Answer :
To solve this problem, we will use Chebychev's Theorem. Chebychev's Theorem provides a way to find the minimum probability that a random variable lies within a certain number of standard deviations from the mean. The theorem states that for any distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the probability that a random variable is within [tex]k[/tex] standard deviations from the mean is at least [tex]1 - \frac{1}{k^2}[/tex].
Given:
- Mean ([tex]\mu[/tex]) = 57
- Standard deviation ([tex]\sigma[/tex]) = 1.4
- Desired probability = 96% or 0.96
The problem asks us to find the value of [tex]c[/tex] such that the probability of an outcome lying between [tex]57 - c[/tex] and [tex]57 + c[/tex] is at least 96%. According to Chebychev's Theorem:
[tex]1 - \frac{1}{k^2} = 0.96[/tex]
Solving for [tex]k[/tex]:
- [tex]1 - \frac{1}{k^2} = 0.96[/tex]
- [tex]\frac{1}{k^2} = 1 - 0.96 = 0.04[/tex]
- [tex]k^2 = \frac{1}{0.04} = 25[/tex]
- [tex]k = \sqrt{25} = 5[/tex]
The value of [tex]k[/tex] is 5. This means that to guarantee at least 96% probability, the outcomes should lie within 5 standard deviations from the mean.
Now, calculate [tex]c[/tex] using the relation [tex]c = k \times \sigma[/tex]:
[tex]c = 5 \times 1.4 = 7[/tex]
Therefore, the value of [tex]c[/tex] should be 7, rounding to the nearest whole number.
Thus, the outcome of the experiment will lie between [tex]57 - 7 = 50[/tex] and [tex]57 + 7 = 64[/tex] with at least 96% probability.
