In a manufacturing process, we are interested in measuring the average length of a certain type of bolt. Past data indicate that the population standard deviation is .25 inches. How many manufactured bolts should be sampled in order to make us 95 percent confident that the sample mean bolt length is within .02 inches of the true mean bolt length?

Question

BlackTom AI · Accepted Answer

We’re asked to determine the required sample size for estimating a population mean with a specified margin of error when the population standard deviation is known. The formula to use is n = (z_{α/2} * σ / E)^2, where z_{α/2} corresponds to the desired confidence level, σ is the population standard deviation, and E is the desired margin of error.

First, identify the given values: σ = 0.25 inches, E = 0.02 inches, and confidence level 95% implies z_{α/2} ≈ 1.96.

Compute the ratio σ / E: 0.25 / 0.02 = 12.5. Next multiply by z_{α/2}: 1.96 * 12.5 = 24.5.

Square the result to obtain the required sample size: (24.5)^2 = 600.25. Since you can’t sample a fraction of a bolt, you round up to ensure the margin of error is within the target; therefore, n = 601.

Now, evaluate the alternative answer choices:
- 25: This would imply a very small sample size, which cannot achieve the requested margin of error given the standard deviation; it ignores the 1.96 multiplier and the 0.02 tolerance.
- 49: Although larger than 25, it still falls short of the calculated 600.25; it would not meet the 95% confidence requirement with the specified σ and E.
- 423: Closer to the real value but still below the 600.25 threshold; would likely produce a margin of error larger than 0.02 inches.
- 1225: This is above the necessity implied by the calculation; while it would certainly meet the margin of error, it is not the smallest sample size that satisfies the condition.

By contrast, 601 is the smallest integer n that satisfies (z_{0.025} * σ / E)^2 ≥ n, ensuring the 95% confidence that the sample mean is within 0.02 inches of the true mean given σ = 0.25.

BU.510.601.K1.FA25 Quiz 2- Requires Respondus LockDown Browser

查看解析

登录即可查看完整答案

类似问题

The sample size needed to estimate a population mean to within 2 units with 95% confidence when the population standard deviation equals 8 is:当总体标准差等于 8 时，以 95% 的置信度将总体均值估计在 2 个单位以内所需的样本量为：

Why is it important to check the sample size of a scientific study when assessing its reliability in media reports?

The minimum sample size to represent a survey of 576 students should be:

The minimum sample size to represent a survey of 1936 students should be:

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单