Harry Roberts, a nâturalist for the Alaska Fish and Game...

Harry Roberts, a nâturalist for the Alaska Fish and Game department, studies grizzly bears with the goal of maintaining a healthy population. Measurements on $n=61$ bears provided the following summary statistics (see also Exercise 8.23): $$ \begin{array}{l|cccccc} \text { Variable } & \begin{array}{c} \text { Weight } \\ (\mathrm{kg}) \end{array} & \begin{array}{c} \text { Body } \\ \text { length } \\ (\mathrm{cm}) \end{array} & \begin{array}{c} \text { Neck } \\ (\mathrm{cm}) \end{array} & \begin{array}{c} \text { Girth } \\ (\mathrm{cm}) \end{array} & \begin{array}{c} \text { Head } \\ \text { length } \\ (\mathrm{cm}) \end{array} & \begin{array}{c} \text { Head } \\ \text { width } \\ (\mathrm{cm}) \end{array} \\ \hline \begin{array}{l} \text { Sample } \\ \text { mean } \bar{x} \end{array} & 95.52 & 164.38 & 55.69 & 93.39 & 17.98 & 31.13 \end{array} $$ $\mathbf{S}=\left[\begin{array}{rrrrrr}3266.46 & 1343.97 & 731.54 & 1175.50 & 162.68 & 238.37 \\ 1343.97 & 721.91 & 324.25 & 537.35 & 80.17 & 117.73 \\ 731.54 & 324.25 & 179.28 & 281.17 & 39.15 & 56.80 \\ 1175.50 & 537.35 & 281.17 & 474.98 & 63.73 & 94.85 \\ 162.68 & 80.17 & 39.15 & 63.73 & 9.95 & 13.88 \\ 238.37 & 117.73 & 56.80 & 94.85 & 13.88 & 21.26\end{array}\right]$ (a) Obtain the large sample $95 \%$ simultaneous confidence intervals for the six population mean body measurements. (b) Obtain the large sample $95 \%$ simultaneous confidence ellipse for mean weight and mean girth. (c) Obtain the $95 \%$ Bonferroni confidence intervals for the six means in Part a. (d) Refer to Part b. Construct the $95 \%$ Bonferroni confidence rectangle for the mean weight and mean girth using $m=6$. Compare this rectangle with the confidence ellipse in Part b. (e) Obtain the $95 \%$ Bonferroni confidence interval for mean head width $\sim$ mean head length using $m=6+1 \approx 7$ to allow for this statement as well as statements about each individual mean.

Harry Roberts, a nâturalist for the Alaska Fish and Game department, studies grizzly bears with the goal of maintaining a healthy population. Measurements on $n=61$ bears provided the following summary statistics (see also Exercise 8.23):
$$
\begin{array}{l|cccccc}
\text { Variable } & \begin{array}{c}
\text { Weight } \\
(\mathrm{kg})
\end{array} & \begin{array}{c}
\text { Body } \\
\text { length } \\
(\mathrm{cm})
\end{array} & \begin{array}{c}
\text { Neck } \\
(\mathrm{cm})
\end{array} & \begin{array}{c}
\text { Girth } \\
(\mathrm{cm})
\end{array} & \begin{array}{c}
\text { Head } \\
\text { length } \\
(\mathrm{cm})
\end{array} & \begin{array}{c}
\text { Head } \\
\text { width } \\
(\mathrm{cm})
\end{array} \\
\hline \begin{array}{l}
\text { Sample } \\
\text { mean } \bar{x}
\end{array} & 95.52 & 164.38 & 55.69 & 93.39 & 17.98 & 31.13
\end{array}
$$
$\mathbf{S}=\left[\begin{array}{rrrrrr}3266.46 & 1343.97 & 731.54 & 1175.50 & 162.68 & 238.37 \\ 1343.97 & 721.91 & 324.25 & 537.35 & 80.17 & 117.73 \\ 731.54 & 324.25 & 179.28 & 281.17 & 39.15 & 56.80 \\ 1175.50 & 537.35 & 281.17 & 474.98 & 63.73 & 94.85 \\ 162.68 & 80.17 & 39.15 & 63.73 & 9.95 & 13.88 \\ 238.37 & 117.73 & 56.80 & 94.85 & 13.88 & 21.26\end{array}\right]$
(a) Obtain the large sample $95 \%$ simultaneous confidence intervals for the six population mean body measurements.
(b) Obtain the large sample $95 \%$ simultaneous confidence ellipse for mean weight and mean girth.
(c) Obtain the $95 \%$ Bonferroni confidence intervals for the six means in Part a.
(d) Refer to Part b. Construct the $95 \%$ Bonferroni confidence rectangle for the mean weight and mean girth using $m=6$. Compare this rectangle with the confidence ellipse in Part b.
(e) Obtain the $95 \%$ Bonferroni confidence interval for mean head width $\sim$ mean head length
using $m=6+1 \approx 7$ to allow for this statement as well as statements about each individual mean.

Key Concepts

Large Sample Theory

Large sample theory, often based on the Central Limit Theorem, allows for the approximation of the sampling distribution of estimators by a normal distribution when the sample size is sufficiently large. This approximation is critical in many statistical methods, including the construction of confidence intervals and regions, as it underpins the use of asymptotic properties to derive critical values and make inference about population parameters.

Covariance Matrix

A covariance matrix is a key tool in multivariate statistics that captures the variances of individual variables along its diagonal and the pairwise covariances off the diagonal. It provides a comprehensive picture of the variability and the linear relationships among the measurements. In the context of constructing confidence regions, the covariance matrix informs the shape and orientation of the regions, such as ellipses, being used.

Bonferroni Correction

The Bonferroni method is a conservative statistical adjustment used when multiple comparisons or simultaneous inferences are made. It adjusts the significance level for each individual test so that the overall family-wise error rate remains at or below the desired level. This is particularly useful for constructing multiple confidence intervals, as it reduces the chance of incorrectly rejecting any of the null hypotheses due to the multiplicity of tests.

Confidence Ellipse

A confidence ellipse is a type of confidence region used to jointly estimate two parameters, typically in a bivariate normal context. It represents the set of values for the pair of parameters that are consistent with the data at a given confidence level. The shape and orientation of the ellipse are determined by the variances of the individual estimates and their covariance, providing insight into both the marginal uncertainty and the correlation between the estimates.

Simultaneous Confidence Intervals

These intervals are constructed to estimate multiple population parameters at once while controlling the overall confidence level. Instead of creating separate confidence intervals for each parameter, simultaneous methods ensure that the probability that all intervals contain their respective true values is at least the specified confidence level. This is crucial in multivariate settings where making multiple inferences increases the risk of Type I errors.

Transcript

Please give Ace some feedback