Question 1161134: 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 0.74%, with a standard deviation of 0.16%. 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 8/9 (about89%) of the relative increases in stock price lie between BLANK % and BLANK %. (round your answer to 2 decimal places.)
B) According to Chebyshev’s theorem, at least BLANK of the relative increases in stock price lie between 0.42% and 1.06%.
C) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately BLANK of the relative increases in stock price lie between 0.42% and 1.06%
D) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the relative increases in stock price lie between BLANK % and BLANK %
Answer by CPhill(2285)   (Show Source):
You can put this solution on YOUR website! Here are the completed statements based on Chebyshev's theorem and the Empirical Rule:
### Given Information:
* Population Mean ($\mu$) = $0.74\%$
* Population Standard Deviation ($\sigma$) = $0.16\%$
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### A) According to Chebyshev’s theorem, at least 8/9 (about 89%) of the relative increases in stock price lie between **0.26%** and **1.22%**.
* **Finding $k$:**
$$1 - \frac{1}{k^2} = \frac{8}{9} \implies \frac{1}{k^2} = \frac{1}{9} \implies k^2 = 9 \implies k = 3$$
* **Lower limit:** $\mu - 3\sigma = 0.74\% - 3(0.16\%) = 0.74\% - 0.48\% = \mathbf{0.26\%}$
* **Upper limit:** $\mu + 3\sigma = 0.74\% + 3(0.16\%) = 0.74\% + 0.48\% = \mathbf{1.22\%}$
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### B) According to Chebyshev’s theorem, at least **3/4 (or 75%)** of the relative increases in stock price lie between 0.42% and 1.06%.
* **Finding $k$:**
$$0.42\% = 0.74\% - k(0.16\%) \implies k(0.16\%) = 0.32\% \implies k = 2$$
* **Applying Chebyshev's formula:**
$$1 - \frac{1}{k^2} = 1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4} = \mathbf{75\%}$$
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### C) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately **95%** of the relative increases in stock price lie between 0.42% and 1.06%.
* As calculated in Part B, $0.42\%$ and $1.06\%$ correspond to $k = 2$ standard deviations from the mean ($\mu \pm 2\sigma$).
* The Empirical Rule states that approximately **95%** of observations in a bell-shaped distribution fall within 2 standard deviations of the mean.
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### D) Suppose that the distribution is bell-shaped. According to the empirical rule, approximately 68% of the relative increases in stock price lie between **0.58%** and **0.90%**.
* The Empirical Rule states that approximately 68% of the data falls within 1 standard deviation of the mean ($\mu \pm 1\sigma$).
* **Lower limit:** $\mu - 1\sigma = 0.74\% - 1(0.16\%) = \mathbf{0.58\%}$
* **Upper limit:** $\mu + 1\sigma = 0.74\% + 1(0.16\%) = \mathbf{0.90\%}$
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