)</td></tr> </table> a) Find the bias of each estimator in each population. b) For the clean population, which estimator has the better long-run center and which is more precise? c) For the right-skewed population, which estimator has the better long-run center and which is more precise? d) Explain why the preferred estimator can change.

Independent observations \(X_1\) and \(X_2\) have mean \(\mu\) and standard deviation \(6\). Compare the estimators \(A=X_1\) and \(B=\frac{X_1+X_2}{2}\) for \(\mu\). a) Determine whether each estimator is unbiased. b) Find each variance and standard deviation. Give the standard deviation of \(B\) exactly and to three decimal places. c) Which estimator is more precise? Explain.

In random samples of size \(20\) from a population with proportion \(p=0.30\), let \(X\) be the number of successes. Two estimators are proposed: \(A=\frac{X}{20}\) and \(B=\frac{X+1}{22}\). Find the mean of the sampling distribution of each estimator. Which estimator is unbiased for \(p\)?

The true parameter is \(\theta=20\). Across repeated samples, estimator A has mean \(20\) and standard deviation \(3\), while estimator B has mean \(22\) and standard deviation \(1\). Mean squared error is variance plus the square of bias. Find the mean squared error of each estimator and decide which is smaller.

A survey method is known to be unbiased for a population mean of \(50\). One random sample gives an estimate of \(53\). A student says, “The method must be biased because this estimate is too high.” Explain the mistake.

In a town, \(80\%\) of households live in apartments and \(20\%\) in houses. Representative surveys estimate the average monthly electricity bill at \(\$90\) for apartments and \(\$150\) for houses. A student averages the two estimates and gets \(\$120\). Find the correctly weighted estimate for the town and explain the student's error.

The population mean is \(100\). An estimator has mean \(102\) for every sample size. Its standard deviation decreases toward \(0\) as the sample size grows. Does the estimator become more precise? Will its estimates approach the true mean as the sample size grows? Explain both answers.

Two methods estimate the same target value, \(20\). Their probability distributions are shown. a) Find the mean estimate for each method from the bars. b) Which method is more likely to miss the target by more than \(8\) units? Find that probability for each.

A school wants to estimate its average number of absences per student, which is \(5\). It compares two possible estimates: - Method A gives \(4\) with probability \(0.75\) and \(9\) with probability \(0.25\). - Method B gives \(2\) or \(8\), each with probability \(0.5\). Mean squared error (MSE) is the average of the squared differences between an estimate and the true value. Find the MSE for each method and decide which is smaller.

A population consists of \(-1\) and \(1\), each equally likely. Two observations are sampled independently with replacement. Let \(S^2\) be the sample variance calculated with denominator \(n-1\). a) Construct the sampling distribution of \(S^2\). b) Find \(E(S^2)\) and compare it with the population variance.

For every possible value of a parameter \(\theta\), an estimator \(A\) satisfies \(E(A)=0.8\theta+2\). a) Create an unbiased estimator of \(\theta\) using \(A\). b) A sample gives \(A=18.4\). What estimate does your corrected estimator produce?

Two estimators are used for a parameter with true value \(\theta=50\). Estimator \(T_1\) has mean \(50\) and standard deviation \(4\). Estimator \(T_2\) has mean \(52\) and standard deviation \(2\). a) Find the bias of each estimator. b) Compute each mean squared error using \(\operatorname{MSE}=\operatorname{variance}+(\operatorname{bias})^2\). c) Which estimator has the smaller mean squared error?

Three independent measurements \(X_1,X_2,X_3\) each have mean \(\mu\) and positive variance \(\sigma^2\). Two estimators of \(\mu\) are \(A=\frac{X_1+X_2+X_3}{3}\) and \(B=\frac{X_1+2X_2+X_3}{4}\). a) Show that both estimators are unbiased. b) Find each variance. c) Which estimator is more precise?

The mean of a population is known to be \(\mu=11\), but its variance \(\sigma^2\) is unknown. For independent observations \(X_1,\ldots,X_n\), consider the estimator \(U=\frac{1}{n}\sum_{i=1}^n(X_i-11)^2\). a) Explain why \(U\) is unbiased for \(\sigma^2\). b) Calculate the estimate from the sample \(8,10,11,15\).

