Respuesta :
Answer:
1) FALSE
2) FALSE
3) TRUE
4) TRUE
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 1
The mean and standard deviation of a normally distributed random variable which has been standardized are one and zero, respectively.
The mean and the deviation for the normal standard distribution are:
[tex]\mu=0, \sigma =1[/tex]
So then that's FALSE.
Part 2
Using the standard normal curve, the z−score representing the 10th percentile is 1.28.
We are looking for a value a, that satisfy this:
[tex]P(X>a)=0.90[/tex] (a)
[tex]P(X<a)=0.10[/tex] (b)
We can find a z value that satisfy the condition with 0.10 of the area on the left and 0.90 of the area on the right it's z=-1.28. On this case P(Z<-1.28)=0.10 and P(Z>-1.28)=0.90
So that's FALSE.
Part 3
Let z1 be a z−score that is unknown but identifiable by position and area. If the symmetrical area between −z1 and +z1 is 0.9544, the value of z1 is 2.0
If we find this probability:
[tex]P(-2.0< Z<2.0)= P(Z<2.0)-P(Z<-2.0)=0.9773-0.0228=0.9545[/tex]
So on this case we can approximate to 0.9544. And that's TRUE.
Part 4
Using the standard normal curve, the z−score representing the 75th percentile is 0.67.
We are looking for a value a, that satisfy this:
[tex]P(X>a)=0.25[/tex] (a)
[tex]P(X<a)=0.75[/tex] (b)
We can find a z value that satisfy the condition with 0.75 of the area on the left and 0.25 of the area on the right it's z=0.67. On this case P(Z<0.67)=0.75 and P(Z>0.67)=0.25
So that's TRUE.
