national park statistics
Can someone please figure these out and explain how you got the answer?
This is a study guide for statistics and I dont know how to figure these out. Could someone work it out and then explain how you got the answer?
1.) The mean value of land and buildings per acre from a sample of farms is $2000 with a standard deviation of $500. The data set has a bell shaped distribution. Estimate the percent of farms whose land and building values per acre are between $1000 and $3000.
2.) Old Faithful is a geyser at Yellowstone National Park. The mean duration of eruptions is 8 minutes with a standard deviation of 2 minutes. Using Chebyshev’s Thereom, at least what percent of eruptions will last between 2 and 14 minutes?
a) Bell curve implies normal distribution
x-bar = $2000
std. dev. = $500
$1000 is 2 std. dev. to the left of mean and $3000 is 2 std. dev. to the right of mean. The percent of farms whose land and building values per acre are between is the area of normal distribution between -2 std. dev. and 2 std. dev.
= P( -2 <= Z <= 2) = 95.45% where Z ~ N(0,1)
2.
x-bar = 8 min
std. dev. = 2 min
2min is 3 std. dev. to the left of mean and 14min is 3 std. dev. to the right of mean.
Chebyshev's Inequality says that
P ( | X - mean | >= 3*std. dev.) <= 1/3^2
<= 1/9
<= 11.11%
or -P ( | X - mean | >= 3*std. dev.) >= -11.11%
or 1-P ( | X – mean | >= 3*std. dev.) >= 88.89%
or P ( |X – mean | <= 3*std. dev.) >= 88.89%
Or the there is at least 88.89% of eruptions will last between 2 and 14 minutes