The weights of ice cream cartons are normally distributed with a mean weight of 9 ounces and a standard deviation of 0,5 ounce. (a) What is the probability that a randomly selected carton has a weight greater than 9,13 ounces? (b) A sample of 36 cartons is randomly selected. What is the probability that their mean weight is greater than 9.13 ounces? *** (a) The probability is 0.3974. (Round to four decimal places as needed.) (b) The probability is urc (Round to four decimal places as needed.) ss ess ibrary tions
The weights of ice cream cartons are normally distributed with a mean weight of 9 ounces and a standard deviation of 0,5 ounce. (a) What is the probability that a randomly selected carton has a weight greater than 9,13 ounces? (b) A sample of 36 cartons is randomly selected. What is the probability that their mean weight is greater than 9.13 ounces? *** (a) The probability is 0.3974. (Round to four decimal places as needed.) (b) The probability is urc (Round to four decimal places as needed.) ss ess ibrary tions
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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I don’t understand how to get the probability when doing the z score I keep getting the question a wrong but question be right please help test is coming up and I’m trying to practice this
It gives me the formula but something is not clicking for me again I get the part b right
![### Probability of Ice Cream Carton Weights
The weights of ice cream cartons are normally distributed with a mean weight of 9 ounces and a standard deviation of 0.5 ounce.
**Problem (a):**
What is the probability that a randomly selected carton has a weight greater than 9.13 ounces?
**Solution for (a):**
The probability is **0.3974**.
*Note: This value is rounded to four decimal places as needed.*
---
**Problem (b):**
A sample of 36 cartons is randomly selected. What is the probability that their mean weight is greater than 9.13 ounces?
**Solution for (b):**
The probability is **[Blank]**.
*Note: This value should be rounded to four decimal places as needed.*
**Instructions:**
- For Problem (a), the answer is provided.
- For Problem (b), compute the probability based on the given parameters and the formula for the sample mean's distribution.
To better understand these problems and solutions, please refer to the related concepts in normal distribution and sampling distribution methodologies.
**Helpful Links:**
- [Help me solve this](#)
- [View an example](#)
- [Get more help](#)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a348127-0366-4d6a-8a8b-87b3575e425d%2Fe69f02f1-34e7-4af8-92ab-08db2fc6685e%2Fpi5re1q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Probability of Ice Cream Carton Weights
The weights of ice cream cartons are normally distributed with a mean weight of 9 ounces and a standard deviation of 0.5 ounce.
**Problem (a):**
What is the probability that a randomly selected carton has a weight greater than 9.13 ounces?
**Solution for (a):**
The probability is **0.3974**.
*Note: This value is rounded to four decimal places as needed.*
---
**Problem (b):**
A sample of 36 cartons is randomly selected. What is the probability that their mean weight is greater than 9.13 ounces?
**Solution for (b):**
The probability is **[Blank]**.
*Note: This value should be rounded to four decimal places as needed.*
**Instructions:**
- For Problem (a), the answer is provided.
- For Problem (b), compute the probability based on the given parameters and the formula for the sample mean's distribution.
To better understand these problems and solutions, please refer to the related concepts in normal distribution and sampling distribution methodologies.
**Helpful Links:**
- [Help me solve this](#)
- [View an example](#)
- [Get more help](#)
![### Understanding Probability in Normal Distribution
**Problem Description:**
The weights of ice cream cartons are normally distributed with:
- **Mean weight (μ):** 9 ounces
- **Standard deviation (σ):** 0.5 ounces
Two scenarios are given:
1. **A single carton is randomly selected.**
- **Question:** What is the probability that this carton has a weight greater than 9.13 ounces?
2. **A sample of 36 cartons is randomly selected.**
- **Question:** What is the probability that their mean weight is greater than 9.13 ounces?
---
**Solution Steps:**
(a) **For a single randomly selected carton:**
- To find the probability that a single carton has a weight greater than 9.13 ounces, we will use the standard normal distribution (Z-distribution).
- **Calculation:**
- Convert the weight to a Z-score using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
- For this problem:
\[
Z = \frac{9.13 - 9}{0.5} = \frac{0.13}{0.5} = 0.26
\]
- Using Z-tables or a standard normal distribution calculator, find the probability corresponding to Z = 0.26.
- **Result:**
- The probability is approximately **0.3974** (rounded to four decimal places).
---
(b) **For the mean weight of a sample of 36 cartons:**
- When dealing with the means of samples, the standard deviation of the sampling distribution (Standard Error) must be calculated using:
\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
\]
- Where \( n \) is the sample size.
- For this problem:
\[
\text{Standard Error} = \frac{0.5}{\sqrt{36}} = \frac{0.5}{6} = 0.0833
\]
- **Calculation:**
- Convert the mean weight to a Z-score using the standard error:
\[
Z = \frac{\bar{X} - \mu}{\text{Standard Error}} = \frac{9.13 - 9}{0.0833} = 1.56](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a348127-0366-4d6a-8a8b-87b3575e425d%2Fe69f02f1-34e7-4af8-92ab-08db2fc6685e%2Fr3xa8g6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Probability in Normal Distribution
**Problem Description:**
The weights of ice cream cartons are normally distributed with:
- **Mean weight (μ):** 9 ounces
- **Standard deviation (σ):** 0.5 ounces
Two scenarios are given:
1. **A single carton is randomly selected.**
- **Question:** What is the probability that this carton has a weight greater than 9.13 ounces?
2. **A sample of 36 cartons is randomly selected.**
- **Question:** What is the probability that their mean weight is greater than 9.13 ounces?
---
**Solution Steps:**
(a) **For a single randomly selected carton:**
- To find the probability that a single carton has a weight greater than 9.13 ounces, we will use the standard normal distribution (Z-distribution).
- **Calculation:**
- Convert the weight to a Z-score using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
- For this problem:
\[
Z = \frac{9.13 - 9}{0.5} = \frac{0.13}{0.5} = 0.26
\]
- Using Z-tables or a standard normal distribution calculator, find the probability corresponding to Z = 0.26.
- **Result:**
- The probability is approximately **0.3974** (rounded to four decimal places).
---
(b) **For the mean weight of a sample of 36 cartons:**
- When dealing with the means of samples, the standard deviation of the sampling distribution (Standard Error) must be calculated using:
\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}}
\]
- Where \( n \) is the sample size.
- For this problem:
\[
\text{Standard Error} = \frac{0.5}{\sqrt{36}} = \frac{0.5}{6} = 0.0833
\]
- **Calculation:**
- Convert the mean weight to a Z-score using the standard error:
\[
Z = \frac{\bar{X} - \mu}{\text{Standard Error}} = \frac{9.13 - 9}{0.0833} = 1.56
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