A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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Solve the following
![### Probability Distribution of Cherries in Cherry Puffs
#### Given Information:
The random variable \( X \), representing the number of cherries in a cherry puff, has the probability distribution shown below:
\[
\begin{array}{c|cccc}
\boldsymbol{x} & 6 & 7 & 8 & 9 \\ \hline
\mathbf{P(X = x)} & 0.1 & 0.3 & 0.4 & 0.2 \\
\end{array}
\]
Complete parts (a) through (c) below. Utilize the provided standard normal distribution tables for calculations:
- [Page 1 of the standard normal distribution table](#)
- [Page 2 of the standard normal distribution table](#)
#### Tasks:
(a) Calculate the following:
1. The expected value (mean) \( \mu \) of \( X \) (Type an integer or a decimal. Do not round.)
2. The variance \( \sigma^2 \) of \( X \) (Type an integer or a decimal. Do not round.)
(b) For random samples of 49 cherry puffs, find:
1. The mean \( \mu_{\overline{X}} \) of the sample mean \( \overline{X} \) (Type an integer or a decimal. Do not round.)
2. The variance \( \sigma^2_{\overline{X}} \) of the sample mean \( \overline{X} \) (Round to three decimal places as needed.)
(c) Determine the probability that the average number of cherries in 49 cherry puffs will be less than 7.8.
- (Round to four decimal places as needed.)
#### Calculation Steps:
1. **Expected Value \( \mu \) and Variance \( \sigma^2 \)**
2. **Sample Mean \( \mu_{\overline{X}} \) and Sample Variance \( \sigma^2_{\overline{X}} \)**
3. **Probability Calculation for the Average Number of Cherries**
By completing the above steps, you will understand how to calculate the expected value, variance, and probability for the given probability distribution, essential concepts in statistics and probability theory.](https://content.bartleby.com/qna-images/question/7167d01a-fc0a-4c3f-ab49-29dc29956a80/18460df6-fff4-4f58-9a27-93f4fa4b1cf2/qhpwr3m_processed.png)
Transcribed Image Text:### Probability Distribution of Cherries in Cherry Puffs
#### Given Information:
The random variable \( X \), representing the number of cherries in a cherry puff, has the probability distribution shown below:
\[
\begin{array}{c|cccc}
\boldsymbol{x} & 6 & 7 & 8 & 9 \\ \hline
\mathbf{P(X = x)} & 0.1 & 0.3 & 0.4 & 0.2 \\
\end{array}
\]
Complete parts (a) through (c) below. Utilize the provided standard normal distribution tables for calculations:
- [Page 1 of the standard normal distribution table](#)
- [Page 2 of the standard normal distribution table](#)
#### Tasks:
(a) Calculate the following:
1. The expected value (mean) \( \mu \) of \( X \) (Type an integer or a decimal. Do not round.)
2. The variance \( \sigma^2 \) of \( X \) (Type an integer or a decimal. Do not round.)
(b) For random samples of 49 cherry puffs, find:
1. The mean \( \mu_{\overline{X}} \) of the sample mean \( \overline{X} \) (Type an integer or a decimal. Do not round.)
2. The variance \( \sigma^2_{\overline{X}} \) of the sample mean \( \overline{X} \) (Round to three decimal places as needed.)
(c) Determine the probability that the average number of cherries in 49 cherry puffs will be less than 7.8.
- (Round to four decimal places as needed.)
#### Calculation Steps:
1. **Expected Value \( \mu \) and Variance \( \sigma^2 \)**
2. **Sample Mean \( \mu_{\overline{X}} \) and Sample Variance \( \sigma^2_{\overline{X}} \)**
3. **Probability Calculation for the Average Number of Cherries**
By completing the above steps, you will understand how to calculate the expected value, variance, and probability for the given probability distribution, essential concepts in statistics and probability theory.
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