Exam 6: Point Estimation

Which of the following statements are true if $X _ { 1 } , X _ { 2 } , \cdots \cdots , X _ { n }$ is a random sample from a distribution with mean $\mu \text { and variance } \sigma ^ { 2 }$ ?

The objective of __________ is to select a single number such as $\bar { x } \text { or } \mathrm { s } ^ { 2 }$ , based on sample data, that represents a sensible value (good guess) for the true value of the population parameter, such as $\mu \text { or } \sigma ^ { 2 }$ .

Which of the following statements are not always true?

Consider a random sample $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ from the shifted exponential pdf $f ( x ; \lambda , \theta ) = \left\{ \begin{array} { l l } \lambda e ^ { - \lambda ( x - \theta ) } & x \geq \theta \\0 & \text { otherwise }\end{array} \right.$ a. Obtain the maximum likelihood estimators of $\theta \text { and } \lambda$ b. A random sample of size $n = 10$ results in the values 3.12, .65, 2.56, 2.21, 5.45, 3.43, 10.40, 8.94, 17.83, and 1.31, calculate the estimates of $\theta \text { and } \lambda$

The accompanying data describe flexural strength (Mpa) for concrete beams of a certain type was introduced in Example 1.2. 9.2 9.7 8.8 10.7 8.4 8.7 10.7 6.9 8.2 8.3 7.3 9.1 7.8 8.0 8.6 7.8 7.5 8.0 7.3 8.9 10.0 8.8 8.7 12.6 12.3 12.8 11.7

Let $\hat { \theta } _ { 1 } , \hat { \theta } _ { 2 } , \cdots \cdots , \hat { \theta } _ { n }$ be the maximum likelihood estimates (mle's) of the parameters $\theta _ { 1 } , \theta _ { 2 } , \cdots \cdots , \theta _ { n }$ . Then the mle of any function h( $\theta _ { 1 } , \theta _ { 2 } , \cdots \cdots , \theta _ { n }$ ) of these parameters is the function $h \left( \hat { \theta } _ { 1 } , \hat { \theta } _ { 2 } , \cdots \cdots , \hat { \theta } _ { m } \right)$ of the mle's. This result is known as the __________ principle.

Consider a random sample $X _ { 1 } , \ldots , X _ { n }$ from the pdf $f ( x ; \theta ) = 5 ( 1 + \theta x ) \quad - 1 \leq x \leq 1$ where $- 1 \leq \theta \leq 1$ (this distribution arises in particle physics). Show that $\hat { \theta } = 3 \bar { X }$ is an unbiased estimator of $\theta$ [ Hint: First determine $\mu = E ( X ) = E ( \bar { X } ) . ]$

Which of the following statements are not true?

The standard error of an estimator $\hat { \theta }$ is the __________ of $\hat { \theta }$ .

Given four observed values: $x _ { 1 } = 4.6 , x _ { 2 } = 6.2 , x _ { 3 } = 3.7 \text {, and } x _ { 4 } = 5.5 \text {, }$ would result in a point estimate for $\mu$ that is equal to __________.

Let $X _ { 1 } , X _ { 2 } , \ldots , X _ { n }$ represent a random sample from a Rayleigh distribution with pdf $f ( x ; \theta ) = \frac { x } { \theta } e ^ { - x ^ { 2 }/ ( 2 \theta ) }\quad x > 0$ a. It can be shown that $E \left( X ^ { 2 } \right) = 2 \theta$ Use this fact to construct an unbiased estimator of $\theta$ based on $\sum x _ { i } ^ { 2 }$ (and use rules of expected value to show that it is unbiased). b. Estimate $\theta$ from the following $n = 10$ observations on vibratory stress of a turbine blade under specified conditions: 17.08 10.43 4.79 6.86 13.88 14.43 20.07 9.60 6.71 11.15

An estimator that has the properties of __________ and __________ will often be regarded as an accurate estimator.

Which of the following statements are not true?

Which of the following statements are true?

Which of the following statements are true?

A random sample of a. Derive the maximum likelihood estimator of $p$ . If $n$ = 25 and $x$ =5, what is the estimate? b. Is the estimator of part (a) unbiased? c. If $n$ = 25 and $x$ =5, what is the mle of the probability $( 1 - p ) ^ { 5 }$ that none of the next five helmets examined is flawed?

A point estimator $\hat { \theta }$ is said to be an __________ estimator of $\hat { \theta }$ if $E ( \hat { \theta } ) = \theta$ for every possible value of $\theta$ .

Among all estimators of parameter $\theta$ that are unbiased, choose the one that has minimum variance. The resulting $\hat { \theta }$ is called the __________ of $\theta$ .

Let $X _ { 1 } , \cdots \cdots , X _ { n }$ be a random sample of size n from an exponential distribution with parameter $\lambda$ . The moment estimator of $\widehat { \lambda } \text { is } \hat { \lambda }$ = __________.

The shear strength of each of ten test spot welds is determined, yielding the following data (psi): 395 379 404 370 392 365 412 418 361 378 a. Assuming that shear strength is normally distributed, estimate the true average shear strength and standard deviation of shear strength using the method of maximum likelihood. b. Again assuming a normal distribution, estimate the strength value below which 95% of all welds will have their strengths. (Hint: What is the 95 percentile in terms of $\mu \text { and } \sigma$ ? Now use the invariance principle.)