Respuesta :
Answer:
68% of the lengths of pregnancies fall between 250 days and 282 days.
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
In this problem, we have that:
Mean = 266
Standard deviation = 16.
What is the proportion of the lengths of pregnancies that fall between 250 days and 282 days?
250 = 266 - 16
So 250 is one standard deviation below the mean.
282 = 266 + 16
So 282 is one standard deviation above the mean.
By the Empirical Rule, 68% of the lengths of pregnancies fall between 250 days and 282 days.
Answer:
[tex]P(250<X<282)=P(\frac{250-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{282-\mu}{\sigma})=P(\frac{250-266}{16}<Z<\frac{282-266}{16})=P(-1<z<1)[/tex]
And we can find this probability with this difference and using the normal standard table or excel:
[tex]P(-1<z<1)=P(z<1)-P(z<-1)= 0.8413-0.1587= 0.6826[/tex]
So then we will have approximatetly 68.26% of the values between 250 and 282 days
Step-by-step explanation:
Let X the random variable that represent the The length of human pregnancies from conception to birth, and for this case we know the distribution for X is given by:
[tex]X \sim N(266,16)[/tex]
Where [tex]\mu=266[/tex] and [tex]\sigma=16[/tex]
We are interested on this probability
[tex]P(250<X<282)[/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(250<X<282)=P(\frac{250-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{282-\mu}{\sigma})=P(\frac{250-266}{16}<Z<\frac{282-266}{16})=P(-1<z<1)[/tex]
And we can find this probability with this difference and using the normal standard table or excel:
[tex]P(-1<z<1)=P(z<1)-P(z<-1)= 0.8413-0.1587= 0.6826[/tex]
So then we will have approximatetly 68.26% of the values between 250 and 282 days
