Try it on paper first. A hint, the key idea and the worked solution are here when you need them.
A spinner shows , or , each with the same chance. A player spins it twice, and is the mean of the two results. List the equally likely ordered results. Find the population variance of a single spin, dividing by . Find the variance of the sample means, dividing by . Find how many times larger the first variance is.
The sampling distribution of is centred at . Its standard error is
For larger samples it is close to a normal curve, so about of sample means lie within standard errors of .
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The population. The mean is . The squared distances are , and , so the variance is .
Step 2. The sample means. Their mean is . Their squared distances from add up to , so their variance is .
Step 3. Compare. .
Answer: , and .
What it means. The variance of a mean of independent spins is the population variance divided by . That is the rule , squared.
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