The random variable X, representing the number of cherries in a cherry puff, has the probability distribution shown. 7 0.3 X P(X-x) 6 0.1 8 0.4 9 0.2 Complete parts (a) through (c) below. Click here to view page 1 of the standard normal distribution table. Click here to view page 2 of the standard normal distribution table. μ=|| (Type an integer or a decimal. Do not round.) 0²= (Type an integer or a decimal. Do not round.) (b) Find the mean µ and the variance of the mean X for random samples of 49 cherry puffs "x (Type an integer or a decimal. Do not round.) (Round to three decimal places as needed.) (c) Find the probability that the average number of cherries in 49 cherry puffs will be less than 7.8. The probability is. (Round to four decimal places as needed.)
The random variable X, representing the number of cherries in a cherry puff, has the probability distribution shown. 7 0.3 X P(X-x) 6 0.1 8 0.4 9 0.2 Complete parts (a) through (c) below. Click here to view page 1 of the standard normal distribution table. Click here to view page 2 of the standard normal distribution table. μ=|| (Type an integer or a decimal. Do not round.) 0²= (Type an integer or a decimal. Do not round.) (b) Find the mean µ and the variance of the mean X for random samples of 49 cherry puffs "x (Type an integer or a decimal. Do not round.) (Round to three decimal places as needed.) (c) Find the probability that the average number of cherries in 49 cherry puffs will be less than 7.8. The probability is. (Round to four decimal places as needed.)
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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![### 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)