Estimators \(A\) and \(B\) are both unbiased for \(\theta\). Their variances are \(9\) and \(16\), and their covariance is \(6\). Consider the average estimator \(C=\frac{A+B}{2}\). a) Show that \(C\) is unbiased. b) Find \(\operatorname{Var}(C)\). c) Is \(C\) more precise than \(A\)?

Independent observations \(X_1,X_2,X_3\) each have mean \(\mu\). A proposed estimator is \(T=0.2X_1+0.3X_2+0.4X_3\). a) Find the bias of \(T\) as an estimator of \(\mu\). b) Modify \(T\) to make it unbiased. c) For observed data \(10,13,16\), calculate the corrected estimate exactly.

A population has unknown mean \(\mu\) and known variance \(16\). A random sample of size \(25\) has sample mean \(\bar X\). Because \(E(\bar X^2)=\mu^2+\frac{16}{25}\), construct an unbiased estimator of \(\mu^2\). Then calculate its value when \(\bar x=10\).

A random sample of size \(n\) is taken from a uniform distribution on \([0,\theta]\). Let \(M\) be the sample maximum. It is known that \(E(M)=\frac{n}{n+1}\theta\). a) Modify \(M\) to obtain an unbiased estimator of \(\theta\). b) For \(n=5\) and observed maximum \(M=18\), calculate the estimate.

An estimator \(R\) reports only whole-number values. For a parameter with true value \(\theta=10.2\), its sampling distribution is <table> <tr><th>\(R\)</th><th>\(9\)</th><th>\(10\)</th><th>\(11\)</th></tr> <tr><th>Probability</th><td>\(0.10\)</td><td>\(0.60\)</td><td>\(0.30\)</td></tr> </table> a) Determine whether \(R\) is unbiased. b) Find its mean squared error. c) Explain how an estimator restricted to integers can still be unbiased for a noninteger parameter.

For a random sample of size \(n\ge2\), an analyst computes \(V=\frac{1}{n}\sum_{i=1}^{n}(X_i-\bar X)^2\). It is known that \(E(V)=\frac{n-1}{n}\sigma^2\). a) Construct an unbiased estimator of the population variance \(\sigma^2\). b) When \(n=10\) and \(V=18\), find the unbiased estimate.

A sample proportion \(\hat p\) is based on \(n\ge2\) independent trials with success probability \(p\). It is known that \(E[\hat p(1-\hat p)]=\frac{n-1}{n}p(1-p)\). a) Construct an unbiased estimator of \(p(1-p)\). b) For \(n=50\) and \(\hat p=0.30\), calculate the estimate. Round to four decimal places.

A population has two groups. Group L is \(60\%\) of the population with mean value \(20\); Group H is \(40\%\) with mean value \(50\). In a very large survey, members of Group L respond independently with probability \(0.80\), while members of Group H respond independently with probability \(0.40\). Treat the respondent group proportions as their large-sample response-weighted proportions. The analyst estimates the population mean using the mean among respondents without adjustment. a) Find the true population mean. b) Find the large-sample center of the respondent-mean estimator. c) Find and interpret its bias.

An estimator \(T\) is used for a parameter with true value \(\theta=10\). Its sampling distribution is <table> <tr><th>\(T\)</th><th>\(8\)</th><th>\(10\)</th><th>\(20\)</th></tr> <tr><th>Probability</th><td>\(0.45\)</td><td>\(0.10\)</td><td>\(0.45\)</td></tr> </table> An estimator is called median-unbiased when at least half its probability is at or below the parameter and at least half is at or above it. Determine whether \(T\) is mean-unbiased and whether it is median-unbiased.

Recorded measurements satisfy \(Y=X+E\), where the true values \(X\) and measurement errors \(E\) are independent. The error variance is known to be \(4\). Suppose \(S_Y^2\) is an unbiased estimator of \(\operatorname{Var}(Y)\). a) Construct an unbiased estimator of \(\operatorname{Var}(X)\). b) Find the estimate when \(S_Y^2=9\). c) The same rule gives \(-1\) when another sample has \(S_Y^2=3\). Explain how an unbiased estimator of a nonnegative parameter can produce a negative estimate.

The ordered finite population is shown. A systematic sample of size \(2\) selects positions three apart. The random start is equally likely to be position \(1,2,\) or \(3\). The sample mean estimates the population mean. a) Find the three possible estimates. b) Determine whether the estimator is unbiased over the random starts. c) Does unbiasedness imply that each possible estimate is close to the population mean?

