By the argument in the last video, if the X_ i are i.i.d. with mean \mu and variance \sigma ^2, and if M_ n=(X_1+\cdots +X_ n)/n, then we have an inequality of the form
\mathbf{P}\big ( |M_ n-\mu | \geq \epsilon \big ) \leq \frac{a\sigma ^2}{n},
for a suitable value of a.
a) If \epsilon =0.1, then the value of a is:
unanswered
b) If we change \epsilon =0.1 to \epsilon =0.1/k, for k\geq 1 (i.e., if we are interested in k times higher accuracy), how should we change n so that the value of the upper bound does not change from the value calculated in part (a)?
n should
stay the same
increase by a factor of k
increase by a factor of k^2
decrease by a factor of k
none of the above
b) If we change \epsilon =0.1 to \epsilon =0.1/k, for k\geq 1, the value of the upper bound does not change from the value calculated in part (a) if we increase n by a factor of k^2.
This is based on the argument in the last video, which states that the probability of the sample mean deviating from the true mean can be controlled by the variance and sample size. Since \epsilon = 0.1/k is k times smaller than \epsilon = 0.1, we need to decrease the probability bound by a factor of k in order to maintain the same level of accuracy. Since the probability bound is inversely proportional to the sample size n, we need to increase n by a factor of k^2 to maintain the same value of the upper bound.
a) If ε = 0.1, then the value of a is not provided in the information given. The specific value of a would depend on the details of the argument or calculation used to arrive at the inequality. Without further information, it is not possible to determine the value of a.
b) If we change ε = 0.1 to ε = 0.1/k, for k ≥ 1, and we want the upper bound to remain the same, we need to find how n should change.
The inequality is given as:
P(|M_n - μ| ≥ ε) ≤ aσ^2/n
To keep the upper bound unchanged, we want the right-hand side of the inequality to remain the same. Since a, σ, and ε remain constant, we need to adjust n.
If we substitute ε = 0.1/k into the inequality, we have:
P(|M_n - μ| ≥ 0.1/k) ≤ aσ^2/n
To keep the upper bound the same, n should decrease by a factor of k. Therefore, the correct answer is: decrease by a factor of k.