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 0.70%, with a standard deviation of 0.15%. 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 0.25 % and 1.15.

(b) According to Chebyshev's theorem, at least of the relative increases in stock price lie between 0.40 % 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 0.40 % and 1.00.

(d) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the relative increases in stock price lie between _% and _%. .

Answer 1

Given:

Chebyshev's theorem: Approximately 1 - (1/k² ) of the observations falls between

a) First, find the value of k, we have given

Solve any one of the equation to find the value of k,

According to Chebyshev's theorem, at least 88.89% of the relative increase in stock price lies between 0.25% and 1.15%

b) Use the same theorem as above, first, find the value of k

Solve any one of the equation to get the value of k,

Therefore,

According to Chebyshev's theorem, at least 75% of the relative increase in stock price lies between 0.40% and 1.00%

Empirical rule: For every bell shape distribution,

Approximately 68% of the data values lie within 1 standard deviation away from the mean.

Approximately 95% of the data values lie within 2 standard deviations away from the mean.

and approximately 99.73% of the data values lie within 3 standard deviations away from the mean.

c) Have to find the percentage between 0.40% and 1.00%, so first have to find these values are how many standard deviations away from mean.

As (b) part find the value of k that solves any one equation to get the value of k.

according to the part b, the value of k = 2

That is the values 0.40% and 1.00% are 2 standard deviations away from the mean.

So according to the Empirical rule, approximately 75% of the of the relative increases in stock price lies between 0.40% and 1.00%

(d) According to the empirical rule, approximately 68% of the data values fall within 1 standard deviation from the mean that is in between

and

Therefore, according to the empirical rule, approximately 68% of the relative increases in stock price lie between 0.55% and 0.85%