Central Limit Theorem - Z-Score Problems

[Herondaleheir]

Graduate students applying for entrance to many universities must take a Miller Analogies Test. It is known that the test scores have a mean of 75 and a variance of 16. In 1990, 50 students applied for entrance into graduate school in physics.

i) Find the probability that the average score of this group of students is higher than the average (i.e., 75).
ii) Find the probability that the sample mean deviates from the population mean by more than 1.5.

So for i) I'm a little confused because I assumed you would have to use the z score and subtract the percentile from 1. But since x would be 75, it would be z = (75 - 75)/0.57 and that gives 1- 0 = 1? That doesn't seem like the right probability.

For the standard deviation, I'm using the central limit theorem. So I did s = 4/(sqrt50) = 0.57

And then for ii) I'm just really confused and would appreciate some hints or help.

 

[Dr.Peterson]

Graduate students applying for entrance to many universities must take a Miller Analogies Test. It is known that the test scores have a mean of 75 and a variance of 16. In 1990, 50 students applied for entrance into graduate school in physics.

i) Find the probability that the average score of this group of students is higher than the average (i.e., 75).
ii) Find the probability that the sample mean deviates from the population mean by more than 1.5.

So for i) I'm a little confused because I assumed you would have to use the z score and subtract the percentile from 1. But since x would be 75, it would be z = (75 - 75)/0.57 and that gives 1- 0 = 1? That doesn't seem like the right probability.

For the standard deviation, I'm using the central limit theorem. So I did s = 4/(sqrt50) = 0.57

And then for ii) I'm just really confused and would appreciate some hints or help.

There is no percentile in this problem! You found that the z-score for the mean is 0 (which is always true, by the way); that 0 is not a percentile, or a probability. The z-score is just the number of standard deviations from the mean. How do you find a probability from a z-score?

I think you just missed a step, but it's a key step.

For (ii), they are asking for the probability that [MATH]|\overline{x} - 75| >1.5[/MATH]. That is, either [MATH]\overline{x} >75 + 1.5[/MATH] or [MATH]\overline{x} <75 - 1.5[/MATH].