Respuesta :
Answer:
a) [tex]P(X<38)=P(\frac{X-\mu}{\sigma}<\frac{38-\mu}{\sigma})=P(Z<\frac{38-40}{1})=P(z<-2)[/tex]
And we can find this probability with the normal standard table or excel and we got:
[tex]P(z<-2)=0.02275[/tex]
And we would expect about 0.02275*10000 =227.5 rejected and 2.75 in the sample of 100 selected
b) [tex]P(39<X<42)=P(\frac{39-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{42-\mu}{\sigma})=P(\frac{39-40}{1}<Z<\frac{42-40}{1})=P(-1<z<2)[/tex]
And we can find this probability with this difference:
[tex]P(-1<z<2)=P(z<2)-P(z<-1)[/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(-1<z<2)=P(z<2)-P(z<-1)=0.97725-0.159=0.819[/tex]
And we expect about 0.819*10000= 8190 and 81.9 in the sample od 100 selected
c) [tex]40-1.64\frac{1}{\sqrt{100}}=39.836[/tex]
[tex]40+1.64\frac{1}{\sqrt{100}}=40.164[/tex]
So on this case the 90% confidence interval would be given by (39.836;40.164)
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".
Part a
Let X the random variable that represent the weights of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(40,1)[/tex]
Where [tex]\mu=40[/tex] and [tex]\sigma=1[/tex]
We are interested on this probability
[tex]P(X<38)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(X<38)=P(\frac{X-\mu}{\sigma}<\frac{38-\mu}{\sigma})=P(Z<\frac{38-40}{1})=P(z<-2)[/tex]
And we can find this probability with the normal standard table or excel and we got:
[tex]P(z<-2)=0.02275[/tex]
And we would expect about 0.02275*10000 =227.5 rejected and 2.75 in the sample of 100 selected
Part b
[tex]P(39<X<42)=P(\frac{39-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{42-\mu}{\sigma})=P(\frac{39-40}{1}<Z<\frac{42-40}{1})=P(-1<z<2)[/tex]
And we can find this probability with this difference:
[tex]P(-1<z<2)=P(z<2)-P(z<-1)[/tex]
And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.
[tex]P(-1<z<2)=P(z<2)-P(z<-1)=0.97725-0.159=0.819[/tex]
And we expect about 0.819*10000= 8190 and 81.9 in the sample od 100 selected
Part c
The confidence interval for the mean is given by the following formula:
[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (1)
Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.05,0,1)".And we see that [tex]z_{\alpha/2}=1.64[/tex]
Now we have everything in order to replace into formula (1):
[tex]40-1.64\frac{1}{\sqrt{100}}=39.836[/tex]
[tex]40+1.64\frac{1}{\sqrt{100}}=40.164[/tex]
So on this case the 90% confidence interval would be given by (39.836;40.164)
