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Chapter 4: 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.

Short Answer

Expert verified
After creating the histogram, evaluations for the credibility of the normal model should be made. Subsequently, the maximum likelihood estimates for \(\mu\), \(\sigma^{2}\), \(\sigma\), and \(\mu / \sigma\) should be calculated and plotted on the histogram. A comparison of the histogram with the overlaid normal pdf helps to visually confirm how well the normal model fits. The proportion \(p\) of professional baseball pitchers who weigh over 215 pounds should be calculated twice: once for the binomial model and once considering the normal model assumption. These results provide the values of \(p\) under both the binomial and normal models.

Step by step solution

01

Create a Histogram

Input data into a statistical software to create a histogram. This plot visually represents the spread and center of the data. Remember to note any skewness or outliers that could affect the normality of the data.
02

Evaluate Normal Probability Model

Based on the histogram, make observations about the symmetry and bell shape of the data to evaluate the credibility of the normal probability model. If the data seems symmetric and bell shaped, it gives more credibility to the normal model.
03

Calculate Maximum Likelihood Estimates

Using package R or any statistical software, calculate the maximum likelihood estimates of \(\mu\) (the mean), \(\sigma^{2}\) (the variance), \(\sigma\) (the standard deviation), and \(\mu / \sigma\) (the mean over standard deviation). Plot \(\mu\) on the histogram.
04

Overlay Normal pdf on Histogram

Using the calculated maximum likelihood estimates as parameters, create a normal probability density function (pdf). Overlay this normal pdf on top of your histogram to visually compare the data with the normal distribution.
05

Calculate p for Binomial Model

Now, using the binomial model, calculate the maximum likelihood estimate of the proportion \(p\) of professional baseball pitchers who weigh over 215 pounds.
06

Calculate p for Normal Model

For the normal probability model \(N\left(\mu, \sigma^{2}\right)\), calculate the maximum likelihood estimate (mle) of \(p\), which is the proportion of professional baseball pitchers who weigh over 215 pounds under the normal distribution assumption.

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Most popular questions from this chapter

Let \(p\) denote the probability that, for a particular tennis player, the first serve is good. Since \(p=0.40\), this player decided to take lessons in order to increase \(p\). When the lessons are completed, the hypothesis \(H_{0}: p=0.40\) is tested against \(H_{1}: p>0.40\) based on \(n=25\) trials. Let \(Y\) equal the number of first serves that are good, and let the critical region be defined by \(C=\\{Y: Y \geq 13\\}\). (a) Show that \(\alpha\) is computed by \(\alpha=1\) -pbinom \((12,25, .4)\). (b) Find \(\beta=P(Y<13)\) when \(p=0.60\); that is, \(\beta=P(Y \leq 12 ; p=0.60)\) so that \(1-\beta\) is the power at \(p=0.60\).

Let \(X_{1}, X_{2}\) be a random sample of size \(n=2\) from the distribution having pdf \(f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0

Suppose \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample drawn from a \(N\left(\mu, \sigma^{2}\right)\) distribution. As discussed in Example 4.2.1, the pivot random variable for a confidence interval is $$ t=\frac{\bar{X}-\mu}{S / \sqrt{n}} $$ where \(\bar{X}\) and \(S\) are the sample mean and standard deviation, respectively. Recall by Theorem \(3.6 .1\) that \(t\) has a Student \(t\) -distribution with \(n-1\) degrees of freedom; hence, its distribution is free of all parameters for this normal situation. In the notation of this section, \(t_{n-1}^{(\gamma)}\) denotes the \(\gamma 100 \%\) percentile of a \(t\) -distribution with \(n-1\) degrees of freedom. Using this notation, show that a \((1-\alpha) 100 \%\) confidence interval for \(\mu\) is $$ \left(\bar{x}-t^{(1-\alpha / 2)} \frac{s}{\sqrt{n}}, \bar{x}-t^{(\alpha / 2)} \frac{s}{\sqrt{n}}\right) $$

Let \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{m}\) be two independent random samples from the respective normal distributions \(N\left(\mu_{1}, \sigma_{1}^{2}\right)\) and \(N\left(\mu_{2}, \sigma_{2}^{2}\right)\), where the four parameters are unknown. To construct a confidence interval for the ratio, \(\sigma_{1}^{2} / \sigma_{2}^{2}\), of the variances, form the quotient of the two independent \(\chi^{2}\) variables, each divided by its degrees of freedom, namely, $$ F=\frac{\frac{(m-1) S_{2}^{2}}{\sigma_{2}^{2}} /(m-1)}{\frac{(n-1) S_{1}^{2}}{\sigma_{1}^{2}} /(n-1)}=\frac{S_{2}^{2} / \sigma_{2}^{2}}{S_{1}^{2} / \sigma_{1}^{2}} $$ where \(S_{1}^{2}\) and \(S_{2}^{2}\) are the respective sample variances. (a) What kind of distribution does \(F\) have? (b) Critical values \(a\) and \(b\) can be found so that \(P(F

In Exercise \(4.2 .27\), in finding a confidence interval for the ratio of the variances of two normal distributions, we used a statistic \(S_{1}^{2} / S_{2}^{2}\), which has an \(F\) distribution when those two variances are equal. If we denote that statistic by \(F\), we can test \(H_{0}: \sigma_{1}^{2}=\sigma_{2}^{2}\) against \(H_{1}: \sigma_{1}^{2}>\sigma_{2}^{2}\) using the critical region \(F \geq c\). If \(n=13, m=11\), and \(\alpha=0.05\), find \(c .\)
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