Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The National Assessment of Educational Progress periodically administers tests on different subjects to high school students. In 2000, the grade 12 students in the sample averaged 301 on the mathematics test; the SD was 30. Can you say what the likely size of the chance error in the 301 is if a simple random sample of 1000 students was tested?

share|cite|improve this question

Let $X_1, X_2,\dots,X_1000$ be the achievement results in the sample, and let $$Y=\frac{X_1+X_2+\cdots+X_{1000}}{1000}.$$ Then the expectation of $Y$ is $301$. The variance of $Y$ is $\frac{1}{1000}$ times the variance of $X$. So the standard deviation of $Y$ is $\dfrac{30}{\sqrt{1000}}$, approximately $0.9847$.

Moreover, because $Y$ is an average of identically distributed essentially independent random variable, we can assume that $Y$ has a nearly normal distribution. (We are presumably not sampling without replacement, but the student population is probably so large in comparison with the sample size that the variance is about the same as for sampling with replacement.)

It seems that you are being asked about how well the sample mean approximates the population mean $301$. More precisely, or imprecisely, you are asked for the "likely" size of the chance error.

Depends what one means by likely. For example, with probability about $0.9$, a normal is within $1.645$ standard deviation units of the true mean. In our case, $1.645$ standard deviation units is about $1.56$, so with probability about $90\%$ the sample mean will be within distance $1.56$ of the population mean of $301$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.