A population has two units with paired values \((x, y)=(1, 1)\) and \((3, 6)\). The parameter of interest is the ratio of population totals, \(R=\frac{\sum y}{\sum x}\). One unit is selected at random with equal probability, and the analyst estimates \(R\) using the sample ratio \(T=\frac{y}{x}\). a) Find the true value of \(R\). b) Find the sampling distribution and expected value of \(T\). c) Determine the bias.

An estimator \(T\) is unbiased for a positive parameter \(\mu=2\). Its sampling distribution places probability \(0.5\) at \(T=1\) and probability \(0.5\) at \(T=3\). An analyst uses \(\frac{1}{T}\) to estimate \(\frac{1}{\mu}\). a) Verify that \(T\) is unbiased for \(\mu\). b) Find \(E\left(\frac{1}{T}\right)\). c) Find the bias of \(\frac{1}{T}\) as an estimator of \(\frac{1}{\mu}\).

An unbiased estimator \(X\) of \(\theta\) has variance \(4\). An analyst adds an independent random value \(C\), equal to \(3\) or \(-3\) with equal probability, and reports \(T=X+C\). a) Show that \(T\) is unbiased. b) Find its variance. c) Explain why replacing \(T\) by the conditional average over the added randomization returns the more precise estimator \(X\).

Independent observations \(X_1\) and \(X_2\) come from a population with unknown mean \(\mu\). a) Show that \(X_1X_2\) is an unbiased estimator of \(\mu^2\). b) Explain why independence is essential. c) Find the estimate when \(X_1=4\) and \(X_2=7\).

An unbiased estimator \(T\) of a population proportion \(p=0.10\) has sampling distribution <table> <tr><th>\(T\)</th><th>\(-0.20\)</th><th>\(0.10\)</th><th>\(0.40\)</th></tr> <tr><th>Probability</th><td>\(0.20\)</td><td>\(0.60\)</td><td>\(0.20\)</td></tr> </table> To avoid impossible reported proportions, an analyst clips negative estimates to \(0\), producing \(C=\max(0,T)\). a) Verify that \(T\) is unbiased. b) Find the bias of \(C\). c) Explain why enforcing the parameter boundary changes the long-run center.

Independent observations \(X_1,\dots,X_n\), with \(n\ge2\), have common mean \(\mu\). Let \(L_i\) be the mean after leaving out \(X_i\), and define \(T=\frac{1}{n}\sum_{i=1}^{n}L_i\). For the sample \(3,7,8,14\): a) Find the four leave-one-out means and \(T\). b) Show that \(T=\bar X\) for any sample with at least two observations. c) Is \(T\) an unbiased estimator of \(\mu\)? d) Does averaging all leave-one-out means give a different estimator from the ordinary sample mean?

For a population with mean \(\mu\) and standard deviation \(\sigma\), consider \(T_n=\frac{n}{n+1}\bar X.\) a) Find the bias and standard deviation of \(T_n\). b) For \(\mu=20\), \(\sigma=8\), and \(n=99\), calculate both quantities. Give the standard deviation exactly and to three decimal places. c) Is \(T_n\) consistent for \(\mu\)? Justify your answer using both the bias and standard deviation.

An estimator \(S^2\) is unbiased for a population variance \(\sigma^2=4\). Its sampling distribution is <table> <tr><th>\(S^2\)</th><th>\(1\)</th><th>\(4\)</th><th>\(9\)</th></tr> <tr><th>Probability</th><td>\(\frac{1}{3}\)</td><td>\(\frac{7}{15}\)</td><td>\(\frac{1}{5}\)</td></tr> </table> An analyst uses \(S=\sqrt{S^2}\) to estimate \(\sigma=2\). a) Verify that \(S^2\) is unbiased for \(\sigma^2\). b) Find the expected value and bias of \(S\). Give exact answers and approximations to three decimal places. c) What does this show about taking the square root of an unbiased variance estimator?

A preserve is divided into \(600\) equal-area plots. A simple random sample of \(30\) plots has a mean of \(18.4\) mature trees per plot. Let \(T\) be the total number of mature trees in the preserve. a) Give a point estimator for \(T\) and calculate the estimate. b) If the standard deviation of the sampling distribution of \(\bar{X}\) is \(1.2\) trees per plot, find the standard deviation of the total estimator. c) Explain why the total estimator is unbiased when \(\bar{X}\) is unbiased for the population mean per plot.

