a.
"95\\%CI=(\\bar{x}-1.96\\frac{\\sigma}{\\sqrt{n}},\\;\\bar{x}+1.96\\frac{\\sigma}{\\sqrt{n}})=\\\\=(200-1.96\\frac{50}{\\sqrt{25}},\\;200+1.96\\frac{50}{\\sqrt{25}})=(180.4,\\;219.6)."
b.
"95\\%CI=(\\bar{x}-1.96\\frac{\\sigma}{\\sqrt{n}},\\;\\bar{x}+1.96\\frac{\\sigma}{\\sqrt{n}})=\\\\=(200-1.96\\frac{25}{\\sqrt{25}},\\;200+1.96\\frac{25}{\\sqrt{25}})=(190.2,\\;209.8)."
c.
"95\\%CI=(\\bar{x}-1.96\\frac{\\sigma}{\\sqrt{n}},\\;\\bar{x}+1.96\\frac{\\sigma}{\\sqrt{n}})=\\\\=(200-1.96\\frac{10}{\\sqrt{25}},\\;200+1.96\\frac{10}{\\sqrt{25}})=(196.08,\\;203.92)."
d. When the standard deviation decreases, the confidence interval becomes narrower.
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