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Introduction to Mathematical Statistics

Robert V. Hogg, Joseph W. McKean, Allen T. Craig

Chapter 4

Some Elementary Statistical Inferences - all with Video Answers

Educators

Section 1

Sampling and Statistics

01:33

Problem 1

Twenty motors were put on test under a high-temperature setting. The lifetimes in hours of the motors under these conditions are given below. Also, the data are in the file lifetimemotor. rda at the site listed in the Preface. Suppose we assume that the lifetime of a motor under these conditions, $X$, has a $\Gamma(1, \theta)$ distribution. $\begin{array}{cccccccccc}1 & 4 & 5 & 21 & 22 & 28 & 40 & 42 & 51 & 53 \\ 58 & 67 & 95 & 124 & 124 & 160 & 202 & 260 & 303 & 363\end{array}$
(a) Obtain a histogram of the data and overlay it with a density estimate, using the code hist $(x, p r=T) ;$ lines (density $(x))$ where the $R$ vector $x$ contains the data. Based on this plot, do you think that the $\Gamma(1, \theta)$ model is credible?
(b) Assuming a $\Gamma(1, \theta)$ model, obtain the maximum likelihood estimate $\widehat{\theta}$ of $\theta$ and locate it on your histogram. Next overlay the pdf of a $\Gamma(1, \hat{\theta})$ distribution on
the histogram. Use the $\mathrm{R}$ function dgamma $(\mathrm{x}$, shape $=1$, scale $=\hat{\theta}$ ) to evaluate the pdf.
(c) Obtain the sample median of the data, which is an estimate of the median lifetime of a motor. What parameter is it estimating (i.e., determine the median of $X$ )?
(d) Based on the mle, what is another estimate of the median of $X ?$

Manik Pulyani
Manik Pulyani
Numerade Educator
22:07

Problem 2

Here are the weights of 26 professional baseball pitchers; [see page 76 of Hettmansperger and McKean (2011) for the complete data set]. The data are in $\mathrm{R}$ file bb. rda. Suppose we assume that the weight of a professional baseball pitcher is normally distributed with mean $\mu$ and variance $\sigma^{2}$.
$\begin{array}{llllllllllllll}160 & 175 & 180 & 185 & 185 & 185 & 190 & 190 & 195 & 195 & 195 & 200 & 200 \\ 200 & 200 & 205 & 205 & 210 & 210 & 218 & 219 & 220 & 222 & 225 & 225 & 232\end{array}$
(a) Obtain a histogram of the data. Based on this plot, is a normal probability model credible?
(b) Obtain the maximum likelihood estimates of $\mu, \sigma^{2}, \sigma$, and $\mu / \sigma .$ Locate your estimate of $\mu$ on your plot in part (a). Then overlay the normal pdf with these estimates on your histogram in Part (a).
(c) Using the binomial model, obtain the maximum likelihood estimate of the proportion $p$ of professional baseball pitchers who weigh over 215 pounds.
(d) Determine the mle of $p$ assuming that the weight of a professional baseball player follows the normal probability model $N\left(\mu, \sigma^{2}\right)$ with $\mu$ and $\sigma$ unknown.

Saeeda Aman
Saeeda Aman
Numerade Educator
13:28

Problem 3

Suppose the number of customers $X$ that enter a store between the hours $9: 00$ a.m. and $10: 00$ a.m. follows a Poisson distribution with parameter $\theta$. Suppose a random sample of the number of customers that enter the store between $9: 00$ a.m. and $10: 00$ a.m. for 10 days results in the values
$$
\begin{array}{llllllllll}
9 & 7 & 9 & 15 & 10 & 13 & 11 & 7 & 2 & 12
\end{array}
$$
(a) Determine the maximum likelihood estimate of $\theta$. Show that it is an unbiased estimator.
(b) Based on these data, obtain the realization of your estimator in part (a). Explain the meaning of this estimate in terms of the number of customers.

Saeeda Aman
Saeeda Aman
Numerade Educator
01:20

Problem 4

For Example 4.1.3, verify equations $(4.1 .4)-(4.1 .8)$.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:00

Problem 5

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample from a continuous-type distribution.
(a) Find $P\left(X_{1} \leq X_{2}\right), P\left(X_{1} \leq X_{2}, X_{1} \leq X_{3}\right), \ldots, P\left(X_{1} \leq X_{i}, i=2,3, \ldots, n\right)$
(b) Suppose the sampling continues until $X_{1}$ is no longer the smallest observation (i.e., $X_{j}<X_{1} \leq X_{i}, i=2,3, \ldots, j-1$ ). Let $Y$ equal the number of trials, not including $X_{1}$, until $X_{1}$ is no longer the smallest observation (i.e., $\left.Y=j-1\right)$. Show that the distribution of $Y$ is
$$
P(Y=y)=\frac{1}{y(y+1)}, \quad y=1,2,3, \ldots
$$
(c) Compute the mean and variance of $Y$ if they exist.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:55

Problem 6

Consider the estimator of the pmf in expression (4.1.10). In equation (4.1.11), we showed that this estimator is unbiased. Find the variance of the estimator and its mgf.

