Solution Found!
Let X equal the amount of butterfat (in pounds) produced
Chapter 9, Problem 8E(choose chapter or problem)
For determining the half-lives of radioactive isotopes, it is important to know what the background radiation is for a given detector over a certain period. A \(\gamma\)-ray detection experiment over 300 one-second intervals yielded the following data:
\(\begin{array}{llllllllllllllllllll}
0 & 2 & 4 & 6 & 6 & 1 & 7 & 4 & 6 & 1 & 1 & 2 & 3 & 6 & 4 & 2 & 7 & 4 & 4 & 2 \\
2 & 5 & 4 & 4 & 4 & 1 & 2 & 4 & 3 & 2 & 2 & 5 & 0 & 3 & 1 & 1 & 0 & 0 & 5 & 2 \\
7 & 1 & 3 & 3 & 3 & 2 & 3 & 1 & 4 & 1 & 3 & 5 & 3 & 5 & 1 & 3 & 3 & 0 & 3 & 2 \\
6 & 1 & 1 & 4 & 6 & 3 & 6 & 4 & 4 & 2 & 2 & 4 & 3 & 3 & 6 & 1 & 6 & 2 & 5 & 0 \\
6 & 3 & 4 & 3 & 1 & 1 & 4 & 6 & 1 & 5 & 1 & 1 & 4 & 1 & 4 & 1 & 1 & 1 & 3 & 3 \\
4 & 3 & 3 & 2 & 5 & 2 & 1 & 3 & 5 & 3 & 2 & 7 & 0 & 4 & 2 & 3 & 3 & 5 & 6 & 1 \\
4 & 2 & 6 & 4 & 2 & 0 & 4 & 4 & 7 & 3 & 5 & 2 & 2 & 3 & 1 & 3 & 1 & 3 & 6 & 5 \\
4 & 8 & 2 & 2 & 4 & 2 & 2 & 1 & 4 & 7 & 5 & 2 & 1 & 1 & 4 & 1 & 4 & 3 & 6 & 2 \\
1 & 1 & 2 & 2 & 2 & 2 & 3 & 5 & 4 & 3 & 2 & 2 & 3 & 3 & 2 & 4 & 4 & 3 & 2 & 2 \\
3 & 6 & 1 & 1 & 3 & 3 & 2 & 1 & 4 & 5 & 5 & 1 & 2 & 3 & 3 & 1 & 3 & 7 & 2 & 5 \\
4 & 2 & 0 & 6 & 2 & 3 & 2 & 3 & 0 & 4 & 4 & 5 & 2 & 5 & 3 & 0 & 4 & 6 & 2 & 2 \\
2 & 2 & 2 & 5 & 2 & 2 & 3 & 4 & 2 & 3 & 7 & 1 & 1 & 7 & 1 & 3 & 6 & 0 & 5 & 3 \\
0 & 0 & 3 & 3 & 0 & 2 & 4 & 3 & 1 & 2 & 3 & 3 & 3 & 4 & 3 & 2 & 2 & 7 & 5 & 3 \\
5 & 1 & 1 & 2 & 2 & 6 & 1 & 3 & 1 & 4 & 4 & 2 & 3 & 4 & 5 & 1 & 3 & 4 & 3 & 1 \\
0 & 3 & 7 & 4 & 0 & 5 & 2 & 5 & 4 & 4 & 2 & 2 & 3 & 2 & 4 & 6 & 5 & 5 & 3 & 4
\end{array}\)
Do these look like observations of a Poisson random variable with mean \(\lambda=3\)? To answer this question, do the following:
(a) Find the frequencies of \(0, 1, 2, \ldots, 8\).
(b) Calculate the sample mean and sample variance. Are they approximately equal to each other?
(c) Construct a probability histogram with \(\lambda=3\) and a relative frequency histogram on the same graph.
(d) Use \(\alpha=0.05\) and a chi-square goodness-of-fit test to answer this question.
