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. P(x) 0.2 0.3 0.4 0.1 (a) Consider a random sample of size n = 2 (two customers), and let x be the sample mean number of packages shipped. Obtain the probability distribution of X. 1 1.5 2.5 3.5 4 P(x) (b) Refer to part (a) and calculate P(XS 2.5). (c) Again consider a random sample of sizen = 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).) 1 2 3 P(R) (d) If a random sample of size n = 4 is selected, what is P(X S 1.5)? [Hint: You should not have to list all possible outcomes, only those for which xs 1.5.]
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. P(x) 0.2 0.3 0.4 0.1 (a) Consider a random sample of size n = 2 (two customers), and let x be the sample mean number of packages shipped. Obtain the probability distribution of X. 1 1.5 2.5 3.5 4 P(x) (b) Refer to part (a) and calculate P(XS 2.5). (c) Again consider a random sample of sizen = 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).) 1 2 3 P(R) (d) If a random sample of size n = 4 is selected, what is P(X S 1.5)? [Hint: You should not have to list all possible outcomes, only those for which xs 1.5.]
A First Course in Probability (10th Edition)
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ISBN:9780134753119
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Chapter1: Combinatorial Analysis
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Please show and explain the steps! Solve (a) and (b) only!
![---
**Title: Probability Distribution of Packages Shipped**
**Context**
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:
\[
\begin{array}{c|ccccc}
x & 1 & 2 & 3 & 4 \\
\hline
p(x) & 0.2 & 0.3 & 0.4 & 0.1 \\
\end{array}
\]
**Tasks:**
**(a) Probability Distribution of Sample Mean**
- 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) Probability Calculation for Sample Mean**
- Refer to part (a) and calculate \( P(\bar{X} \leq 2.5) \).
**(c) Distribution of Sample Range**
- 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 \( R \).
\[
\begin{array}{c|cccc}
R & 0 & 1 & 2 & 3 \\
\hline
P(R) & & & & \\
\end{array}
\]
**(d) Probability of Sample Mean for \( n = 4 \)**
- 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 \( \bar{X} \leq 1.5 \).
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Transcribed Image Text:---
**Title: Probability Distribution of Packages Shipped**
**Context**
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:
\[
\begin{array}{c|ccccc}
x & 1 & 2 & 3 & 4 \\
\hline
p(x) & 0.2 & 0.3 & 0.4 & 0.1 \\
\end{array}
\]
**Tasks:**
**(a) Probability Distribution of Sample Mean**
- 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) Probability Calculation for Sample Mean**
- Refer to part (a) and calculate \( P(\bar{X} \leq 2.5) \).
**(c) Distribution of Sample Range**
- 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 \( R \).
\[
\begin{array}{c|cccc}
R & 0 & 1 & 2 & 3 \\
\hline
P(R) & & & & \\
\end{array}
\]
**(d) Probability of Sample Mean for \( n = 4 \)**
- 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 \( \bar{X} \leq 1.5 \).
---
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