Location: A. Jacobshal 03
Question 1 2 3 4 5 6 7 8 TOT
Max points 12 5 6 14 10 8 6 10 71
Scored points
1. (12 pts) Consider the variable ?, which measures the balance in Euro of the current acc
...
Location: A. Jacobshal 03
Question 1 2 3 4 5 6 7 8 TOT
Max points 12 5 6 14 10 8 6 10 71
Scored points
1. (12 pts) Consider the variable ?, which measures the balance in Euro of the current account of a randomly selected customer at a major Dutch bank. The balance of this bank account can be either positive or negative, since customers are allowed to be in debt to the bank.
a) What are the cases in this example?
The cases are the units that we investigate, which are customers at the bank. (2pt)
b) Is ? categorical or quantitative? Explain.
X is quantitative, since it takes numerical values. It makes sense to do arithmetic with amounts that are measured in euros. (2pt, both given for correct answer PLUS explanation)
Individual 1 2 3 4 5 6 7
x (balance in Euro) -348 -20 510 201 -9 43 …
Table 1: simple random sample of current account balances
c) Table 1 shows a simple random sample of size 7 for the bank account balance ?. However, the seventh observation has been deleted. Calculate the missing value of ? if it is given that
?̅ = 68.143.
?̅ = 1 �∑6 ?? + ?7�; ?7 = 7 �?̅ − 1 ∑6
??� = 100.001 (2pt)
7 ?=1 7 ?=1
d) Look again at the incomplete sample in Table 1, but now assume that we have no additional information (we do not know ?̅). Without any assumptions on the missing value of ?, can you give a value ?? for which you are certain that ?̅ ≤ ??? That is: can you give an upper bound on the sample average? If possible: give the bound and explain. If not possible: explain why not.
No, you cannot. The sample average can be arbitrarily large if the missing value of ? is arbitrarily large. Extreme observations pull the sample average towards them. Alternatively, students can explain this by means of examples. (3pt, points given for correct intuition)
e) For the incomplete sample in Table 1: can you give an upper bound on the sample median of ? without making any assumption regarding the missing value? If possible: give the bound and explain. If not possible: explain why not.
The sample median is the middle observation if we rank them from low to high values. In this case that is the fourth observation. If the missing value is very small, so that the missing account runs a large debt, the fourth value would be -9. On the other hand, if the missing account has a large and positive balance, the fourth value would be 43, regardless of the exact value of the missing observation. Hence, the upper bound on the median can be given
~ 2~
and it is equal to 43 euro. Students may also give examples as an explanation. (3pt, points given for correct intuition)
2. (5 pts)
a) Sketch the density curves for the following two normal distributions in a single figure:
?(2,3) and ?(10,4). Pay attention to the location and spread of each curve and show how the parameters of the distributions relate to the curves in your figure (draw lines to visualize the parameters of each distribution).
The general shape should be clear: distribution 1 is less dispersed and to the left of distribution 2. Students should include vertical lines indicating the peaks of the normal curves at 2 and 10 and they should indicate a distance of 1 sd from the means to the inflection points. (3pt, 1 for correct location of peaks; 1 for distribution 1 being less dispersed than distribution 2; 1 for indication of the inflection points one sd from mean)
~ 3~
Figure 1: histogram of variable X
b) Is the distribution of the variable X in Figure 1 symmetrical? Would the normal distribution be a good model for the data from the histogram in Figure 1? Explain.
Yes the distribution is symmetrical, but no the normal is not a good model, because the distribution has multiple peaks. The normal distribution is unimodal (has only one peak) and is therefore unable to capture the peculiar shape of the distribution. (2pt, both if the student mentions that )
3. (6 pts) Suppose that the grades for last year’s Data Analysis exam follow a normal distribution with mean 7 and standard deviation 1.2. Call this variable Y.
a) Calculate the fraction of students that scored at least an 8.
?(? ≥ 8) = ? �?−7 ≥ 8−7� = ?(? ≥ 0.8333) = 1 − ?(? < 0.8333) = 1 − 0.7967 =
1.2 1.2
0.2033. (2pt, 1 for correct standardization and 1 for correct probability)
b) Calculate the probability that a randomly selected student scores more than a 4.5 but less than a 6.3.
?(4.5 < ? < 6.3) = ?(−2.0833 < ? < −0.5833) = ?(? < −0.5833) − ?(? ≤
−2.0833 = 0.2810 − 0.0188 = 0.2622. (2pt, 1pt for correct standardization and 1 for correct probability)
c) Students enter the Hall of Eternal Glory by scoring among the top 2 percent of the grade distribution for Data Analysis. How high does one need to score to win a place in this coveted group? Approximate that score with statistics table A.
According to Table A, the corresponding Z-score is approximately 2.06. So, ?(? > 2.06) = 0.02; ?−7 > 2.06; ? > 9.472. So students must score higher than a 9.472. (2pt, 1pt for correct standardization and 1 for correct probability)
~ 4~
4. (14 pts) Suppose that you are a government advisor working on policies to improve the educational outcomes of children in primary school. In particular, you are interested in the effect of providing school lunches to children from poor families. Educational outcomes are measured by the performance at a standardized test. In order to investigate the relationship between free lunches and learning, you collect data on two variables for a simple random sample of 100 schools:
1) X1: the percentage of children in school who receive a free lunch (0 to 100%).
2) X2: the average test score of children in the school (on a scale from 0 to 100).
a) Which is the explanatory and which is the response variable in this example? Explain.
The researcher is interested in the effect of the free lunches (X1) on average test scores (X2). Hence, X2 is the response variable and X1 is the explanatory variable. (2pt)
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