Wildlife researchers mark \(60\) turtles and release them. Later, they capture \(50\) turtles, of which \(12\) are marked. A common capture-recapture estimator of the population size is \(\widehat{N}=\frac{n_1n_2}{m}\), where \(n_1\) is the number initially marked, \(n_2\) is the second sample size, and \(m\) is the number of marked turtles recaptured. a) Estimate the turtle population. b) Suppose marked turtles become easier to catch than unmarked turtles. Would this tend to make the estimate too high or too low? Explain.

A population is \(40\%\) urban and \(60\%\) rural. Independent samples from the two groups give mean weekly travel distances of \(18\) miles and \(27\) miles. The sampling standard deviations of the two sample means are \(1.2\) miles and \(0.8\) mile. a) Give the correctly weighted estimator of the overall population mean and calculate the estimate. b) Find the estimator's standard deviation. Give the exact value and a decimal approximation to three decimal places. c) Explain why simply averaging \(18\) and \(27\) is not the appropriate estimator.

A college population is \(20\%\) graduate students and \(80\%\) undergraduate students. A researcher repeatedly samples \(100\) graduate students and \(100\) undergraduate students, then computes the unweighted proportion of sampled students who use a particular service. a) Explain why this unweighted sample proportion need not be centered at the service-use proportion for the full college population. b) Under what condition would the unweighted sample proportion happen to be unbiased for the full-population proportion? c) Describe how the researcher should combine the two group sample proportions to estimate the full-population proportion.

In a population, the true proportion of people with a certain condition is \(0.20\). A screening system correctly records a person with the condition as positive with probability \(0.90\), but it also records a person without the condition as positive with probability \(0.05\). A random sample of \(400\) people is screened, with screening outcomes treated as independent, and \(\hat p_{\text{recorded}}\) is the sample proportion recorded as positive. a) Find the probability that a randomly selected person is recorded as positive. b) Find the mean and standard deviation of the sampling distribution of \(\hat p_{\text{recorded}}\). Give the standard deviation exactly and to five decimal places. c) Find the bias of \(\hat p_{\text{recorded}}\) as an estimator of the true condition proportion, and interpret its sign.

Independent estimators \(A\) and \(B\) are both unbiased for the same parameter \(\theta\). Their variances are \(9\) and \(4\), respectively. Consider \(W=wA+(1-w)B\). a) Explain why \(W\) is unbiased for every real value of \(w\). b) Find the value of \(w\) that gives \(W\) the smallest variance. c) Find that minimum variance.

A box contains tickets numbered consecutively from \(1\) through an unknown largest number \(N\). To study the estimator that uses the largest observed ticket, suppose the true value is \(N=4\) and two tickets are drawn independently with replacement. Let \(M\) be the larger ticket number drawn. a) Construct the sampling distribution of \(M\). b) Find the bias of \(M\) as an estimator of \(N\).

A population variable equals \(0\) with probability \(0.75\) and \(8\) with probability \(0.25\), so the population mean is \(\mu=2\). Compare two estimators of \(\mu\): \(X\), one randomly selected observation, and \(M\), the median of three independent observations. a) Find the bias of each estimator. b) Find the mean squared error of each estimator. c) Which estimator has the smaller mean squared error?

A screening test is positive for \(90\%\) of people who have a condition and for \(10\%\) of people who do not. Let \(p\) be the population proportion with the condition, and let \(\hat r\) be the positive-test proportion in a random sample. a) Show that the population positive-test rate is \(r=0.10+0.80p\). b) Construct an unbiased estimator of \(p\) from \(\hat r\). c) If \(\hat r=0.34\), find the estimate of \(p\).

Estimator \(A_n\) has bias \(0\) and standard deviation \(\frac{5}{\sqrt n}\). Estimator \(B_n\) has bias \(1\) and standard deviation \(\frac{1}{\sqrt n}\). a) Compare their mean squared errors at \(n=16\). b) Compare their mean squared errors at \(n=100\). c) For which integer sample sizes does \(A_n\) have the smaller mean squared error?

