2022 Spring. Solutions of M262 Final Exam problems.
1. Aslı constructed a machine producing plastic balls with average weight µ
and standard deviation σ. Knowing these two parameters, she states that the
probability that the total weight of 81 balls produced by this machine is less
than 180 is equal to 0.44 and the probability that the total weight of 81 balls
produced by this machine is greater than 90 is equal to 0.33. Based on this
information find µ and σ.
Solution 1:
180
P (x1 + · · · x81 < 180) = P (x̄ <
)
81
x̄ − µ 2.22 − µ 2.22 − µ
= P( √ < √ ) = P (Z < √ ) = 0.44
σ/ 81 σ/ 81 σ/ 81
2.22 − µ
Therefore from z− table we get √ = −0.15.
σ/ 81
90
P (x1 + · · · x81 > 90) = P (x̄ >
)
81
x̄ − µ 1.11 − µ 1.11 − µ
= P( √ > √ ) = P (Z > √ ) = 0.33
σ/ 81 σ/ 81 σ/ 81
1.11 − µ
Therefore from z− table we get √ = 0.44.
σ/ 81
Solving two equation with two unknowns we find µ = 1.89 and σ = −16.03.
Note: Since σ should be positive, Aslı’s statement is not correct and one can
not find true µ and σ.
Solution 2: P ( 81 81
i=1 xi > 90) + P ( i=1 xi < 180) should be greater than 1.
P P
Thus, without any calculation one can say that Aslı’s statement is not correct
and one can not find true µ and σ.
2. The following sample of 8 measurements was selected from a population
that is approximately normally distributed:
7, 5, 3, 8, 2, 11, 5, 7
a) Construct a 95% confidence interval for the population standard deviation.
b) Test to see whether there is sufficient evidence to indicate that the
population standard deviation is less than 4? Use α = 0.1.
1 09-25-2025 13:06:16 GMT -05:00
This study source was downloaded by 100000899606396 from CourseHero.com on
https://www.coursehero.com/file/249824300/22262finalsolpdf/
1. Aslı constructed a machine producing plastic balls with average weight µ
and standard deviation σ. Knowing these two parameters, she states that the
probability that the total weight of 81 balls produced by this machine is less
than 180 is equal to 0.44 and the probability that the total weight of 81 balls
produced by this machine is greater than 90 is equal to 0.33. Based on this
information find µ and σ.
Solution 1:
180
P (x1 + · · · x81 < 180) = P (x̄ <
)
81
x̄ − µ 2.22 − µ 2.22 − µ
= P( √ < √ ) = P (Z < √ ) = 0.44
σ/ 81 σ/ 81 σ/ 81
2.22 − µ
Therefore from z− table we get √ = −0.15.
σ/ 81
90
P (x1 + · · · x81 > 90) = P (x̄ >
)
81
x̄ − µ 1.11 − µ 1.11 − µ
= P( √ > √ ) = P (Z > √ ) = 0.33
σ/ 81 σ/ 81 σ/ 81
1.11 − µ
Therefore from z− table we get √ = 0.44.
σ/ 81
Solving two equation with two unknowns we find µ = 1.89 and σ = −16.03.
Note: Since σ should be positive, Aslı’s statement is not correct and one can
not find true µ and σ.
Solution 2: P ( 81 81
i=1 xi > 90) + P ( i=1 xi < 180) should be greater than 1.
P P
Thus, without any calculation one can say that Aslı’s statement is not correct
and one can not find true µ and σ.
2. The following sample of 8 measurements was selected from a population
that is approximately normally distributed:
7, 5, 3, 8, 2, 11, 5, 7
a) Construct a 95% confidence interval for the population standard deviation.
b) Test to see whether there is sufficient evidence to indicate that the
population standard deviation is less than 4? Use α = 0.1.
1 09-25-2025 13:06:16 GMT -05:00
This study source was downloaded by 100000899606396 from CourseHero.com on
https://www.coursehero.com/file/249824300/22262finalsolpdf/
