Respuesta :
Answer:
a) [tex]P(71<x<73)=P(\frac{71-72}{6}<Z<\frac{73-72}{6})=P(-0.17<z<0.17)=P(Z<0.17)-P(Z<-0.17)=0.5675-0.4325=0.1350[/tex]
b) [tex]P(71<\bar x<73)=P(\frac{71-72}{\frac{6}{\sqrt{27}}}<Z<\frac{73-72}{\frac{6}{\sqrt{27}}})=P(-0.867<z<0.867)=P(Z<0.867)-P(Z<-0.867)=0.8068-0.1930=0.6141[/tex]
c) Is the probability in part (b) much higher? YES very high compared with part a
The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.
iii. The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.
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 that represent variable in the population, and for this case we know the distribution for X is given by:
[tex]X \sim N(72,6)[/tex]
Where [tex]\mu=72[/tex] and [tex]\sigma=6[/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]
On this case [tex]\bar X \sim N(72,\frac{6}{\sqrt{27}})[/tex]
Part a
(a) What is the probability that an 18-year-old man selected at random is between 71 and 73 inches tall? (Round your answer to four decimal places.)
[tex]P(71<x<73)=P(\frac{71-72}{6}<Z<\frac{73-72}{6})=P(-0.17<z<0.17)=P(Z<0.17)-P(Z<-0.17)=0.5675-0.4325=0.1350[/tex]
(b) If a random sample of twenty-seven 18-year-old men is selected, what is the probability that the mean height x is between 71 and 73 inches? (Round your answer to four decimal places.)
[tex]P(71<\bar x<73)=P(\frac{71-72}{\frac{6}{\sqrt{27}}}<Z<\frac{73-72}{\frac{6}{\sqrt{27}}})=P(-0.867<z<0.867)=P(Z<0.867)-P(Z<-0.867)=0.8070-0.1930=0.6141[/tex]
c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher?
Is the probability in part (b) much higher? YES very high compared with part a
The probability in part (b) is much higher because the standard deviation is smaller for the x distribution.
iii. The probability in part (b) is much lower because the standard deviation is smaller for the x distribution.
