{
  "problem": "You have found the following ages (in years) of all 6 bears at your local zoo: $ 31,\\enspace 16,\\enspace 19,\\enspace 13,\\enspace 38,\\enspace 18$ What is the average age of the bears at your zoo? What is the variance? You may round your answers to the nearest tenth.",
  "hints": [
    "Because we have data for all 6 bears at the zoo, we are able to calculate the",
    "population mean",
    "$({\\mu})$",
    "and",
    "population variance",
    "$({\\sigma^2})$",
    "To find the",
    "population mean",
    ", add up the values of all",
    "$6$",
    "ages and divide by",
    "$6$",
    "$ {\\mu} = \\dfrac{\\sum\\limits_{i=1}^{{N}} x_i}{{N}} = \\dfrac{\\sum\\limits_{i=1}^{{6}} x_i}{{6}} $",
    "$ {\\mu} = \\dfrac{31 + 16 + 19 + 13 + 38 + 18}{{6}} = {22.5\\text{ years old}} $",
    "Find the",
    "squared deviations from the mean",
    "for each bear.",
    "Age",
    "$x_i$",
    "Distance from the mean",
    "$(x_i - {\\mu})$",
    "$(x_i - {\\mu})^2$",
    "$31$",
    "years",
    "$8.5$",
    "years",
    "$72.25$",
    "years",
    "$^2$",
    "$16$",
    "years",
    "$-6.5$",
    "years",
    "$42.25$",
    "years",
    "$^2$",
    "$19$",
    "years",
    "$-3.5$",
    "years",
    "$12.25$",
    "years",
    "$^2$",
    "$13$",
    "years",
    "$-9.5$",
    "years",
    "$90.25$",
    "years",
    "$^2$",
    "$38$",
    "years",
    "$15.5$",
    "years",
    "$240.25$",
    "years",
    "$^2$",
    "$18$",
    "years",
    "$-4.5$",
    "years",
    "$20.25$",
    "years",
    "$^2$",
    "Because we used the",
    "population mean",
    "$({\\mu})$",
    "to compute the",
    "squared deviations from the mean",
    ", we can find the",
    "variance",
    "$({\\sigma^2})$",
    ", without introducing any bias, by simply averaging the",
    "squared deviations from the mean",
    "$ {\\sigma^2} = \\dfrac{\\sum\\limits_{i=1}^{{N}} (x_i - {\\mu})^2}{{N}} $",
    "$ {\\sigma^2} = \\dfrac{{72.25} + {42.25} + {12.25} + {90.25} + {240.25} + {20.25}} {{6}} $",
    "$ {\\sigma^2} = \\dfrac{{477.5}}{{6}} = {79.58\\text{ years}^2} $",
    "The average bear at the zoo is 22.5 years old. The population variance is 79.58 years $^2$."
  ]
}
{'source': 'AMPS', 'file': 'amps/khan/519/1608126843.json'}