SEES0083 Quantitative Methods 2018-2019
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MA EXAMINATION 2018-2019 academic Year
Course: Quantitative Methods
Course Code: SEES0083
PART A : Please answer THREE out of FOUR Exercises (24 points)
Exercise 1 (8 points)
Suppose a large population describing the variable X has mean μ = 80 and standard deviation σ = 30. Suppose a random sample of size n = 100 is selected. The probability that the sample mean is between a and b is 50%.
a) (2 points): Can you apply the Central Limit Theorem to the aforementioned sample? Why or why not? Explain the logic.
b) (3 points) find value of a and b for the Standard Normal distribution below
Z = X 一 px σ X |
c) (3 points) find value of a and b for the distribution of the mean of X.
Exercise 2 (8 points)
A collection of 41 paired observations was obtained from a survey of firms in Albania and Kosovo (two transition countries in South-Eastern Europe). Each pair contains information about one firm from Kosovo and one from Albania. The pairing was made in such a way that the two paired firms were as similar as possible concerning such factors as size and age (i.e. the samples are dependent). The ratio of ‘bad’ loans to total loans was calculated for each of the firms. The sample mean difference (between Kosovar and Albanian firms) was 5, and the sample Standard Deviation of the difference was 1. The population Standard Deviation of the difference is unknown.
Test, against a two-sided alternative, the null hypothesis that the two populations’ means are identical. Please show:
a) (2 points) The exact formulation of H0 and H1
b) (2 points) The graph of the distribution that you need to use in order to identify whether you are in the rejection region or not. On the graph, identify the rejection region for the following three cases: α=0.10, α=0.05, α=0.01.
c) (1 points) Show the formula for the test statistic: does this formula depend on α (alpha)?
d) (3 points) Write what the test results are for the three cases: 10%, 5% and 1% significance level (you will need to look at the relevant statistical table(s)). Interpret your results.
Exercise 3 (8 points)
Find the percentage of observations that fall within the interval [μ ± 3σ] in the following three cases:
a) (3 points) For any population with mean 10 and standard deviation 2. Is this result a lower bound, upper bound or exact?
b) (3 points) For a NORMAL population with mean 10 and standard deviation 2. Is this result a lower bound, upper bound or exact?
c) (2 points) For a UNIFORM population with mean 10 and standard deviation 2. Is this result a lower bound, upper bound or exact?
Exercise 4 (8 points)
Consider a standard deck of 52 cards, with four suits: v ♣ ♦ ♠
Let event A = card is strictly less than four (i.e. ace, 2 or 3)
Let event B = card is from a red suit
By showing the complete cross-tab of these events calculate:
a) (4 points) P(Red U Less-than-four) = P(Red) + P(Less-than-four) - P(Red ∩ Less-than-four)
b) (4 points) By applying the conditional probability show that P(Red ∩ Less-than-four) = P(Red| Less-than-four)P(Less-than-four)
PART B : Answer ALL THREE questions (60 points)
Exercise 5 (20 points)
A researcher has obtained a random sample of 227 industrial-sector firms in Albania and Kosovo. The researcher is interested in knowing whether the firms located in the two capital cities (Tirana and Pristina) perform better than the firms located in other cities do. Firm performance is measured by the percentage growth rate of firm sales (dy_per). She presumes that better performance in cities may result from agglomeration economies.
Note that capital is a dummy variable that takes the value 1 if the firm is located in a capital city, and 0 otherwise.
The first STATA output analysed by the researcher is:
. **Firms located in capital cities
. sum dy_per if capital==1
Variable
dy_per 78
. **Firms located elsewhere
. sum dy_per if capital==0
Obs
149 9.251342 16.37798
a) (1 point) What is the difference between the two sample means (the mean growth rate of firms located in capital cities and the mean growth rate of firms located elsewhere)? Do the results conform to the researcher’s expectations?
b) (3 points) Can the researcher conclude that there is a systematic (statistically significant) difference in performance between firms located in capital cities and firms located elsewhere? Why?
