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로그인 회원가입 You are solving a Korean CSAT mathematics problem. Reason carefully, but the final response must contain exactly one final answer inside \boxed{}.

For multiple-choice problems, put only one of ①, ②, ③, ④, ⑤ inside \boxed{}. For short-answer problems, put only the final integer inside \boxed{}. Do not use external tools.

Problem 26 A population is normally distributed with meanmand standard deviation 5. From this population, a random sample of size 36 is taken, and a 99% confidence interval for the population meanmis constructed using the sample mean. The resulting confidence in- terval is 1.2≤m≤a. Find the value ofa. (Here,Zdenotes a standard normal random variable, and use P(|Z| ≤2.58) = 0.99.) [3 points] ①5.1 ②5.2 ③5.3 ④5.4 ⑤5.5 더보기

We are given a normally distributed population with unknown mean 𝑚 m and known standard deviation 𝜎

5 σ=5. A random sample of size 𝑛

36 n=36 is taken, and a 99% confidence interval for 𝑚 m is constructed. The resulting interval is 1.2 ≤ 𝑚 ≤ 𝑎 1.2≤m≤a. We need to find 𝑎 a.

For a population with known standard deviation, the confidence interval for the mean is:

𝑥 ˉ ± 𝑧 𝛼 / 2 ⋅ 𝜎 𝑛 x ˉ ±z α/2 ​

⋅ n ​

σ ​

where 𝑥 ˉ x ˉ is the sample mean, 𝑧 𝛼 / 2 z α/2 ​

is the critical value from the standard normal distribution, 𝜎 σ is the population standard deviation, and 𝑛 n is the sample size.

Given:

𝜎

5 σ=5 𝑛

36 n=36 Confidence level = 99%, so 𝛼

0.01 α=0.01 and 𝑧 𝛼 / 2

𝑧 0.005 z α/2 ​

=z 0.005 ​

. We are told 𝑃 ( ∣ 𝑍 ∣ ≤ 2.58 )

0.99 P(∣Z∣≤2.58)=0.99, so 𝑧 0.005

2.58 z 0.005 ​

=2.58.

Compute the standard error:

𝜎 𝑛

5 36

5 6 ≈ 0.8333 n ​

σ ​

= 36 ​

5 ​

= 6 5 ​

≈0.8333

Compute the margin of error:

𝐸

𝑧 𝛼 / 2 ⋅ 𝜎 𝑛

2.58 ⋅ 5 6

12.9 6

2.15 E=z α/2 ​

⋅ n ​

σ ​

=2.58⋅ 6 5 ​

= 6 12.9 ​

=2.15

The confidence interval is:

𝑥 ˉ − 𝐸 ≤ 𝑚 ≤ 𝑥 ˉ + 𝐸 x ˉ −E≤m≤ x ˉ +E

We are given the lower bound: 𝑥 ˉ − 𝐸

1.2 x ˉ −E=1.2. Therefore:

𝑥 ˉ

1.2 + 𝐸

1.2 + 2.15

3.35 x ˉ =1.2+E=1.2+2.15=3.35

Then the upper bound 𝑎 a is:

𝑎

𝑥 ˉ + 𝐸

3.35 + 2.15

5.5 a= x ˉ +E=3.35+2.15=5.5

Thus 𝑎

5.5 a=5.5, which corresponds to choice ⑤.

\boxed{⑤}

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