Questions & Answers
QUESTION:
For determining the half-lives of radioactive isotopes, it is important to know what the background radiation is for a given detector over a certain period. A \(\gamma\)-ray detection experiment over 300 one-second intervals yielded the following data:
\(\begin{array}{llllllllllllllllllll}
0 & 2 & 4 & 6 & 6 & 1 & 7 & 4 & 6 & 1 & 1 & 2 & 3 & 6 & 4 & 2 & 7 & 4 & 4 & 2 \\
2 & 5 & 4 & 4 & 4 & 1 & 2 & 4 & 3 & 2 & 2 & 5 & 0 & 3 & 1 & 1 & 0 & 0 & 5 & 2 \\
7 & 1 & 3 & 3 & 3 & 2 & 3 & 1 & 4 & 1 & 3 & 5 & 3 & 5 & 1 & 3 & 3 & 0 & 3 & 2 \\
6 & 1 & 1 & 4 & 6 & 3 & 6 & 4 & 4 & 2 & 2 & 4 & 3 & 3 & 6 & 1 & 6 & 2 & 5 & 0 \\
6 & 3 & 4 & 3 & 1 & 1 & 4 & 6 & 1 & 5 & 1 & 1 & 4 & 1 & 4 & 1 & 1 & 1 & 3 & 3 \\
4 & 3 & 3 & 2 & 5 & 2 & 1 & 3 & 5 & 3 & 2 & 7 & 0 & 4 & 2 & 3 & 3 & 5 & 6 & 1 \\
4 & 2 & 6 & 4 & 2 & 0 & 4 & 4 & 7 & 3 & 5 & 2 & 2 & 3 & 1 & 3 & 1 & 3 & 6 & 5 \\
4 & 8 & 2 & 2 & 4 & 2 & 2 & 1 & 4 & 7 & 5 & 2 & 1 & 1 & 4 & 1 & 4 & 3 & 6 & 2 \\
1 & 1 & 2 & 2 & 2 & 2 & 3 & 5 & 4 & 3 & 2 & 2 & 3 & 3 & 2 & 4 & 4 & 3 & 2 & 2 \\
3 & 6 & 1 & 1 & 3 & 3 & 2 & 1 & 4 & 5 & 5 & 1 & 2 & 3 & 3 & 1 & 3 & 7 & 2 & 5 \\
4 & 2 & 0 & 6 & 2 & 3 & 2 & 3 & 0 & 4 & 4 & 5 & 2 & 5 & 3 & 0 & 4 & 6 & 2 & 2 \\
2 & 2 & 2 & 5 & 2 & 2 & 3 & 4 & 2 & 3 & 7 & 1 & 1 & 7 & 1 & 3 & 6 & 0 & 5 & 3 \\
0 & 0 & 3 & 3 & 0 & 2 & 4 & 3 & 1 & 2 & 3 & 3 & 3 & 4 & 3 & 2 & 2 & 7 & 5 & 3 \\
5 & 1 & 1 & 2 & 2 & 6 & 1 & 3 & 1 & 4 & 4 & 2 & 3 & 4 & 5 & 1 & 3 & 4 & 3 & 1 \\
0 & 3 & 7 & 4 & 0 & 5 & 2 & 5 & 4 & 4 & 2 & 2 & 3 & 2 & 4 & 6 & 5 & 5 & 3 & 4
\end{array}\)
Do these look like observations of a Poisson random variable with mean \(\lambda=3\)? To answer this question, do the following:
(a) Find the frequencies of \(0, 1, 2, \ldots, 8\).
(b) Calculate the sample mean and sample variance. Are they approximately equal to each other?
(c) Construct a probability histogram with \(\lambda=3\) and a relative frequency histogram on the same graph.
(d) Use \(\alpha=0.05\) and a chi-square goodness-of-fit test to answer this question.
ANSWER:Step 1 of 4
Given:
The sample number of cows is \(n=90\).
The sample mean amount of butterfat (in pounds) produced is \(\bar{x}=511.633\).
The sample standard deviation is \(s_{x}=87.576\).
The random variable X represents the amount of butterfat (in pounds) produced by 90 cows during a 305-day milk production period following the birth of their first calf.