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Chapter 5: Problem 61

Anabolic steroid abuse has been increasing despite increased press reports of adverse medical and psychiatric consequences. In a recent study, medical researchers studied the potential for addiction to testosterone in hamsters (Neuroscience [2004]: \(971-981\) ). Hamsters were allowed to self-administer testosterone over a period of days, resulting in the death of some of the animals. The data below show the proportion of hamsters surviving versus the peak self-administration of testosterone \((\mu \mathrm{g})\). Fit a logistic regression equation and use the equation to predict the probability of survival for a hamster with a peak intake of \(40 \mu \mathrm{g} . \quad \ln \left(\frac{P}{1-p}\right)=4.589-0.0659 x ; 0.876\)

Short Answer

Expert verified
The estimated survival probability for a hamster with a peak testosterone intake of 40 micrograms is approximately 87.57%.

Step by step solution

01

Identify and Understand the Regression Equation

The given logistic regression equation is \(\ln \left(\frac{P}{1-P}\right) = 4.589 - 0.0659x\), where \(P\) is the survival probability and \(x\) is the peak self-administration of testosterone.
02

Substitute the Given Value

Next, substitute the value \(x = 40\) micrograms into the equation to obtain \(\ln \left(\frac{P}{1-P}\right) = 4.589 - 0.0659(40)\).
03

Simplify the Equation

Simplify the equation to get \(\ln \left(\frac{P}{1-P}\right) = 4.589 - 2.636 = 1.953\).
04

Solve for P

The next step is to solve the equation for \(P\). This is done by first eliminating the natural logarithm \(\ln\). By taking the exponent of both sides of the equation, we get the expression \(\frac{P}{1-P} = e^{1.953}\). Next, rearrange the equation to solve for \(P\): \(P = \frac{e^{1.953}}{1 + e^{1.953}}\).
05

Calculate the value of P

Finally, evaluate the expression to get the value of \(P\). The exponential of 1.953 is approximately 7.046. Divide this by 1 plus itself to get \(P = \frac{7.046}{1 + 7.046} = 0.8757\), which indicates a 87.57% survival probability for a hamster with a peak intake of 40 micrograms of testosterone.

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Most popular questions from this chapter

Data on high school GPA \((x)\) and first-year college GPA \((y)\) collected from a southeastern public research university can be summarized as follows ("First-Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students," Journal of College Student Development [1999]: \(599-605\) ): $$ \begin{array}{clc} n=2600 & \sum x=9620 & \sum y=7436 \\ \sum x y=27,918 & \sum x^{2}=36,168 & \sum y^{2}=23,145 \end{array} $$ a. Find the equation of the least-squares regression line. b. Interpret the value of \(b\), the slope of the least-squares line, in the context of this problem. c. What first-year GPA would you predict for a student with a \(4.0\) high school GPA?

Is the following statement correct? Explain why or why not. A correlation coefficient of 0 implies that no relationship exists between the two variables under study.

In the article "Reproductive Biology of the Aquatic Salamander Amphiuma tridactylum in Louisiana" (Journal of Herpetology [1999]: \(100-105\) ), 14 female salamanders were studied. Using regression, the researchers predicted \(y=\) clutch size (number of salamander eggs) from \(x=\) snout-vent length (in centimeters) as follows: $$ \hat{y}=-147+6.175 x $$ For the salamanders in the study, the range of snout-vent lengths was approximately 30 to \(70 \mathrm{~cm}\). a. What is the value of the \(y\) intercept of the least-squares line? What is the value of the slope of the least-squares line? Interpret the slope in the context of this problem. b. Would you be reluctant to predict the clutch size when snout-vent length is \(22 \mathrm{~cm}\) ? Explain.

'The article "Reduction in Soluble Protein and Chlorophyll Contents in a Few Plants as Indicators of Automobile Exhaust Pollution" (International Journal of Environmental Studies [1983]: \(239-244\) ) reported the following data on \(x=\) distance from a highway (in meters) and \(y=\) lead content of soil at that distance (in parts per million): a. Use a statistical computer package to construct scatterplots of \(y\) versus \(x, y\) versus \(\log (x), \log (y)\) versus \(\log (x)\) and \(\frac{1}{y}\) versus \(\frac{1}{x}\). b. Which transformation considered in Part (a) does the best job of producing an approximately linear relationship? Use the selected transformation to predict lead content when distance is \(25 \mathrm{~m}\).

According to the article "First-Year Academic Success: A Prediction Combining Cognitive and Psychosocial Variables for Caucasian and African American Students" \((\) Journal of College Student Development \([1999]: 599-\) 605), there is a mild correlation between high school GPA \((x)\) and first-year college GPA \((y)\). The data can be summarized as follows: $$ \begin{array}{clc} n=2600 & \sum x=9620 & \sum y=7436 \\ \sum x y=27,918 & \sum x^{2}=36,168 & \sum y^{2}=23,145 \end{array} $$ An alternative formula for computing the correlation coefficient that is based on raw data and is algebraically equivalent to the one given in the text is $$ r=\frac{\sum x y-\frac{\left(\sum x\right)\left(\sum y\right)}{n}}{\sqrt{\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}} \sqrt{\sum y^{2}-\frac{\left(\sum y\right)^{2}}{n}}} $$ Use this formula to compute the value of the correlation coefficient, and interpret this value.
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