Roger has read a report that the weights of adult male Siberian tigers have a distribution which is?
approximately normal with mean = 390lb and std dev = 65 lb.
PLEASE SHOW WORK GETTING READY FOR AN EXAM
a). Find the probability that an individual male Siberian tiger will weigh more than 450lb.
b). Find the probability that a random sample of 4 male Siberian tigers will have a sample mean weight more than 450lb.
(a)
P(X > 450) = 1 - P(X < 450) = 1 - P((X-390)/65 < (450-390)/65) = 1 - P(Z < 0.9231)
(That is, convert to standard normal: Z = (X-mu)/sigma )
From standard normal tables (using interpolation),
P(Z < 0.9231) = 0.8220. So
P(X > 450) = 1 - 0.8220 = 0.1780
(b)
The sample mean is approximately normally distributed with mean 390 and standard deviation 65/sqrt(4)
Otherwise, the calculation proceeds exactly as for part (a)
You should get 1 - 0.9675 = 0.0325
For any normal random variable X with mean ? and standard deviation ? , X ~ Normal( ? , ? ), (note that in most textbooks and literature the notation is with the variance, i.e., X ~ Normal( ? , ?² ). Most software denotes the normal with just the standard deviation.)
You can translate into standard normal units by:
Z = ( X - ? ) / ?
Where Z ~ Normal( ? = 0, ? = 1). You can then use the standard normal cdf tables to get probabilities.
If you are looking at the mean of a sample, then remember that for any sample with a large enough sample size the mean will be normally distributed. This is called the Central Limit Theorem.
If a sample of size is is drawn from a population with mean ? and standard deviation ? then the sample average xBar is normally distributed
with mean ? and standard deviation ? /?(n)
An applet for finding the values
calculator
how to read the tables
In this question we have
X ~ Normal( ?x = 390 , ?x² = 4225 )
X ~ Normal( ?x = 390 , ?x = 65 )
Find P( X > 450 )
P( ( X - ? ) / ? > ( 450 - 390 ) / 65 )
= P( Z > 0.923077 )
= P( Z < -0.923077 )
= 0.1779836
In this question we have
Xbar ~ Normal( ? = 390 , ?² = (65^2) / 4 )
Xbar ~ Normal( ? = 390 , ?² = 1056.25 )
Xbar ~ Normal( ? = 390 , ? = 32.5 )
Find P( Xbar > 450 )
P( ( Xbar - ? ) / ? > ( 450 - 390 ) / 32.5 )
= P( Z > 1.846154 )
= P( Z < -1.846154 )
= 0.03243494