Respuesta :
Answer:
[tex]P(28.6<x<43.0)=P(\frac{28.6-35.8}{2.4}<Z<\frac{43-35.8}{2.4})=P(-3<z<3)=P(Z<3)-P(Z<-3)=0.99865-0.00135=0.9973[/tex]
And on this case since we are within 3 deviations from the mean the result obtained using the z score agrees with the empirical rule.
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Let X the random variable who represent the parts from a supplier of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(35.8,2.4)[/tex]
Where [tex]\mu=35.8[/tex] and [tex]\sigma=2.4[/tex]
And let [tex]\bar X[/tex] represent the sample mean, the distribution for the sample mean is given by:
[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]
Find the probability that a randomly selected part from this supplier will have a value between 28.6 and 43.0 inches
[tex]P(28.6<x<43.0)=P(\frac{28.6-35.8}{2.4}<Z<\frac{43-35.8}{2.4})=P(-3<z<3)=P(Z<3)-P(Z<-3)=0.99865-0.00135=0.9973[/tex]
And that correspond with the 99.73% of the data.
The empirical rule, also referred to as "the three-sigma rule or 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, almost all data falls within three standard deviations (denoted by σ) of the mean (denoted by µ)". The empirical rule shows that 68% falls within the first standard deviation (µ ± σ), 95% within the first two standard deviations (µ ± 2σ), and 99.7% within the first three standard deviations (µ ± 3σ).
And on this case since we are within 3 deviations the result obtained using the z score agrees with the empirical rule.
