Respuesta :
Answer:
a) The problem says that this represent the values within 3 deviations from the mean and using the empirical rule we know that on this case we have 68% of the data on this interval.
b) For this case we can use the z score formula again:
[tex] z_1= \frac{98.73-98.38}{0.48}=0.729[/tex]
[tex] z_2= \frac{97.89-98.38}{0.48}=-1.020[/tex]
For this case we want this probability:
[tex] P(97.89<X<98.73) =P(-1.02<Z<0.729)= P(Z<0.729)-P(Z<-1.02)=0.767-0.154= 0.613[/tex]
So the approximate percentage of temperatures between 97.89F and 98.73F is 61.3%
Step-by-step explanation:
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).
For this case we know that the body temperatures for a group of heatlhy adults represented with the random variable X follows this distribution:
[tex] X \sim N(\mu =98.38F, \sigma=0.48 F)[/tex]
Part a
For this case we can use the z score formula to measure how many deviations we are within the mean, given by:
[tex] z=\frac{x-\mu}{\sigma}[/tex]
If we find the z score for the values given we got:
[tex] z_1= \frac{99.57-98.38}{0.48}=2.479[/tex]
[tex] z_2= \frac{97.05-98.38}{0.48}=-2.771[/tex]
The problem says that this represent the values within 3 deviations from the mean and using the empirical rule we know that on this case we have 68% of the data on this interval.
Part b
For this case we can use the z score formula again:
[tex] z_1= \frac{98.73-98.38}{0.48}=0.729[/tex]
[tex] z_2= \frac{97.89-98.38}{0.48}=-1.020[/tex]
For this case we want this probability:
[tex] P(97.89<X<98.73) =P(-1.02<Z<0.729)= P(Z<0.729)-P(Z<-1.02)=0.767-0.154= 0.613[/tex]
So the approximate percentage of temperatures between 97.89F and 98.73F is 61.3%
