The quality control manager at a light bulb factory needs to estimate the mean life of a large shipment of light bulbs. The standard deviation is 98 hours. A random sample of 49 light bulbs indicated a sample mean life of 370 hours. Complete parts (a) through (d) below. a. Construct a 95% confidence interval estimate for the population mean life of light bulbs in this shipment. The 95% confidence interval estimate is from a lower limit of 342.6 hours to an upper limit of 397.4 hours. (Round to one decimal place as needed.) b. Do you think that the manufacturer has the right to state that the lightbulbs have a mean life of 420 hours? Explain. Based on the sample data, the manufacturer does not have the right to state that the lightbulbs have a mean life of 420 hours. A mean of 420 hours is more than 3 standard errors above the sample mean, so it is highly unlikely that the lightbulbs have a mean life of 420 hours. c. Must you assume that the population light bulb life is normally distributed? Explain. O A. Yes, the sample size is not large enough for the sampling distribution of the mean to be approximately normal by the Central Limit Theorem. O B. No, since o is known, the sampling distribution of the mean does not need to be approximately normally distributed. OC. Yes, the sample size is too large for the sampling distribution of the mean to be approximately normal by the Central Limit Theorem. O D. No, since o is known and the sample size is large enough, the sampling distribution of the mean is approximately normal by the Central Limit Theorem. d. Suppose the standard deviation changes to 77 hours. What are your answers in (a) and (b)? The 95% confidence interval estimate would be from a lower limit of hours to an upper limit of hours. (Round to one decimal place as needed.) Based on the sample data and a standard deviation of 77 hours, the manufacturer V the right to state that the lightbulbs have a mean life of 420 hours. A mean of 420 hours is standard errors V the sample mean, so it is V that the lightbulbs have a mean life of 420 hours.

MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Question

Part D, please.

# Confidence Interval and Population Mean Estimation

## Overview

The quality control manager at a light bulb factory is tasked with estimating the mean life of a large shipment of light bulbs. With a standard deviation of 98 hours, a random sample of 49 bulbs yields a sample mean life of 370 hours. Below is the analysis and interpretation of the results.

### a. Constructing a 95% Confidence Interval

- **Confidence Interval Calculation:**
  - **Lower Limit:** 342.6 hours 
  - **Upper Limit:** 397.4 hours 
  - **Note:** The interval is rounded to one decimal place.

### b. Manufacturer's Right to Claim Mean Life of 420 Hours

- **Conclusion:** The manufacturer does not have the right to claim a mean life of 420 hours.
  - **Reasoning:** A mean of 420 hours is more than 3 standard errors above the sample mean, making it highly unlikely for the light bulbs to have a mean life of 420 hours.

### c. Assumption of Normal Distribution

- **Analysis:**
  - **Correct Statement:** No, since the sample size is large enough, the sampling distribution of the mean is approximately normal by the Central Limit Theorem (Option D).

### d. Impact of Changing Standard Deviation

- **New Standard Deviation:** 77 hours
- **Revised 95% Confidence Interval:**
  - **Lower Limit:** Calculation required
  - **Upper Limit:** Calculation required
  - **Note:** Values are to be rounded to one decimal place.
- **Manufacturer's Right Analysis:**
  - With a standard deviation of 77 hours, the manufacturer's claim is evaluated based on how the mean of 420 hours compares to the adjusted sample mean and standard errors.

## Additional Notes

This example illustrates the application of confidence intervals and normal distribution assumptions in quality control and statistical analysis. Understanding these concepts aids in making informed decisions about product claims and quality assurance.
Transcribed Image Text:# Confidence Interval and Population Mean Estimation ## Overview The quality control manager at a light bulb factory is tasked with estimating the mean life of a large shipment of light bulbs. With a standard deviation of 98 hours, a random sample of 49 bulbs yields a sample mean life of 370 hours. Below is the analysis and interpretation of the results. ### a. Constructing a 95% Confidence Interval - **Confidence Interval Calculation:** - **Lower Limit:** 342.6 hours - **Upper Limit:** 397.4 hours - **Note:** The interval is rounded to one decimal place. ### b. Manufacturer's Right to Claim Mean Life of 420 Hours - **Conclusion:** The manufacturer does not have the right to claim a mean life of 420 hours. - **Reasoning:** A mean of 420 hours is more than 3 standard errors above the sample mean, making it highly unlikely for the light bulbs to have a mean life of 420 hours. ### c. Assumption of Normal Distribution - **Analysis:** - **Correct Statement:** No, since the sample size is large enough, the sampling distribution of the mean is approximately normal by the Central Limit Theorem (Option D). ### d. Impact of Changing Standard Deviation - **New Standard Deviation:** 77 hours - **Revised 95% Confidence Interval:** - **Lower Limit:** Calculation required - **Upper Limit:** Calculation required - **Note:** Values are to be rounded to one decimal place. - **Manufacturer's Right Analysis:** - With a standard deviation of 77 hours, the manufacturer's claim is evaluated based on how the mean of 420 hours compares to the adjusted sample mean and standard errors. ## Additional Notes This example illustrates the application of confidence intervals and normal distribution assumptions in quality control and statistical analysis. Understanding these concepts aids in making informed decisions about product claims and quality assurance.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Recommended textbooks for you
MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman