A certain type of flashlight is sold with the four batteries included. A random sample of 150 flashlights is obtained, and the number of defective batteries in each is determined, resulting in the following data? $$\begin{array} { c | c c c c c }
\hline \text { Number of defective } & 0 & 1 & 2 & 3 & 4 \
\hline \text { Frequency } & 26 & 51 & 47 & 16 & 10 \
\hline
\end{array}$$ Let X be the number of defective batteries in a randomly selected flashlight. Test the null hypothesis that the distribution of X is Bin $( 4 , \theta )$ That is, with $p _ {i } = P ( i \text { defectives } )$ test $H _ { 0 } : p _ { i } = \left( \begin{array} { l }
4 \
i
\end{array} \right) \theta^{t} ( 1 - \theta ) ^ { 4 - i }$ i=0,1,2,3,4
[Hint: To obtain the MLE of $\theta$ write the likelihood (the function to be maximized) as $\theta ^ { u } ( 1 - \theta ) ^ { \nu }$ where the exponents $u \text { and } v$ are linear functions of the cell counts. Then take the natural log, differentiate with respect to $\theta$ equate the result to 0, and solve for $\hat { \theta } .$ ]

Question

A certain type of flashlight is sold with the four batteries included. A random sample of 150 flashlights is obtained, and the number of defective batteries in each is determined, resulting in the following data? $$\begin{array} { c | c c c c c } 
\hline \text { Number of defective } & 0 & 1 & 2 & 3 & 4 \
\hline \text { Frequency } & 26 & 51 & 47 & 16 & 10 \
\hline
\end{array}$$ Let X be the number of defective batteries in a randomly selected flashlight. Test the null hypothesis that the distribution of X is Bin $( 4 , \theta )$ That is, with $p _ {i } = P ( i \text { defectives } )$ test $H _ { 0 } : p _ { i } = \left( \begin{array} { l } 
4 \
i
\end{array} \right) \theta^{t} ( 1 - \theta ) ^ { 4 - i }$ i=0,1,2,3,4
[Hint: To obtain the MLE of $\theta$ write the likelihood (the function to be maximized) as $\theta ^ { u } ( 1 - \theta ) ^ { \nu }$ where the exponents $u \text { and } v$ are linear functions of the cell counts. Then take the natural log, differentiate with respect to $\theta$ equate the result to 0, and solve for $\hat { \theta } .$ ]

Quizplus · Accepted Answer

The Answer of A certain type of flashlight is sold with the four batteries included. A random sample of 150 flashlights is obtained, and the number of defective batteries in each is determined, resulting in the following data? $$\begin{array} { c | c c c c c } 
\hline \text { Number of defective } & 0 & 1 & 2 & 3 & 4 \
\hline \text { Frequency } & 26 & 51 & 47 & 16 & 10 \
\hline
\end{array}$$ Let X be the number of defective batteries in a randomly selected flashlight. Test the null hypothesis that the distribution of X is Bin $( 4 , \theta )$ That is, with $p _ {i } = P ( i \text { defectives } )$ test $H _ { 0 } : p _ { i } = \left( \begin{array} { l } 
4 \
i
\end{array} \right) \theta^{t} ( 1 - \theta ) ^ { 4 - i }$ i=0,1,2,3,4
[Hint: To obtain the MLE of $\theta$ write the likelihood (the function to be maximized) as $\theta ^ { u } ( 1 - \theta ) ^ { \nu }$ where the exponents $u \text { and } v$ are linear functions of the cell counts. Then take the natural log, differentiate with respect to $\theta$ equate the result to 0, and solve for $\hat { \theta } .$ ]

A Certain Type of Flashlight Is Sold with the Four