Ameer Said
Ameer Said
Numerade Educator
04:20

Problem 7

The data set on Scottish schoolchildren discussed in Example 4.1.5 included the eye colors of the children also. The frequencies of their eye colors are
$\begin{array}{lccc}\text { Blue } & \text { Light } & \text { Medium } & \text { Dark } \\ 2978 & 6697 & 7511 & 5175\end{array}$
Use these frequencies to obtain a bar chart and an estimate of the associated pmf.

Saeeda Aman
Saeeda Aman
Numerade Educator
03:31

Problem 8

Recall that for the parameter $\eta=g(\theta)$, the mle of $\eta$ is $g(\theta)$, where $\hat{\theta}$ is the mle of $\theta$. Assuming that the data in Example $4.1 .6$ were drawn from a Poisson distribution with mean $\lambda$, obtain the mle of $\lambda$ and then use it to obtain the mle of the pmf. Compare the mle of the pmf to the nonparametric estimate. Note: For the domain value 6 , obtain the mle of $P(X \geq 6)$.

Ameer Said
Ameer Said
Numerade Educator
11:42

Problem 9

Consider the nonparametric estimator, (4.1.14), of a pdf.
(a) Obtain its mean and determine the bias of the estimator.
(b) Obtain the variance of the estimator.

Philomena Marfo
Philomena Marfo
Numerade Educator
01:13

Problem 10

This data set was downloaded from the site http://lib.stat.cmu.edu/DASL/ at Carnegie-Melon university. The original source is Willerman et al. (1991). The data consist of a sample of brain information recorded on 40 college students. The variables include gender, height, weight, three IQ measurements, and Magnetic Resonance Imaging (MRI) counts, as a determination of brain size. The data are in the rda file braindata. rda at the sites referenced in the Preface. For this exercise, consider the MRI counts.
(a) Load the rda file braindata.rda and print the MRI data, using the code:
$\mathrm{mri}<-$ braindata $[, 7] ;$ print(mri).
(b) Obtain a histogram of the data, hist $(m r i, p r=T)$. Comment on the shape.
(c) Overlay the default density estimator, lines (density(mri)). Comment on the shape.
4.1.10. This data set was downloaded from the site http://lib.stat.cmu.edu/DASL/ at Carnegie-Melon university. The original source is Willerman et al. (1991). The data consist of a sample of brain information recorded on 40 college students. The variables include gender, height, weight, three IQ measurements, and Magnetic Resonance Imaging (MRI) counts, as a determination of brain size. The data are in the rda file braindata. rda at the sites referenced in the Preface. For this exercise, consider the MRI counts.
(a) Load the rda file braindata.rda and print the MRI data, using the code:
$\mathrm{mri}<-$ braindata $[, 7] ;$ print(mri).
(b) Obtain a histogram of the data, hist $(m r i, p r=T)$. Comment on the shape.
(c) Overlay the default density estimator, lines (density(mri)). Comment on the shape.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:12

Problem 11

This is a famous data set on the speed of light recorded by the scientist Simon Newcomb. The data set was obtained at the Carnegie Melon site given in Exercise 4.1.10 and it can also be found in the rda file speedlight. rda at the sites referenced in the Preface. Stigler (1977) presents an informative discussion of this data set.
(a) Load the rda file and type the command print(speed). As Stigler notes, the data values $\times 10^{-3}+24.8$ are Newcomb's data values; hence, negative items can occur. Also, in the unit of the data the "true value" is $33.02$. Discuss the data.
(b) Obtain a histogram of the data. Comment on the shape.
(c) On the histogram overlay the default density estimator. Comment on the shape.
(d) Obtain the sample mean and standard deviation and on the histogram overlay the normal pdf with these estimates as parameters. Comment on the fit.
(e) Determine the proportions of the data within 1 and 2 standard deviations of the sample mean and compare these with the empirical rule.

Manik Pulyani
Manik Pulyani
Numerade Educator