(Solved): (8 points) The National Assessment of Educational Progress (NAEP) includes a ...

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(8 points) The National Assessment of Educational Progress (NAEP) includes a mathematics test for eighth-grade students. The mathematics test scores follow Normal distribution with (unknown) mean \( \mu \) and standard deviation \( \sigma=110 \). a) [2] The sampling distribution of \( \bar{x} \) with a simple random sample of size 25 from the population is normal, what are its mean and standard deviation? Hint: one of these answers involves the unknown mean \( \mu \). b) [4] Suppose the sample mean \( \bar{x}=285 \) with \( \mathrm{n}=25 \). Construct a \( 90 \% \) confidence interval for \( \mu \). c) [1] What is the margin of error for the \( 90 \% \) confidence interval in b)? d) [1] Suppose we took another simple random sample with \( \mathrm{n}=20 \) and its \( \bar{x}=285 \). Compared with the sample in b) with \( \mathrm{n}=25 \), the margin of error for a \( 90 \% \) confidence interval for \( \mu \) is i) Smaller ii) larger iii) either smaller or larger, but we can't say which.