Respuesta :
Answer:
[tex]x = 0.132 + \mu[/tex]
is the required relation of the diameter of an orange at the 67th percentile compare with the mean diameter.
Step-by-step explanation:
We are given the following information in the question:
Standard Deviation, σ = 0.3 inch
We are given that the distribution of diameters is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
We have to find the value of x such that the probability is 0.67.
[tex]P( X \leq x) = P( z \leq \displaystyle\frac{x - \mu}{0.3})=0.67[/tex]
Calculation the value from standard normal z table, we have,
[tex]P( z \leq 0.440) = 0.67[/tex]
[tex]\displaystyle\frac{x - \mu}{0.3} = 0.440\\x - \mu = 0.132\\ x = 0.132 + \mu[/tex]
which is the required relation of the diameter of an orange at the 67th percentile compare with the mean diameter.
Using the normal distribution, we have that diameters at the 0.67th percentile are 0.44 standard deviations, that is, 0.132 inches above the mean.
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 X 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.
In this problem:
- 67th percentile is X when Z has a p-value of 0.67, so X when Z = 0.44.
- It means that the diameter is 0.44 standard deviations = 0.44x0.3 = 0.132 inches above the mean.
A similar problem is given at https://brainly.com/question/14744708
