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The overhead reach distances of adult females are normally distributed with a mean of 205 cm and a standard deviation of 8.9 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 203.50 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 0.8438 (Round to four decimal places as needed.)

The overhead reach distances of adult females are normally distributed with a mean of 205 cm and a standard deviation of 8.9 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 203.50 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 0.8438 (Round to four decimal places as needed.)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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**Transcription and Explanation for Educational Use** --- **Title: Understanding Normal Distribution in Statistical Analysis** The overhead reach distances of adult females are normally distributed with a mean of 205 cm and a standard deviation of 8.9 cm. ### Problems to Solve **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 203.50 cm. **c.** Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? --- ### Solutions **a.** The probability is **0.8438**. (Note: This probability is rounded to four decimal places as needed.) --- ### Discussion In part (a), the task is to determine the likelihood of a single value exceeding a specified threshold in a normally distributed dataset. Here, standard normal distribution techniques or Z-scores are likely employed. In part (b), although the sample size is 20, the problem refers to using a sample mean. This often invokes the Central Limit Theorem, which suggests that the distribution of sample means will be approximately normal, regardless of the shape of the original distribution, provided the sample size is sufficiently large (commonly n > 30 is used, but due to the normality of the original distribution, smaller sample sizes may suffice). The graph illustrates: - A bell-shaped curve representing the normal distribution. - The area under the curve to the right of the specified value depicts the required probabilities. These principles are crucial in statistical analysis for determining probabilities and making inferences about populations from samples. ---
Transcribed Image Text:**Transcription and Explanation for Educational Use** --- **Title: Understanding Normal Distribution in Statistical Analysis** The overhead reach distances of adult females are normally distributed with a mean of 205 cm and a standard deviation of 8.9 cm. ### Problems to Solve **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 203.50 cm. **c.** Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? --- ### Solutions **a.** The probability is **0.8438**. (Note: This probability is rounded to four decimal places as needed.) --- ### Discussion In part (a), the task is to determine the likelihood of a single value exceeding a specified threshold in a normally distributed dataset. Here, standard normal distribution techniques or Z-scores are likely employed. In part (b), although the sample size is 20, the problem refers to using a sample mean. This often invokes the Central Limit Theorem, which suggests that the distribution of sample means will be approximately normal, regardless of the shape of the original distribution, provided the sample size is sufficiently large (commonly n > 30 is used, but due to the normality of the original distribution, smaller sample sizes may suffice). The graph illustrates: - A bell-shaped curve representing the normal distribution. - The area under the curve to the right of the specified value depicts the required probabilities. These principles are crucial in statistical analysis for determining probabilities and making inferences about populations from samples. ---
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Mean = M =  205 , Standard deviation = SD = 8.9

 

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