Problem Analysis #35

\boxed{55}

[THOUGHT] [PYTHON] from fractions import Fraction

z = Fraction(258, 100) # 2.58 = 258/100 sigma = Fraction(5) n = 36 E = z * (sigma / 6) # since sqrt(36) = 6 a = 2 * E + Fraction(6, 5) # 1.2 = 6/5 result = 10 * a

print("a =", a) print("10a =", result) [/PYTHON] \boxed{55}

To solve the problem, we begin by understanding how a confidence interval for the population mean $ from a normal distribution is constructed.

Given:

• Population standard deviation: $\sigma = 5$
• Sample size: $n = 36$
• Confidence level: 99%
• Z-score corresponding to 99% confidence: $z^* = 2.58$
• The confidence interval is given as $1.2 \leq m \leq a$

This implies that the lower bound of the interval is 1.2 and the upper bound is $a$. Since the confidence interval for the mean is symmetric about the sample mean $\bar{x}$, we can write:

Where $E$ is the margin of error, given by:

Now, since the lower bound is 1.2:

Then the upper bound becomes:

Finally, compute:

Final Answer: