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: [ \text{Sample Mean} \pm Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} ] 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: [ \text{Margin of Error} = Z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} = 2.58 \cdot \frac{5}{6} = \frac{12.9}{6} = 2.15 ]

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: [ \bar{x} - 2.15 = 1.2 \implies \bar{x} = 1.2 + 2.15 = 3.35 ]

4. Find the Upper Bound ( a ): The upper bound is calculated as: [ a = \bar{x} + 2.15 = 3.35 + 2.15 = 5.5 ]

5. Compute ( 10a ): [ 10a = 10 \times 5.5 = 55 ]

[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}