The overhead reach distances of adult females are normally distributed with a mean of 205 cm and a standard deviation of 8.6 cm. a. Find the probability that an individual distance is greater than 218.40 cm. b. Find the probability that the mean for 20 randomly selected distances is greater than 203.70 cm. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? a. The probability is. (Round to four decimal places as needed.)

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## Overview

The overhead reach distances of adult females are normally distributed with a mean of 205 cm and a standard deviation of 8.6 cm. This information sets the foundation for three key probability questions, which involve determining specific probabilities and understanding the application of the normal distribution.

### Questions

#### a. Find the probability that an individual distance is greater than 218.40 cm.
1. **Objective**: Calculate the probability that a single adult female's overhead reach distance is more than 218.4 cm.
2. **Methodology**: Utilize the properties of a normal distribution with the given mean (205 cm) and standard deviation (8.6 cm) to find the probability.

#### b. Find the probability that the mean for 20 randomly selected distances is greater than 203.70 cm.
1. **Objective**: Determine the probability that the average overhead reach distance of a sample of 20 adult females exceeds 203.7 cm.
2. **Methodology**: Utilize the sampling distribution of the sample mean. The mean remains at 205 cm, and the standard error of the mean can be calculated using the standard deviation divided by the square root of the sample size (n).

#### c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?
1. **Objective**: Explain the applicability of the normal distribution for a sample size less than 30.
2. **Methodology**: Discuss the circumstances under which the normal distribution is applicable based on the Central Limit Theorem, especially given that the original population is normally distributed.

### Detailed Calculations

**To be filled and calculated manually or using statistical software:**

#### a. The probability is [____]. 
*(Round to four decimal places as needed.)*

### Graph and Diagram Explanation

No graphs or diagrams are included in the provided text. However, for illustration purposes, if a graph were to be included:
- **Graph**: A bell curve representing the normal distribution with a mean of 205 cm and a standard deviation of 8.6 cm.
- **Annotations**: The area representing the probability of exceeding 218.4 cm would be shaded, and a vertical line at 218.4 cm would indicate where the probability is calculated beyond.

### Conclusion

The given problem involves understanding and applying the properties of the normal distribution to specific probability questions regarding overhead reach distances of adult females. The steps involve the use
Transcribed Image Text:## Overview The overhead reach distances of adult females are normally distributed with a mean of 205 cm and a standard deviation of 8.6 cm. This information sets the foundation for three key probability questions, which involve determining specific probabilities and understanding the application of the normal distribution. ### Questions #### a. Find the probability that an individual distance is greater than 218.40 cm. 1. **Objective**: Calculate the probability that a single adult female's overhead reach distance is more than 218.4 cm. 2. **Methodology**: Utilize the properties of a normal distribution with the given mean (205 cm) and standard deviation (8.6 cm) to find the probability. #### b. Find the probability that the mean for 20 randomly selected distances is greater than 203.70 cm. 1. **Objective**: Determine the probability that the average overhead reach distance of a sample of 20 adult females exceeds 203.7 cm. 2. **Methodology**: Utilize the sampling distribution of the sample mean. The mean remains at 205 cm, and the standard error of the mean can be calculated using the standard deviation divided by the square root of the sample size (n). #### c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? 1. **Objective**: Explain the applicability of the normal distribution for a sample size less than 30. 2. **Methodology**: Discuss the circumstances under which the normal distribution is applicable based on the Central Limit Theorem, especially given that the original population is normally distributed. ### Detailed Calculations **To be filled and calculated manually or using statistical software:** #### a. The probability is [____]. *(Round to four decimal places as needed.)* ### Graph and Diagram Explanation No graphs or diagrams are included in the provided text. However, for illustration purposes, if a graph were to be included: - **Graph**: A bell curve representing the normal distribution with a mean of 205 cm and a standard deviation of 8.6 cm. - **Annotations**: The area representing the probability of exceeding 218.4 cm would be shaded, and a vertical line at 218.4 cm would indicate where the probability is calculated beyond. ### Conclusion The given problem involves understanding and applying the properties of the normal distribution to specific probability questions regarding overhead reach distances of adult females. The steps involve the use
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