ECON 120B Midterm Exam, Summer-I 2022
1. (4 points) For the following claims, determine if they are true/false (write T or F in the boxes).
(a) The size of a test is r[rejecting the null hypothesis].
(b) The power of a test is 1 - r[Type-2 error].
(c) Let X Bernoulli(p). Then 1 - X also has a Bernoulli distribution.
(d) If the correlation between X and Y is zero, then they are independent.
2. (4 points) Assume X Ⅵ (2, 5).
(a) Find the expectation of (X - 1)/2.
(b) Find the standard deviation of 2(X + 4).
(c) Find r[X > 2 or X < 0].
(d) Let Y Ⅵ (-1, 3). Find the expectation of X + 2Y.
3. (4 points) Given a sample, X1 , X2 , . . . , Xn , a researcher computes the sample mean and sample standard deviation. To test the hypothesis H0 : μ = 1 vs. H1 : μ 1, she computes the t-statistic, which is jTj = 1.87. For the following claims, determine if they are true/false (write T or F in the boxes).
(a) She will reject the hypothesis at the 5% level.
(b) The 90% conidence interval contains 1.
(c) It is not possible to determine whether the 99% conidence interval contains 0.
(d) The p-value for testing the hypothesis H0 : μ = 1 is larger than 0.025.
4. (5 points) The following table gives the joint probability distribution between X and Y.
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Y = 0
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Y = 1
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Y = 2
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X = 0
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0.20
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0
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0
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X = 1
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0
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0.15
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0.30
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X = 2
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0
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0
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0.35
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(a) Find r[Y = 0] and r[Y = 2]. (1 point)
(b) Find the conditional distribution of X given Y = 1. (1 point)
(c) Find the distribution of Y - X . (1 point)
(d) Find the expectation and variance of Y - X . (2 points)
5. (5 points) Consider the following Stata output. Find (a), (b), (c), (d), (e).
. ttest X == 0.5
One-sample t test
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Variable | Obs Mean Std . err . Std . dev . [95% conf . interval]
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X | 326 - .0458342 (a) (b) (c) (d)
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mean = mean(X) t = -9.5711
H0: mean = 0.5 Degrees of freedom = 325
Ha: mean < 0.5 Ha: mean != 0.5 Ha: mean > 0.5
Pr(T < t) = 0.0000 Pr(|T| > |t|) = (e) Pr(T > t) = 1.0000
6. (4 points) A dataset contains 124 observations for a variable X . Consider the following Stata output.
. summarize X
Variable | Obs Mean Std . dev . Min Max
-------------+---------------------------------------------------------
X | 124 .826025 1.490026 -2.849437 4.654474
(a) Find the sample variance of X. (1 point)
(b) Construct the t-statistic for testing the hypothesis H0 : E[X] = 0.8 vs. H1 : E[X] 0.8. Do we reject the hypothesis at the 5% level? (2 points)
(c) Compute the 99% conidence interval for E[X]. (1 point)
7. (4 points) Let X , Y and Z be three mutually independent random variables following the distri- butions
X ~ Bernoulli(0.99), Y ~ N (1, 1), Z ~ N (-10, 100).
Deine a new variable, S, by
To ind the expectation of S, someone proposes the following method:
(a) In words, explain the reasoning behind the two equalities labeled by (i) and (ii). (3 points)
(b) Find E[S]. (1 point)