Answer:
(a). The mean speed is 760 m/s.
(b). The standard deviation for our normal model is 142.1 m/s.
(c). The unusual speed is 1127 m/s.
(d). The average molecule speed is 929.48 m/s.
Step-by-step explanation:
Given that,
Molecules in the air are moving with speeds based on the normal model.
Let x be the speed of the molecules.
(a). If half of the air molecules are moving faster than 760 m/s.
We can say speed of a molecules is normally distributed.
So, the mean, median and mode of this distribution is at least greater than 760 m/s.
(b). If exactly 95% of the molecules are moving with speeds between 475.8 m/s and 1044.2 m/s,
If the standard deviation is σ and the mean is μ,
We need to calculate the standard deviation for our normal model
Using empirical rule
[tex]\mu-2\sigma=475.8[/tex]...(I)
[tex]\mu+2\sigma=1044.2[/tex]....(II)
Put the value of μ in equation (I)
[tex]760-2\sigma=475.8[/tex]
[tex]-\sigma=\dfrac{475.8-760}{2}[/tex]
[tex]\sigma=142.1\ m/s[/tex]
(c). One molecule is found to be moving with a speed of 1127 m/s.
We know that,
1127 is the between 1126 and 1128.
Since, it is a continuous distribution,
We need to find the probability that speed is between this
Using excel function
[tex]P(1126<x<1128)=P(x<1128)-P(x<1126)[/tex]
[tex]P(1126<x<1128)=0.0002[/tex]
We know that,
Any probability less than 0.05 is considered as unusual
So, 0.0002<0.05
So, 1127 will be considered as the unusual speed.
(d). If the temperature increase caused the z-score of a particle moving at 1100 m/s particle to drop to 1.2 after the temperature increase.
We need to calculate the average molecule speed
Using formula of average molecule speed
[tex]z=\dfrac{x-\mu'}{\sigma}[/tex]
[tex]-\mu'=z\sigma-x[/tex]
Put the value in to the formula
[tex]-\mu'=1.2\times142.1-1100[/tex]
[tex]\mu'=929.48\ m/s[/tex]
Hence, (a). The mean speed is 760 m/s.
(b). The standard deviation for our normal model is 142.1 m/s.
(c). The unusual speed is 1127 m/s.
(d). The average molecule speed is 929.48 m/s.
