Word form and expanded form for 3.4
2 answers:
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Compute the gradient of .
Set this equal to the zero vector and solve for the critical points.
The last case has no real solution, so we can ignore it.
Now,
so we have two critical points (0, 0) and (2, 2).
Compute the Hessian matrix (i.e. Jacobian of the gradient).
Check the sign of the determinant of the Hessian at each of the critical points.
which indicates a saddle point at (0, 0);
We also have , which together indicate a local minimum at (2, 2).
Answer:
a) 37.31 b) 42.70 c) 0.57 d) 0.09
Step-by-step explaanation:
We are regarding a normal distribution with a mean of 35 and a standard deviation of 6, i.e., = 35 and
= 6. We know that the probability density function for a normal distribution with a mean of
and a standard deviation of
is given by
in this case we have
Let be the random variable that represents a row score, we find the values we are seeking in the following way
a) we are looking for a number such that
=
, this number is
=37.31
you can find this answer using the R statistical programming languange and the instruction qnorm(0.65, mean = 35, sd = 6)
b) we are looking for a number such that
=
, this number is
=42.70
you can find this answer using the R statistical programming languange and the instruction qnorm(0.9, mean = 35, sd = 6)
c) we find this probability as
=
you can find this answer using the R statistical programming languange and the instruction pnorm(38, mean = 35, sd = 6) -pnorm(28, mean = 35, sd = 6)
d) we find this probability as
=
you can find this answer using the R statistical programming languange and the instruction pnorm(44, mean = 35, sd = 6) -pnorm(41, mean = 35, sd = 6)