a) The SAT population distribution has a mean of μ = 1060, a standard deviation of σ = 195, and is normally distributed. Use the Z-table to find the probability of a randomly chosen student scoring lower than 1000 on the exam. (This doesn’t use the sampling distribution, it’s a regular Z-table problem.)
a) The SAT population distribution has a mean of μ = 1060, a standard deviation
of σ = 195, and is
randomly chosen student scoring lower than 1000 on the exam. (This doesn’t use the
sampling distribution, it’s a regular Z-table problem.)
b) Suppose we take a sample of 50 SAT takers. That’s a large sample. According
to the Central Limit Theorem, what is the mean and standard error of the sampling
distribution with n = 50?
c) We have a large sample (over 30), so the Central Limit Theorem guarantees that
our sampling distribution is normal. Use the Z-table to calculate the probability that
your sample mean will be below 1000. It will help to sketch a picture of the sampling
distribution.
d) Unlike SAT scores, the population distribution of household incomes in the US is
not normally distributed. It is skewed right.
• If I know the mean and standard deviation of household income (i.e. μ and σ), can
I use the Z-table to estimate the probability of getting an individual household
with an income below $60,000 (i.e. could I perform a similar calculation as you
did in part a)? Briefly say why or why not. Note that I don’t want you to actually
calculate anything.
• Can I use the Z-table to estimate the probability of getting a sample mean below
$60, 000 when n = 50 (i.e. could I perform a similar calculation as you did in
part c)? Briefly explain why or why not. Note that I don’t want you to actually
calculate anything.
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