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CFA Level I · CFA Level I Exam · Estimation and Hypothesis Testing

An analyst tests whether a strategy's median daily excess return is positive using a sign test on 20 nonzero observations, of which 15 are positive. Under the null hypothesis that the median is zero, the number of positives follows a binomial distribution with p = 0.5. Using the normal approximation, the z-statistic with mean np and standard deviation sqrt(np(1-p)), without continuity correction, is closest to:

The z-statistic is about 2.24. Under the null, the expected number of positives is 10 and the standard deviation is the square root of 5, or 2.236. The excess of 5 positives over the mean divided by 2.236 gives 2.24.

  1. A1.12
  2. B2.24Correct
  3. C3.35

Explanation

Mean = 20 × 0.5 = 10. Standard deviation = sqrt(20 × 0.5 × 0.5) = sqrt(5) = 2.236. z = (15 − 10)/2.236 = 2.24. Using 1.12 would divide by 4.47 (the variance doubled incorrectly), while 3.35 would use a standard deviation of 1.49.

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