Next, the researcher performs a two-sample t-test for the equality between the two means.
. ttest dy_per, by(capital) unequal
Two-sample t test with unequal variances
Group |
Obs |
Mean |
Std . |
Err . |
Std . Dev . |
[95% |
Conf . |
Interval] |
0 |
149 |
9.251342 |
1.34 |
1736 |
16.37798 |
6.59 |
9907 |
11.90278 |
1 |
78 |
5.861538 |
1.77 |
5572 |
15.68142 |
2.32 |
5924 |
9.397153 |
combined |
227 |
8.086564 |
1.07 |
4424 |
16.18783 |
5.96 |
9393 |
10.20373 |
diff |
|
3.389804 |
2.22 |
5514 |
|
-1 .00 |
4855 |
7.784463 |
diff = mean(0) - mean(1) t = 1.5232
Ho: diff = 0 Satterthwaite's degrees of freedom = 162.482
Ha: diff < 0
Pr(T < t) = 0.9352
Ha: diff != 0
Pr( |T | > |t |) = 0.1297
Ha: diff > 0
Pr(T > t) = 0.0648
c) (2 points) What are the null hypotheses and the t-stat?
d) (4 points) If the researcher’s assumption is that firms located in capital cities perform better than firms located elsewhere do, which is the relevant alternative hypothesis (Ha) against which the null should be tested? (Hint: this is a one-tailed test)
e) (4 points) Write down and interpret the P-value for this alternative hypothesis.
f) (6 points) Based on a 10% significance level, can the researcher reject the null that the two population means are equal against the alternative hypothesis that they are different? Why? (Hint: this is a two-tailed test)
Exercise 6 (20 points)
Next, the researcher from Exercise 5 decides to investigate whether firm performance depends on the firm’s age at the time that the survey was conducted. Age is defined as the number of years that elapsed since the firm’s establishment. She runs two regressions:
1) The growth rate of firm sales (in percentages) on age in LEVELS.
2) The growth rate of firm sales (in percentages) on the LOG of age.
The Stata outputs are, respectively:
. reg dy_per Age
Source |
SS df MS |
||||
Model Residual |
|
||||
Total |
59222.3915 226 262.04598 |
Number of obs F(1, 225) Prob > F
R-squared Adj R-squared
=
=
=
=
=
=
227
3.69
0.0561
0.0161
0.0118
16.092
dy_per |
Coef . |
Std . Err. |
t |
P> |t | |
[95% Conf . |
Interval] |
Age |
- .2899098 |
.1509604 |
-1 .92 |
0.056 |
- .587387 |
.0075673 |
_cons |
11.81069 |
2.2139 |
5.33 |
0.000 |
7.448061 |
16.17332 |
. reg dy_per lnAge
Source |
SS df MS |
Model Residual |
1413.833 1 1413.833 57808.5585 225 256.926927 |
Total |
59222.3915 226 262.04598 |
Number of obs F(1, 225) Prob > F
R-squared Adj R-squared
=
=
=
=
=
=
227
5.50
0.0199
0.0239
0.0195
16.029
dy_per |
Coef . |
Std . Err. |
t |
P> |t | |
[95% Conf . |
Interval] |
lnAge |
-3 .7701 |
1.607158 |
-2 .35 |
0.020 |
-6 .937107 |
- .6030929 |
_cons |
17.05481 |
3.968345 |
4.30 |
0.000 |
9.234932 |
24.87468 |
a) (5 points) For BOTH models, interpret the estimated slope coefficients (Hint: for the level- log model, consider what happens when the firm’s age doubles)
b) (5 points) For BOTH models, what is the predicted growth rate of sales for a company that is 5 years old?
c) (5 points) Based on the R-squared and the significance of the coefficients, which of these two models would you suggest using, and why? Explain in detail the logic of your answer.
2022-08-11