Problem Analysis #35

[THOUGHT] To find the value of $a$ in the confidence interval $1.2 \leq m \leq a$, we follow these steps:

1. Understand the Confidence Interval Formula: The confidence interval for the population mean $m$ is given by:

where:

• $Z_{\alpha/2} = 2.58$ (since $P(|Z| \leq 2.58) = 0.99$),
• $\sigma = 5$ (standard deviation),
• $n = 36$ (sample size).

2. Calculate the Margin of Error:

3. Determine the Sample Mean: The confidence interval is symmetric around the sample mean. Given the lower bound $1.2$, we can find the sample mean ($\bar{x}$) as:

4. Find the Upper Bound $a$: The upper bound is calculated as:

5. Compute $10a$:

[PYTHON]

# Given values
Z_alpha_2 = 2.58
sigma = 5
n = 36
lower_bound = 1.2
margin_of_error = Z_alpha_2 * sigma / (n ** 0.5)
sample_mean = lower_bound + margin_of_error
a = sample_mean + margin_of_error
result = int(10 * a)
print(result)

[THOUGHT]
The value of $10a$ is confirmed to be 55 based on the calculations and verified by Python execution.

$\boxed{55}$