Q1. If the random variable x is normally distributed, what percentage of all possible observed values of x will be :
a. within one standard deviation of the mean?
b. within two standard deviation of the mean?
c. within three standard deviation of the mean?
Q.2 The amount of sales tax paid on a purchase is rounded to the nearest cent. Assume that the round-off error is uniformly distributed in the interval -.5 to .5 cents.
a. write the formula for the probability curve describing the round-off error.
b.graph the probability curve describing the round-off error.
c. what is the probability that the round-off error exceeds .3 cents or is less than -.3 cents?
d.what is the probability that the round-off error exceeds .1 cent or is less than -.1 cent?
e. find the mean and standard deviation of the round-off error.
f. find the probability that the round-off error will be within one standard deviation of the mean.
please help me :) these question are important for my quiz on tuesday and the homework is due tomorrow =) !
a. within one standard deviation of the mean?
b. within two standard deviation of the mean?
c. within three standard deviation of the mean?
Q.2 The amount of sales tax paid on a purchase is rounded to the nearest cent. Assume that the round-off error is uniformly distributed in the interval -.5 to .5 cents.
a. write the formula for the probability curve describing the round-off error.
b.graph the probability curve describing the round-off error.
c. what is the probability that the round-off error exceeds .3 cents or is less than -.3 cents?
d.what is the probability that the round-off error exceeds .1 cent or is less than -.1 cent?
e. find the mean and standard deviation of the round-off error.
f. find the probability that the round-off error will be within one standard deviation of the mean.
please help me :) these question are important for my quiz on tuesday and the homework is due tomorrow =) !
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1)
a)68%
b)95%
c)99.7%
2)
a)
f(x) = 1/ (.5-(-.5)) , -.5 < x < .5
f(x) = 1 , -.5 < x < .5
c)
P( x < -0.3) = Integral dx , -.5 < x < -.3
= (-.3)-(-.5) = .2
P( x > .3) = integral dx , .3 < x < .5
= .5-.3 = .2
.2+.2 = .4
d)
P( x < -.1 ) = integral dx , -.5 < x < -.1
= (-.1)-(-.5) = .4
P( x > .1) = integral dx , .3 < x < .5
= .5-.3 = .2
.4+.2 =.6
e)
Mean = integral x dx , -.5 < x < .5
= x^2/2
= (.5)^2 /2 - (-.5)^2 /2 = 0
E(x^2) = integral x^2 dx = x^3/3
= (.5)^3 /3 - (-.5)^3 / 3 = .125/3 + .125/3 = .25/3 = .08333
Var(x) = E(x^2)-E^2(x) = .08333
f)
SD(x) = sqrt[0.08333] =.288675
within 1 sd of the mean = P( -.288675 < x < .288675)
= .288675 - (-.288675) = .57735
a)68%
b)95%
c)99.7%
2)
a)
f(x) = 1/ (.5-(-.5)) , -.5 < x < .5
f(x) = 1 , -.5 < x < .5
c)
P( x < -0.3) = Integral dx , -.5 < x < -.3
= (-.3)-(-.5) = .2
P( x > .3) = integral dx , .3 < x < .5
= .5-.3 = .2
.2+.2 = .4
d)
P( x < -.1 ) = integral dx , -.5 < x < -.1
= (-.1)-(-.5) = .4
P( x > .1) = integral dx , .3 < x < .5
= .5-.3 = .2
.4+.2 =.6
e)
Mean = integral x dx , -.5 < x < .5
= x^2/2
= (.5)^2 /2 - (-.5)^2 /2 = 0
E(x^2) = integral x^2 dx = x^3/3
= (.5)^3 /3 - (-.5)^3 / 3 = .125/3 + .125/3 = .25/3 = .08333
Var(x) = E(x^2)-E^2(x) = .08333
f)
SD(x) = sqrt[0.08333] =.288675
within 1 sd of the mean = P( -.288675 < x < .288675)
= .288675 - (-.288675) = .57735
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Emphirical Rule: In statistics, the 68-95-99.7 rule, or three-sigma rule, or empirical rule, states that for a normal distribution, nearly all values lie within 3 standard deviations of the mean.
About 68.27% of the values lie within 1 standard deviation of the mean. Similarly, about 95.45% of the values lie within 2 standard deviations of the mean. Nearly all (99.73%) of the values lie within 3 standard deviations of the mean.
From the above rule we can answer that
a; 68.27%
b: 95.45%
c: 99.73%
About 68.27% of the values lie within 1 standard deviation of the mean. Similarly, about 95.45% of the values lie within 2 standard deviations of the mean. Nearly all (99.73%) of the values lie within 3 standard deviations of the mean.
From the above rule we can answer that
a; 68.27%
b: 95.45%
c: 99.73%