Question

[Red Cross Red Shield] The distribution of the amount paid in health claims by Red Cross Red Shield (RCRS), a medical insurance company, on individual health insurance plans for college students has mean $500, median $400, and standard deviation $550.

Consider the variation of the amount paid of a sample of 40 plan members. Do you think that the standard deviation of the sample mean of participants (Stdev[ ]) is smaller than, larger than, or equal to the standard deviation of individual annual amount paid? Why?

	a.	Stdev[  ]  should be larger than the standard deviation of individual annual amount paid, since the sample has multiple individuals.
	b.	Stdev[  ] should be smaller than the standard deviation of individual annual amount paid: We take a sample of people (as opposed to picking one person) and compute the average. The variation of the average is much smaller than the variation of a single value because individual variations will be evened out when we compute the average.
	c.	Not enough information is given to answer this question
	d.	Stdev[  ] should be the same as the standard deviation of individual annual amount paid.

Answer 1

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Answer

Stdev[ ] should be smaller than the standard deviation of individual annual amount paid: We take a sample of people (as opposed to picking one person) and compute the average. The variation of the average is much smaller than the variation of a single value because individual variations will be evened out when we compute the average.

Reason : We can utilize Central Limit Theorem to figure out it . If X follows N (mu, sigma) then sample mean x bar follows N (mu , sigma/✓n).

Moreover if we take a random sample then always the following hold:

E(xbar) = E(x) = mu

S.D.(xbar) = S.D(x)/✓n =sigma/✓n