The population of SAT scores forms a normal distribution with a mean of m=500 an

Question

The population of SAT scores forms a normal distribution with a mean of m=500 and a standard deviation of s=100. If the average SAT score is calculated for a sample of n=25 students,
a. What is the probability that the sample mean will be greater than M=510? In symbols, what is p( M> 510)?

b. What is the probability that the sample mean will be greater than M=520? In symbols, what is p( M> 520)?

c. What is the probability that the sample mean will be between M=510 and M=520? In symbols, what is p( 510<520)?

Explanation / Answer

a. What is the probability that the sample mean will be greater than M=510? In symbols, what is p( M> 510)?

P(M>510) = P((M-500)/(100/25) > (510-500)/(100/5))

=P(Z>0.5)

= 0.3085 (check standard normal table)

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b. What is the probability that the sample mean will be greater than M=520? In symbols, what is p( M> 520)?

p( M> 520) = P((M-500)/(100/25) > (520-500)/(100/5))

=P(Z>1)

= 0.1587 (check standard normal table)

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c. What is the probability that the sample mean will be between M=510 and M=520? In symbols, what is p( 510<520)?

p( 510<M<520) = P((510-500)/(100/5) <M< (520-500)/(100/5))

=P(0.5<Z<1)

= 0.1498(check standard normal table)

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