Respuesta :
Using the normal distribution and the central limit theorem, it is found that the sample of 36 Great Basin rattlesnakes is more likely to be shorter than 29.65 inches.
Normal Probability Distribution
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For the Sample of 36 Great Basin rattlesnakes:
- The mean is [tex]\mu = 32.3[/tex].
- The standard deviation is [tex]\sigma = 5.3[/tex].
- Sample of 36, hence [tex]n = 36, s = \frac{5.3}{\sqrt{36}} = 0.88[/tex]
- We are interested in the percentile of 29.65 inches, hence [tex]X = 29.65[/tex].
Then:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{29.65 - 32.3}{0.88}[/tex]
[tex]Z = -3.01[/tex]
For the Sample of 100 Southern Pacific rattlesnakes:
- The mean is [tex]\mu = 32.3[/tex].
- The standard deviation is [tex]\sigma = 7.95[/tex].
- Sample of 100, hence [tex]n = 100, s = \frac{7.95}{\sqrt{100}} = 0.795[/tex]
- We are interested in the percentile of 29.65 inches, hence [tex]X = 29.65[/tex].
Then:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{29.65 - 32.2}{0.795}[/tex]
[tex]Z = -3.33[/tex]
Due to the higher z-score, the random sample of 36 Great Basin rattlesnakes is at a higher percentile, hence it is more likely to be shorter than 29.65 inches,
To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213
