Let x be a continuous random variable that is normally distributed with a mean of 56 and a standard deviation of 14 places the probability that x assumes a value a. less than 38 i b. greater than 72 i c. greater than 38 d. less than 63
Let x be a continuous random variable that is normally distributed with a mean of 56 and a standard deviation of 14 places the probability that x assumes a value a. less than 38 i b. greater than 72 i c. greater than 38 d. less than 63
Chapter9: Sequences, Probability And Counting Theory
Section9.7: Probability
Problem 5SE: The union of two sets is defined as a set of elements that are present in at least one of the sets....
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![### Probability Calculation for Normally Distributed Random Variables
Let \( x \) be a continuous random variable that is normally distributed with a mean of 56 and a standard deviation of 14. Find to 4 decimal places the probability that \( x \) assumes a value:
1. **Less than 38**
- Probability: [Insert answer here]
2. **Greater than 72**
- Probability: [Insert answer here]
3. **Greater than 38**
- Probability: [Insert answer here]
4. **Less than 63**
- Probability: [Insert answer here]
#### Explanation:
To solve these problems, you can use the standard normal distribution (z-score) formula:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
- \( X \) is the value from the distribution
- \( \mu \) is the mean of the distribution
- \( \sigma \) is the standard deviation of the distribution
- \( Z \) is the z-score corresponding to the value \( X \)
Once the z-score is calculated, you can use standard normal distribution tables or software/calculators to find the corresponding probabilities.
For each given condition (less than 38, greater than 72, etc.), calculate the z-score and then determine the cumulative probability from the Z-table or appropriate function in a statistical software/calculator.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcce67d60-ba9c-4b65-a73f-671ae19935ed%2F45f9092d-f542-4da1-a52e-ff0e2144d579%2Fhpol7eg_processed.png&w=3840&q=75)
Transcribed Image Text:### Probability Calculation for Normally Distributed Random Variables
Let \( x \) be a continuous random variable that is normally distributed with a mean of 56 and a standard deviation of 14. Find to 4 decimal places the probability that \( x \) assumes a value:
1. **Less than 38**
- Probability: [Insert answer here]
2. **Greater than 72**
- Probability: [Insert answer here]
3. **Greater than 38**
- Probability: [Insert answer here]
4. **Less than 63**
- Probability: [Insert answer here]
#### Explanation:
To solve these problems, you can use the standard normal distribution (z-score) formula:
\[ Z = \frac{X - \mu}{\sigma} \]
Where:
- \( X \) is the value from the distribution
- \( \mu \) is the mean of the distribution
- \( \sigma \) is the standard deviation of the distribution
- \( Z \) is the z-score corresponding to the value \( X \)
Once the z-score is calculated, you can use standard normal distribution tables or software/calculators to find the corresponding probabilities.
For each given condition (less than 38, greater than 72, etc.), calculate the z-score and then determine the cumulative probability from the Z-table or appropriate function in a statistical software/calculator.
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