Normal Distribution Homework: Expectation and Variance for Sample of 25 Students

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SUMMARY

The discussion focuses on calculating the expectation and variance for a sample of 25 students from a normally distributed SAT score population with a mean of 600 and a standard deviation of 75. The expectation for the sample mean, Y, is equal to the population mean, which is 600. The variance of the sample mean is calculated as the population variance divided by the sample size, resulting in a variance of 225. To find the probability that Y exceeds 610, the z-score is computed, leading to the use of standard normal distribution tables or calculators to determine the probability.

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  • Understanding of normal distribution properties
  • Knowledge of expectation and variance calculations
  • Familiarity with z-scores and standard normal distribution
  • Ability to use statistical tables or calculators
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Homework Statement



Students of a US university have an average SAT score of 600 and a standard deviation of 75. Assume the scores are distributed as a normal distribution.

If X is the score of a randomly selected student, derive the expectation and variance of the mean score, Y, of n randomly selected students.

If the sample is of n=25 students, what is the probability Y exceeds 610?

Homework Equations





The Attempt at a Solution



For a normal distribution, the expectation is the mean, and the variance is the standard deviation squared, so am I correct in saying for n students it would be n times this value?

As for the second part, I'm lost
 
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excuse lack of latex code here.

from my general statistics book:

1. Yes, the mean of x(bar) is the population mean.
2. The standard deviation of the sample is sigma/sqrt(n).

You need to calculate the z score of 610. Then use this with standard tables, or calculator to find P(.67<Z<Infinity).
 

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