Two unbiased estimators of \(\theta\) are available. Estimator \(A\) has standard deviation \(2\), and estimator \(B\) has standard deviation \(5\). A procedure randomly uses \(A\) with probability \(0.70\) and \(B\) with probability \(0.30\), independently of the data. Let \(T\) be the selected estimate. a) Is \(T\) unbiased? b) Find its variance. c) Compare its precision with always using \(A\).

Two independent Bernoulli observations have success probability \(p\). Let \(D=1\) if the two outcomes are different and \(D=0\) if they are the same. a) Show that \(\frac{D}{2}\) is an unbiased estimator of \(p(1-p)\). b) What estimate is produced when the two observed outcomes are different?

Two weather stations send daily data files: Pine Creek sends \(40\) files, and Lake Point sends \(100\). Only one station can be checked. Pine Creek is chosen with probability \(0.25\), and Lake Point with probability \(0.75\). The estimate of the total number of files is the chosen station's count divided by its selection probability. a) Give the two possible estimates and their probabilities. Give exact values. b) Show that this estimator is unbiased for the total. c) What exact estimate is reported if Lake Point is selected?

A study estimates an unknown population mean \(\mu_Y\) using a sample mean \(\bar Y\). The same sample also gives \(\bar X\) for an auxiliary variable whose population mean is known to be \(50\). Suppose \(\operatorname{Var}(\bar Y)=9\), \(\operatorname{Var}(\bar X)=4\), and \(\operatorname{Cov}(\bar Y,\bar X)=5\). Consider \(T=\bar Y+(50-\bar X)\). a) Show that \(T\) is unbiased for \(\mu_Y\). b) Find its variance and compare it with using \(\bar Y\) alone. c) Calculate the estimate when \(\bar y=68\) and \(\bar x=47\).

A population has mean \(\mu\) and standard deviation \(6\). For a sample of size \(n\), let \(\bar X_n\) be the sample mean. An analyst reports \(T_n=\bar X_n+Z\), where \(Z\) is independent of the sample and equals \(2\) or \(-2\) with equal probability. a) Is \(T_n\) unbiased for \(\mu\)? b) Find its standard deviation when \(n=36\). c) Is \(T_n\) consistent for \(\mu\)? Explain.

For \(X\) successes in \(n\) independent trials with success probability \(p\), an analyst uses the smoothed estimator \(S=\frac{X+1}{n+2}\) instead of \(\hat p=\frac{X}{n}\). a) Show that the bias of \(S\) is \(\frac{1-2p}{n+2}\). b) Find its expected value and bias when \(n=18\) and \(p=0.70\). c) Find the observed estimate when \(X=14\).

A fair coin is flipped until the first head appears or until three flips have been made, whichever comes first. An analyst estimates the head probability \(p\) by the observed proportion of heads among the flips actually made. a) List the possible stopped sequences, their probabilities, and the resulting estimates. b) Find the expected estimate. c) Determine the bias when the true value is \(p=0.5\).

Measurements taken at times \(t=1,2,3,4,5\) satisfy \(E(X_t)=\mu+\beta t\), where \(\mu\) is the baseline value and \(\beta\) is an unknown linear trend. a) Show that the simple average of the five measurements is not generally unbiased for \(\mu\). b) Construct an unbiased estimator of \(\mu\) using only \(X_1\) and \(X_5\). c) Calculate it when \(X_1=12\) and \(X_5=20\).

A sample mean \(\bar X\) estimates a population mean \(\mu\). For the population being studied, \(\mu=12\) and \(\operatorname{Var}(\bar X)=9\). An analyst considers the shrinkage estimator \(T_a=a\bar X+(1-a)10\), where \(0\le a\le1\). a) Write the bias, variance, and mean squared error of \(T_a\). b) Find the value of \(a\) that minimizes mean squared error. c) Find the minimum mean squared error and the estimate produced when \(\bar x=15\).

Independent estimators \(A\) and \(B\) have \(E(A)=\theta+2\), \(E(B)=2\theta-1\), \(\operatorname{Var}(A)=9\), and \(\operatorname{Var}(B)=16\). Consider an estimator with no added constant, \(T=aA+bB\). a) Find \(a\) and \(b\) so that \(T\) is unbiased for every value of \(\theta\). b) Find the variance of \(T\). c) Calculate the estimate when \(A=14\) and \(B=21\).