Exam One Answers
Econ 180: Quantitative Methods
Professor Robert J. Lemke
Spring 2006


  1. Consider a sample of data: S = {4, 6, 7, 4, 5, 3, 2, 6, 11, 8, 10}.

    1. What is the sample mean?
      Add up all of the elements in the set to get 66. As there are 11 elements in the set, the sample mean is then 66 / 11 = 6.

    2. What is the sample variance?
      Square all of the elements in the set, and then add them up: 16 + 36 + ... + 100 = 476. The sample variance is then [ 476 - 11*(6*6) ] / 10 = 8.

    3. What is the coefficient of variation?
      The coefficient of variation = the sample standard deviation divided by the sample mean = sqrt(8) / 6 = 47.14%.

    4. What is the z-score associated with the sample value of 8?
      The z-score for x = 8 is [ 8 - 6 ] / sqrt(8) = sqrt(2) = 0.7071.

    5. What is the second element of the ordered sample?
      The second element of the ordered sample is the second smallest element in the sample, which is 3.


  2. Let X be a discrete random variable with the following probability distribution function:

    x12345
    p(x)0.400.300.150.100.05

    1. What is the probability that X takes a value above 3?
      P ( X > 3 ) = p(4) + p(5) = 0.10 + 0.05 = 0.15.

    2. What is the probability that X takes a value of at most 2?
      P ( X <= 2 ) = p(1) + p(2) = 0.40 + 0.30 = 0.70.

    3. What is the probability that X takes a value of 5?
      P ( X = 5 ) = p(5) = 0.05.

    4. What is the mean of X?
      To begin, add two rows to the table.
      x12345Total
      p(x)0.400.300.150.100.051.00
      xp(x)0.400.600.450.400.252.10
      x2p(x)0.401.201.351.601.255.80

      So now we are in a position to answer this and the next question. The mean of the distibution is the sum of the third row, which equals 2.1.

    5. What is the variance of X?
      The variance of the distribution is the sum of the fourth row less the mean squared, which equals 5.8 - 2.1*2.1 = 1.39.


  3. At a particular college, it is known that 70 percent of students receive financial aid, while 30 percent do not receive financial aid. A random sample of 8 students is obtained.

    1. What is the probability that exactly 7 of the students receive financial aid?
      This is a standard Binomial question where the number of students who receive financial aid is distributed Binomial with a probability of success of 0.70 and 8 draws. Let X be the number of students in the sample who receive financial aid. Then X ~ Bin(0.7,8). Finally, P(X=7) = (8 choose 7) * (0.7)7 * (0.3)1 = 0.197 = 19.7%.

    2. What is the probability that at least 6 of the students receive financial aid?
      With the same set-up as the previous part, the question is P(X>=6) = p(6) + P(7) + P(8) = (8 choose 6) * (0.7)6 * (0.3)2 + (8 choose 7) * (0.7)7 * (0.3)1 + (8 choose 0) * (0.7)8 * (0.3)0 = 0.296 + 0.197 + 0.058 = 0.551 = 55.1%.


  4. In a random sample of 200 households, it is determined that 80 households have at least one person living in it with a college degree (so that 120 households have no college degree). It is also determined that 150 households give charitable contributions on a regular basis (so that 50 do not give to charity on a regular basis). Finally, it is known that education (i.e., having a college degree) and charitable giving are independent traits across households.

    1. How many of the 200 households do not have a college degree and give to charity on a regular basis?
      Note that education and charitable giving are independent. Thus, P(NE and C) = P(NE)xP(C|NE) = (0.6)(0.75) = 0.45, where C = gives to charity and NE = not educated with a college diploma. Thus, 200(0.45) = 90 households give to charity and are not college educated.

    2. How many of the 80 college-educated households give to charity on a regular basis?
      Because of independence,P(C|E) = P(C) = 0.75, where E means college educated. Thus, 80(0.75) = 60 of the 80 college-educated households give to charity on a regular basis.


