To analyze the changes in stock prices and use Chebyshev's theorem to provide bounds on the data, follow these steps:
### (a) Calculating the Range for 84% Confidence Interval
First, determine the boundaries for the stock price increase where at least 84% of the data will lie based on Chebyshev's theorem.
The mean relative increase in stock price is [tex]\( \mu = 0.73\% \)[/tex], and the standard deviation is [tex]\( \sigma = 0.20\% \)[/tex].
Chebyshev's theorem states that for any distribution, at least [tex]\( \frac{1 - \frac{1}{k^2}}{100} \)[/tex]% of the data lies within [tex]\( k \)[/tex] standard deviations of the mean, where [tex]\( k \)[/tex] is a positive number.
To find [tex]\( k \)[/tex] for an 84% confidence interval:
[tex]\[ 1 - \frac{1}{k^2} = 0.84 \][/tex]
[tex]\[ \frac{1}{k^2} = 1 - 0.84 = 0.16 \][/tex]
[tex]\[ k^2 = \frac{1}{0.16} = 6.25 \][/tex]
[tex]\[ k = \sqrt{6.25} = 2.5 \][/tex]
Hence, we need to find the boundaries that lie within [tex]\( 2.5 \)[/tex] standard deviations of the mean:
[tex]\[ \text{Lower bound} = \mu - k \sigma = 0.73\% - 2.5 \times 0.20\% \][/tex]
[tex]\[ \text{Lower bound} = 0.73\% - 0.50\% = 0.23\% \][/tex]
[tex]\[ \text{Upper bound} = \mu + k \sigma = 0.73\% + 2.5 \times 0.20\% \][/tex]
[tex]\[ \text{Upper bound} = 0.73\% + 0.50\% = 1.23\% \][/tex]
Thus, according to Chebyshev's theorem, at least 84% of the relative increases in stock price lie between [tex]\( 0.23\% \)[/tex] and [tex]\( 1.23\% \)[/tex].
Statement (a):
According to Chebyshev's theorem, at least 84% of the relative increases in stock price lie between [tex]\( \mathbf{0.23\%} \)[/tex] and [tex]\( \mathbf{1.23\%} \)[/tex].
### (b) Percentage of Relative Increases in Given Interval
To determine the percentage of relative increases in stock price that lie between [tex]\( 0.33\% \)[/tex] and [tex]\( 1.13\% \)[/tex]:
Here, we need to find [tex]\( k \)[/tex] such that the interval [tex]\( [0.33\%, 1.13\%] \)[/tex] is within [tex]\( k \)[/tex] standard deviations of the mean.
First, calculate [tex]\( k \)[/tex]:
[tex]\[ \text{Upper bound} = 1.13\% \quad \text{and} \quad \text{Lower bound} = 0.33\% \][/tex]
[tex]\[ \text{Width from mean to upper bound} = 1.13\% - 0.73\% = 0.40\% \][/tex]
[tex]\[ k = \frac{0.40\%}{0.20\%} = 2 \][/tex]
Using Chebyshev's theorem:
[tex]\[ 1 - \frac{1}{k^2} = 1 - \frac{1}{2^2} = 1 - \frac{1}{4} = 0.75 \][/tex]
Thus, at least 75% of the relative increases in stock price lie within the interval from [tex]\( 0.33\% \)[/tex] to [tex]\( 1.13\% \)[/tex].
Statement (b):
According to Chebyshev's theorem, at least [tex]\( \mathbf{75\%} \)[/tex] of the relative increases in stock price lie between [tex]\( 0.33\% \)[/tex] and [tex]\( 1.13\% \)[/tex].
By understanding and applying Chebyshev's theorem, we can conclude that:
- At least 84% of the increases lie between 0.23% and 1.23%.
- At least 75% of the increases lie between 0.33% and 1.13%.
