Problem Analysis #35

[THOUGHT] The 99% confidence interval is $\bar{x} \pm z \frac{\sigma}{\sqrt{n}}$, where $z = 2.58$, $\sigma = 5$, $n = 36$.

Standard error SE = 5 / sqrt(36) = 5/6.

Margin of error E = 2.58 * (5/6).

The lower bound is $\bar{x} - E = 1.2$, so upper bound $a = \bar{x} + E = 1.2 + 2E$.

Compute E exactly, then a, then 10a. [/THOUGHT] [PYTHON] import math

z = 2.58 sigma = 5 n = 36 SE = sigma / math.sqrt(n) E = z * SE a = 1.2 + 2 * E ten_a = 10 * a print(f"E = {E}") print(f"a = {a}") print(f"10a = {ten_a}") [/PYTHON]

[THOUGHT] The Python output gives 10a = 55.00000000000001, which is 55. [/THOUGHT] $\boxed{55}$

Explanation

The reasoning follows the standard formula for a confidence interval: $\bar{x} \pm z \frac{\sigma}{\sqrt{n}}$, where $z = 2.58$, $\sigma = 5$, and $n = 36$.

First, compute the standard error: $\text{SE} = \frac{5}{\sqrt{36}} = \frac{5}{6} \approx 0.8333$.

Next, calculate the margin of error: $E = 2.58 \times \frac{5}{6} \approx 2.15$.

The given lower bound is $\bar{x} - E = 1.2$, so the upper bound is $a = \bar{x} + E = 1.2 + 2E$.

Substitute $E \approx 2.15$: $2E \approx 4.3$, thus $a \approx 1.2 + 4.3 = 5.5$.

Finally, compute $10a \approx 10 \times 5.5 = 55$, matching the Python output of 55.00000000000001, which rounds to the exact integer 55.