Question

There is some evidence that, in the years 1981 - 85, a simple name change resulted in a short-term increase in the price of certain business firms' stocks (relative to the prices of similar stocks). (See D. Horsky and P. Swyngedouw, "Does it pay to change your company's name? A stock market perspective," Marketing Science v.6, pp. 320- 35, 1987.)

Suppose that, to test the profitability of name changes in the more recent market (the past five years), we analyze the stock prices of a large sample of corporations shortly after they changed names, and we find that the mean relative increase in stock price was about 0.72%, with a standard deviation of 0.14%. Suppose that this mean and standard deviation apply to the population of all companies that changed names during the past five years. Complete the following statements about the distribution of relative increases in stock price for all companies that changed names during the past five years.

(a) According to Chebyshev's theorem, at least____ of the relative increases in stock price lie between 0.51 % and 0.93 %.

(b) According to Chebyshev's theorem, at least ____ of the relative increases in stock price lie between 0.44 % and 1.00 %.

(c) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately____ of the relative increases in stock price lie between 0.44 % and 1.00 %.

(d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 99.7% of the relative increases in stock price lie between___%
and ____%.

Answer 1

We are given and .

According to Chebyshev's theorem, at least percent of the relative increases in stock price lie within k standard deviations from mean.

(a) Here we will first find 0.51 and 0.93 are how many standard deviations (value of k) away from mean.

Now we know that 0.51 and 0.93 are 1.5 standard deviations away from mean (0.51 is 1.5 standard deviations below mean and 0.93 is 1.5 standard deviations above mean). Therefore, .

According to Chebyshev's theorem, at least percent of the relative increases in stock price lie between 0.51 % and 0.93 %.

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(b) Here we will first find 0.44 and 1.00 are how many standard deviations (value of k) away from mean.

Now we know that 0.44 and 1.00 are 2 standard deviations away from mean (0.44 is 2 standard deviations below mean and 1.00 is 2 standard deviations above mean). Therefore, .

According to Chebyshev's theorem, at least percent of the relative increases in stock price lie between 0.44% and 1.00%.

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(c) According to Empirical rule, 95% of the data lies within 2 standard deviations from mean. From part b, 0.44 and 1.00 are 2 standard deviations away from mean (0.44 is 2 standard deviations below mean and 1.00 is 2 standard deviations above mean). Therefore:

According to the empirical rule, approximately of the relative increases in stock price lie between 0.44 % and 1.00 %.

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(d) According to Empirical rule, 99.7% of the data lies within 3 standard deviations from mean, that is between and .

According to the empirical rule, approximately 99.7% of the relative increases in stock price lie between %
and %.