  5. The Career Development Center of an elite liberal arts college surveyed its 347 graduating seniors concerning their GPA and the number of job offers they received. The CDC created the following cross-tabulation of the data:

    Number of Job Offers
    01234
    GPA3.50 - 4.002826343
    3.00 - 3.4992578628
    2.00 - 2.9910291341
    0.00 - 1.991314800

    1. How many students graduated with at least a 2.00 grade point average?
      Summing up the counts in the top three rows, we find that 2 + 8 + ... + 4 + 1 = 312 students graduated with at least a 2.00 grade point average. Alternatively, we know that 347 students were surveyed. As 13 + 14 + 8 = 35 students had a GPA under 2.0, we immediately know that 347 - 35 = 312 students graduated with at least a 2.00 grade point average.

    2. The CDC analyzes the data and concludes the over-riding theme is that students with higher grade point averages tend to receive more job offers. A student with a 1.5 grade point average looks at the table and responds that the CDC is wrong because 29 students with a GPA between 2.00 and 2.99 received one job offer while only 25 students with a GPA between 3.00 and 3.49 and only 8 students with a GPA between 3.5 and 4.0 received one job offer. How would you respond to this analysis?
      The CDC's analysis is correct, while the student's analysis is wrong. The student is making two errors. First, he is looking at raw counts, which is always a bad idea. One should look at probabilities (or percentages). The second mistake being made is that the goal is not necessarily to get one job offer, but to get many job offers. Looking at the middle two rows, for example, reveals that there are 9 + 25 + 78 + 62 + 8 = 182 students in the 3.0 to 3.49 range and there are 10 + 29 + 13 + 4 + 1 = 57 students in the 2.00 to 2.99 range. Thus, of the first group, 25 / 182 = 13.7% received one job offer, 78 / 182 = 42.9% received two job offers, 62 / 182 = 34.1% received three job offers, and 8 / 182 = 4.4% received four job offers. In comparison, in the 2.00 to 2.99 GPA range, 29 / 57 = 50.1% received one job offer, 13 / 57 = 22.8% received two job offers, 4 / 57 = 7.0% received three job offers, and only 1 / 57 = 1.8% received four job offers. Clearly one is more likely to be offered more jobs if one is in the higher GPA range.

    3. What is the probability of receiving 2 or more job offers if one's GPA is at least 3.5?
      The probability of receiving 2 or more job offers if one's GPA is at least 3.5 is (26+34+3) / (2+8+26+34+3) = 63 / 73 = 86.3%.

    4. For each GPA class, what is the probability of receiving at least one job offer?
      The probability of receiving at least one job offer in the range of 3.50 to 4.00 GPAs is (8+26+34+3) / (2+8+26+34+3) = 71 / 73 = 97.3%. The probability of receiving at least one job offer in the range of 3.00 to 3.49 GPAs is (25+78+62+8) / (9+25+78+62+8) = 173 / 182 = 95.1%. The probability of receiving at least one job offer in the range of 2.00 to 2.99 GPAs is (29+13+4+1) / (10+29+13+4+1) = 47 / 57 = 82.4%. The probability of receiving at least one job offer in the range of 0.00 to 1.99 GPAs is (14+8+0+0) / (13+14+8+0+0) = 22 / 35 = 62.9%.

    5. What is the expected number of job offers for a student whose GPA is between 3.00 and 3.49? What is the expected number of job offers for a student whose GPA is between 2.00 and 2.99?
      One way to calculate these probabilities is to create pdf tables. Consider the following:
      GPA: 3.00 to 3.49
      x01234Total
      p(x)9/182 = 0.04925/182 = 0.13778/182 = 0.42962/182 = 0.3418/182 = 0.0441.00
      xp(x)0.00.1370.8581.0230.1762.194
      GPA: 2.00 to 2.99
      x01234Total
      p(x)10/57 = 0.17529/57 = 0.50913/57 = 0.2284/57 = 0.0701/57 = 0.0181.00
      xp(x)0.00.5090.4560.2100.0.721.247
      Thus, students in the 3.00 to 3.49 range receive 2.194 job offers on average, while students in the 2.00 to 2.99 range receive 1.247 job offers on average.