A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Please solve (a), (b), and (c) only!
(a) is not all 1/7
(b) is not 0.2, 0.6, or 0.4285.
(c) is not 0.25, 0.375, 0.25, and 0.125
Thank you!
![**Image Transcription for Educational Website:**
Let \(X\) be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of \(X\) is as follows:
\[
\begin{array}{c|ccccc}
x & 1 & 2 & 3 & 4 \\
\hline
p(x) & 0.2 & 0.3 & 0.4 & 0.1 \\
\end{array}
\]
**(a)** Consider a random sample of size \(n = 2\) (two customers), and let \(\bar{X}\) be the sample mean number of packages shipped. Obtain the probability distribution of \(\bar{X}\).
\[
\begin{array}{c|cccccc}
\bar{X} & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\
\hline
P(\bar{X}) & & & & & & & \\
\end{array}
\]
**(b)** Refer to part (a) and calculate \(P(\bar{X} \leq 2.5)\).
**(c)** Again consider a random sample of size \(n = 2\), but now focus on the statistic \(R\) = the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of \(R\). [Hint: Calculate the value of \(R\) for each outcome and use the probabilities from part (a).]
\[
\begin{array}{c|cccc}
R & 0 & 1 & 2 & 3 \\
\hline
P(R) & & & & \\
\end{array}
\]
**(d)** If a random sample of size \(n = 4\) is selected, what is \(P(\bar{X} \leq 1.5)\)? [Hint: You should not have to list all possible outcomes, only those for which \(\bar{X} \leq 1.5\).]
---
*Note:* The table headers for \(\bar{X}\) and \(R\) indicate different possible outcomes. The probability distributions \(P(\bar{X})\) and \(P(R)\) need to be calculated based on](https://content.bartleby.com/qna-images/question/306db17a-c2a3-4020-aada-7a92dff91485/538bf9da-6d6b-420d-be98-e4babbaeae80/rpjeuj_processed.png)
Transcribed Image Text:**Image Transcription for Educational Website:**
Let \(X\) be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of \(X\) is as follows:
\[
\begin{array}{c|ccccc}
x & 1 & 2 & 3 & 4 \\
\hline
p(x) & 0.2 & 0.3 & 0.4 & 0.1 \\
\end{array}
\]
**(a)** Consider a random sample of size \(n = 2\) (two customers), and let \(\bar{X}\) be the sample mean number of packages shipped. Obtain the probability distribution of \(\bar{X}\).
\[
\begin{array}{c|cccccc}
\bar{X} & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\
\hline
P(\bar{X}) & & & & & & & \\
\end{array}
\]
**(b)** Refer to part (a) and calculate \(P(\bar{X} \leq 2.5)\).
**(c)** Again consider a random sample of size \(n = 2\), but now focus on the statistic \(R\) = the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of \(R\). [Hint: Calculate the value of \(R\) for each outcome and use the probabilities from part (a).]
\[
\begin{array}{c|cccc}
R & 0 & 1 & 2 & 3 \\
\hline
P(R) & & & & \\
\end{array}
\]
**(d)** If a random sample of size \(n = 4\) is selected, what is \(P(\bar{X} \leq 1.5)\)? [Hint: You should not have to list all possible outcomes, only those for which \(\bar{X} \leq 1.5\).]
---
*Note:* The table headers for \(\bar{X}\) and \(R\) indicate different possible outcomes. The probability distributions \(P(\bar{X})\) and \(P(R)\) need to be calculated based on
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