Stat1 Resit Exam 2021-06-30 Questions

Statistics 1 for Economics
Resit exam – two hours – 2021-06-30

highlighted gray = different value for different versions
Question 1 (45 points = 20+11+10+4)

In a factory, machine A produces components for smartphones. After inspection, 10% of all produced
components are rejected since their quality is not sufficient.
Machine A is replaced by a new machine B that makes similar components, and the manufacturer
hopes that the percentage of rejected components has now decreased, i.e. is less than 10%. a.
A random sample of 40 components produced by the machine B has been checked, and only one
product is rejected. Test, using the p-value and a significance level of 5%, whether it can be proven
that machine B is better than machine A. (p-value in 3 decimals)
* If for this kind of hypothesis tests we can choose from different test statistics, then show in
1. Conditions and assumptions why you did not choose the other test statistic.
b.
In a new random sample of 160 products from the new machine B, 8 components are rejected
because their quality is insufficient. Find, as accurately as possible, the p-value for the hypothesis
test whether the percentage of defective components for machine B is less than 10%. (you do not
need to do a complete hypothesis test, but the definitions and your calculations of the p-value do need to be
complete)
c.
Management decides to estimate the proportion of rejected products for the new machine B. It is
now assumed that this proportion cannot be higher than 10%. If a 99%-Confidence Interval is
required with a maximum width of 2%, then at least how many components should be checked?
d.
Without numerically calculating it, what is the algebraic definition of Power of the Test and how is it
algebraically related to the probability of a Type II Error ?
Question 2 (27 points = 6+6+6+6+3)

A private investor has a portfolio P consisting of securities of two funds A and B: 70% of the portfolio
is invested in fund A and 30% in fund B. The two funds generate a random return of 𝑅𝐴 and 𝑅𝐵
respectively. Their possible values are given in the joint probabilities table below. The total return of
the portfolio is given by 𝑅𝑝 = 0.70 ∙ 𝑅𝐴 + 0.30 ∙ 𝑅𝐵.
𝑅𝐴
2 4 8
4 0.14 0.30 0.16

𝑅𝐵
6 0.26 0.10 0.04
a. How much is the expected return of the portfolio? (three decimals)

b. What is the coefficient of correlation between the returns? You may use that the variance of
𝑅𝐴 is 4.8 and the variance of 𝑅𝐵 is 0.96. (three decimals)
c. Calculate the risk (standard deviation of return) of portfolio P. (three decimals)
d. If it is known that the return 𝑅𝑅𝐴𝐴 will be more than 2, what is the probability that 𝑅𝑅𝐵𝐵
will be equal to 6? (three decimals)
e. The investor choses to change his portfolio: 50% in fund A and 50% in fund B. What will
happen to the expected return on the total investment? Increase, decrease, or no change?
Explain, mathematically.
Question 3 (18 points = 4+7+7)

For a specific online shop, it is known that the number of incoming orders during the weekend
follows a Poisson distribution with an expectation of 𝜇 = 8 per hour.
a. Find the probability that during a specific hour (in the weekend) more than 4 orders are
placed. (three decimals)
b. Find the probability that the number of orders deviates more than one standard deviation
from the mean number of orders. (three decimals)
c. We consider the 24 numbers of orders during 24 hours on a Saturday as 24 independent
random variables, each Poisson distributed with parameter 𝜇 = 8. Then what is the
probability that in exactly 20 of these 24 hours, the number of orders will be more than 4 ?
(three decimals)
Question 4 (10 points, -2 points per error)

TRUE or FALSE
1. If the sample size is large enough (at least 30), then we are allowed to use test statistic 𝜒2 in
a hypothesis test regarding 𝜎2.
When the data is not quantitative but ordinal, then we can still perform a sign test.
2. If we want to construct a new confidence interval with half the width of the confidence
interval that we already found, then we need a sample size that is twice as large
If we want to construct a new confidence interval with a third of the width of the confidence
interval that we already found, then we need a sample size that is three as large.
3. The random variables 𝑋 and 𝑌 in the next table are independent:
1 4 5
1 0.01 0.04 0.05
𝑌 4 0.04 0.16 0.20
6 0.05 0.20 0.25
4. The uniform distribution between boundaries a and 𝑏 is noted by 𝑈(a,𝑏).
In a boxplot of a uniformly distributed data set, the length of the whiskers are 1.5× the Inter
Quartile Range.
In a boxplot of a uniformly distributed data set, the length of each whisker is at least 1.0×
the Inter Quartile Range.
In a boxplot of a uniformly distributed data set, the total length of both whiskers is 1.0× the
Inter Quartile Range.
5. The Power of the Test is independent of the probability of an Error of Type I
The Power of the Test is not independent of the probability of an Error of Type I.
6. Sign tests can only be one-sided, because, in essence, the sign “plus” is located to only
one side of the sign “minus”.
* *
*

Stat1 Resit Exam 2021-06-30 Questions

Uploaded by

Copyright:

Available Formats

Stat1 Resit Exam 2021-06-30 Questions

Uploaded by

Document Information

Original Description:

Original Title

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Stat1 Resit Exam 2021-06-30 Questions

Uploaded by

Copyright:

Available Formats

Statistics 1 for Economics

Resit exam – two hours – 2021-06-30

Question 1 (45 points = 20+11+10+4)

Question 2 (27 points = 6+6+6+6+3)

4 0.14 0.30 0.16

a. How much is the expected return of the portfolio? (three decimals)

Question 3 (18 points = 4+7+7)

Question 4 (10 points, -2 points per error)

3. The random variables 𝑋 and 𝑌 in the next table are independent:

1 0.01 0.04 0.05

𝑌 4 0.04 0.16 0.20

6 0.05 0.20 0.25

4. The uniform distribution between boundaries a and 𝑏 is noted by 𝑈(a,𝑏).

5. The Power of the Test is independent of the probability of an Error of Type I

You might also like