Economic 2122A – Econometrics I Assignment 2
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Question 1
A random sample for five exam scores produced the following data:
| Hours Studied (x) | Test Grade (y) |
| 3.5 | 85 |
| 2 | 74 |
| 5 | 88 |
| 4.5 | 90 |
| 1.5 | 70 |
- Compute the correlation coefficient
- Comment on the correlation between and according to the covariance and the correlation coefficient you have got.
Question 2
A corporation takes delivery of some new machinery that must be installed and checked before it becomes available to use. The corporation is sure that it will take no more than 7 days for this installation and check to take place. Let be the event “it will be more than 4 days before the machinery becomes available” and be the event “it will be less than 6 days before the machinery becomes available.”
- Are events and mutually exclusive?
- Are events and collectively exhaustive?
- Draw the Venn diagram for ̅in the sample space (indicate ̅by a shaded area).
- Draw the Venn diagram for ̅∩ in the sample space (indicate ̅∩ by a shaded area).
- ∪ ( ̅∩ ) = ? (Write down the resulting event.)
- ∩ ( ̅∩ ) = ? (Write down the resulting event.)
Question 3 (Show the formula you use, not only the final result.)
Suppose you can choose any 4 different digits from the ten digits 0 to 9 to form a password to your account. How many possible passwords can you form?
Question 4 (Show your steps, not only the final results.)
A factory uses two machines, A and B, to produce a certain type of output. Machine A produces 80% of the daily output, and machine B produces 20% of the daily output. 4% of machine A’s output is defective, and 2% of machine B’s output is defective. Suppose an item was found to be defective while you are running a random inspection.
- What is the probability that this item was produced by machine A?
- What is the probability that this item was produced by machine B?
Related: (Solution) Economics 2122A Econometrics I Assignment 3
Solution – Econometrics I Assignment 2
Compute the correlation coefficient
| X-Mx | Y-My | (X-Mx)2 | (Y-My)2 | (X-Mx)(Y-My) | |
| 0.200 -1.300 1.700 1.200 -1.800 Mx=3.300 | 3.600 -7.40 6.600 8.600 -11.400 My=81.400 | 0.040 1.690 2.890 1.440 3.240 Sum=9.300 | 12.960 54.760 43.560 73.960 129.960 Sum=315.200 | 0.720 9.620 11.220 10.320 20.520 Sum=52.400 | X: X Values Y: Y Values Mx: Mean of X Values My: Mean of Y Values X – Mx & Y – My: Deviation scores (X – Mx)2 & (Y – My)2: Deviation Squared (X – Mx)(Y – My): Product of Deviation Scores |
Correlation coefficient calculation
X Values
∑ = 16.5
Mean = 3.3
∑(X – Mx)2 = SSx = 9.3
Y Values
∑ = 407
Mean = 81.4
∑(Y – My)2 = SSy = 315.2
X and Y Combined
N = 5
∑(X – Mx)(Y – My) = 52.4
R Calculation
r = ∑((X – My)(Y – Mx)) / √((SSx)(SSy))
r = 52.4 / √((9.3)(315.2)) = 0.9678
R (correlation coefficient)= 0.9678
Answer:
𝜎𝑋2=1/(5−1)[(3.5−3.3)^2+(2−3.3)^2+(5−3.3)^2+(4.5−3.3)^2+(1.5−3.3)^2]=2.325
𝜎𝑋=√2.325≈1.525
𝜎𝑌2=1/(5−1)[(85−81.4)^2+(74−81.4)^2+(88−81.4)^2+(90−81.4)^2+(70−81.4)^2]=78.8
𝜎𝑌 = √78.8 ≈ 8.877
𝑐𝑜𝑟𝑟(𝑋,𝑌)=𝑐𝑜𝑣(𝑋,𝑌)/𝜎𝑋𝜎𝑌=0.968
Comment on the correlation between 𝑥 and 𝑦 according to the covariance and the correlation coefficient you have got. (3 points)
Correct Answer: covariance indicates that hours studied and test grade are positively correlated; correlation coefficient indicates that hours studied and test grade are strongly positively correlated.
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