1. Let "X" denote the score of an exam: "X\\sim N(\\mu,\\sigma^2)"
Then
Given that "\\mu=86, \\sigma=14."
The probability that a random student scored below "72" is "0.1587\\ (15.87\\%)."
2. A confidence interval for a population mean with a known population standard deviation is based on the conclusion of the Central Limit Theorem that the sampling distribution of the sample means follow an approximately normal distribution.
Hence the sample mean will have the distribution "N(\\mu,\\sigma^2\/n)."
The confidence interval we should use, then, is
Given that "\\bar{x}=101.82,n=6,\\sigma=1.3,\\alpha=0.05"
"CI=101.82\\pm 1.96\\cdot{1.3 \\over \\sqrt{6}}"
"CI=(100.78,102.86)"
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