Data is collected from a large city. The growth of adolescents over the course of puberty is normally distributed with a mean of 5.26 and a standard deviation of 0.50 inches.
a) what percentage of adolescents in this city grew less than 4.5 inches?
b) what percentage grew more than 5.12?
c) a random sample of 100 adolescents is gathered and the mean growth during puberty was 5.12 If another sample of 100 is taken what is the probability that the sample mean will be greater than5.12 inches?
d) why is the z score used in answering a,b, and c?
e. why is the formula for z used in c different from that used in a and b?
I think you can calculate the Z-scores for yourself.
The Z-score assumes a normal distribution, taking into account the mean and standard deviation. From calculus, we know the proportions/percentages in the normal distribution. (See a table in the back of your statistics text.)
A and B involve a distribution of raw scores, while C involves a distribution of means. With the first two, the divisor is the standard deviation. With the third, the divisor is the standard error. For larger samples, the standard error is the standard deviation divided by the square root of N.
I hope this helps. Thanks for asking.