The lifetime of certain type of light bulb is normally distributed with a mean of 1000 hours and a standard deviation of 110 hours. A hardware store manager claims that the new light bulb model has a longer average lifetime. A sample of 10 from the new light bulb model is obtained for a test. Consider a rejection region After testing hypotheses, suppose that a further study establishes that, in fact, the average lifetime of the new lightbulb is 1130 hours. Find the probability of a type II error (round off to second decimal place).

Question

BlackTom AI · Accepted Answer

The problem describes a hypothesis test for the mean lifetime of a bulb, with a known standard deviation. The given rejection region is { xbar >= 1100 }. The true mean under the alternative (the scenario described after testing) is mu = 1130, and we are asked for the probability of a Type II error (failing to reject H0 when mu = 1130).

First, note the sampling distribution of the sample mean when the true mean is mu = 1130: xbar ~ N(1130, (sigma^2 / n)) with sigma = 110 and n = 10, so the standard deviation of xbar is sigma_xbar = 110 / sqrt(10) ≈ 34.785.

Compute the probability of not rejecting H0, i.e., P(xbar < 1100) under mu = 1130:
Z = (1100 - 1130) / (110 / sqrt(10)) = (-30) / 34.785 ≈ -0.862.
P(xbar < 1100) = P(Z < -0.862) = Phi(-0.862) ≈ 0.194.

This is the probability of a Type II error (beta) for mu = 1130. Rounding to two decimals gives beta ≈ 0.19.

Why the result makes sense: the rejection region starts at 1100 hours for the sample mean. Since the true mean is 1130 (well above 1100) we would expect a fairly high chance of observing xbar ≥ 1100 and thus rejecting H0; however, there is still a nonzero probability that the sample mean falls below 1100 due to sampling variability, which is exactly the beta we calculated (about 19%).

If there were multiple answer choices, each option would be assessed similarly: compute the sampling distribution under the specified true mean, translate the rejection region into a z-threshold, and evaluate the tail probability corresponding to failing to reject H0. In this case the computed beta is 0.19 when rounded to two decimals.

AMS 310 Section 7.2 Quiz

查看解析

登录即可查看完整答案

类似问题

A Type II error occurs when:

A Type II error occurs when:

What is the probability of making a Type II Error in a test?

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