The college board reported the following mean scores for the three parts of the scholastic aptitude test (sat) (the world almanac, 2009): critical reading 502 mathematics 515 writing 494assume that the population standard deviation on each part of the test is = 100.a. what is the probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 502 on the critical reading part of the test (to 4 decimals)? b. what is the probability a sample of 90 test takers will provide a sample mean test score within 10 points of the population mean of 515 on the mathematics part of the test (to 4 decimals)? c. what is the probability a sample of 100 test takers will provide a sample mean test score within 10 of the population mean of 494 on the writing part of the test (to 4 decimals)?

a)

And we can use excel or the normal standard table to find this probability:

b)

And we can use excel or the normal standard table to find this probability:

c)

And we can use excel or the normal standard table to find this probability:

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 a

Let X the random variable that represent the scores for critical reading of a population, and for this case we know the distribution for X is given by:

Where and

We select a sample of size n=90, since the distribution for X is normal then the distribution for the sample size is also normal

And for this case we want this probability:

And for this case we can use the z score given by:

And if we use this formula we got:

And we can use excel or the normal standard table to find this probability:

Part b

Let X the random variable that represent the scores for Math of a population, and for this case we know the distribution for X is given by:

Where and

We select a sample of size n=90, since the distribution for X is normal then the distribution for the sample size is also normal

And for this case we want this probability:

And for this case we can use the z score given by:

And if we use this formula we got:

And we can use excel or the normal standard table to find this probability:

Part c

Let X the random variable that represent the scores for Writing of a population, and for this case we know the distribution for X is given by:

Where and

We select a sample of size n=100, since the distribution for X is normal then the distribution for the sample size is also normal

And for this case we want this probability:

And for this case we can use the z score given by:

And if we use this formula we got:

And we can use excel or the normal standard table to find this probability:

answer; test-retest;