Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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%.
### (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%.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
