The Quality Control Manager At A Light Bulb Factory Needs To Estima
The quality control manager at a light-bulb factory needs to estimate the mean life of a new type of light-bulb. The population standard deviation is assumed to be 50 hours. A random sample of 30 light-bulbs shows a sample mean life of 475 hours. Construct and explain a 90% confidence interval estimate of the population mean life of the new light-bulb.
A survey of first-time home buyers found that the sample mean annual income was $46,000. Assume that the survey used a sample of 20 first-time home buyers and that the sample standard deviation was $1,200. Compute and explain a 95% confidence interval estimate of the population mean.
A telephone poll of 750 American adults asked "where would you rather go in your spare time?" One response, by 275 adults, was "the Mall". Compute and explain a 99% confidence interval estimate of the proportion of all American adults who would respond "the Mall".
For problem #1 above, what size sample would be needed to achieve a margin of error of 15 hours or less?
Paper For Above instruction
Estimation of population parameters using confidence intervals is a fundamental aspect of inferential statistics, enabling researchers and managers to make informed decisions about populations based on sample data. In the case of the light bulb factory, the goal was to estimate the mean life span of a new light bulb type with a specified confidence level. Similarly, confidence interval calculations are employed in diverse fields, including economics and public opinion polling, as demonstrated in the subsequent examples.
Confidence Interval for Mean Life of Light Bulb
The first scenario involves estimating the average lifespan of a new light bulb, where the sample mean is 475 hours, and the population standard deviation is known to be 50 hours. The sample size is 30, which provides sufficient data for a z-interval (since the population standard deviation is known). The confidence level is 90%, corresponding to a z-value of approximately 1.645 for a two-tailed test (ZTable, 2023).
The formula for the confidence interval when the population standard deviation is known is:
CI = sample mean ± z*(σ/√n)
Substituting the known values:
CI = 475 ± 1.645 * (50 / √30) ≈ 475 ± 1.645 * 9.128 ≈ 475 ± 15.0
The 90% confidence interval is approximately (460.0 hours, 490.0 hours). This interval suggests we are 90% confident that the true mean lifespan of this light bulb type falls within this range. This information helps quality control managers assess whether the new product meets durability standards and design expectations.
Confidence Interval for Mean Income of First-Time Home Buyers
The second example deals with estimating the average annual income of first-time home buyers based on a sample of 20 individuals. Since the sample standard deviation ($1,200) is used, and the sample size is small, the t-distribution is appropriate.
The sample mean is $46,000, with a sample standard deviation (s) of $1,200, and degrees of freedom (df) = 19. For a 95% confidence level, the critical t-value (t_{0.025,19}) is approximately 2.093 (t-Distribution table, 2023).
The confidence interval formula is:
CI = x■ ± t*(s/√n)
Calculating:
CI = 46000 ± 2.093 * (1200 / √20) ≈ 46000 ± 2.093 * 268.3 ≈ 46000 ± 561.0
The 95% confidence interval is approximately ($45,439, $45,561). This narrow interval gives a precise estimate of the average income of first-time home buyers, valuable for lenders and real estate developers in gauging neighborhood economic profiles.
Confidence Interval for Proportion Responding "the Mall"
The third example estimates the proportion of American adults who prefer to go to the mall during leisure time. From a sample of 750, 275 respondents chose "the Mall". The sample proportion p■ is:
p■ = 275 / 750 ≈ 0.3667
Using a 99% confidence level, the critical z-value is approximately 2.576. The standard error (SE) is calculated as:
SE = √[p■(1 - p■)/n] ≈ √[0.3667 * 0.6333 / 750] ≈ √(0.232 / 750) ≈ 0.0176
The confidence interval is:
p■ ± z * SE ≈ 0.3667 ± 2.576 * 0.0176 ≈ 0.3667 ± 0.0454
Thus, the 99% confidence interval for the true proportion of adults favoring the mall is approximately (0.3213, 0.4121). This interval informs marketers and retail planners about the popularity of mall shopping among adults, aiding strategic decisions.
Determining Sample Size for Desired Margin of Error
The final question asks what sample size is necessary to achieve a margin of error (E) of 15 hours or less in estimating the mean life of the light bulbs. Using the same confidence level (90%) and population standard deviation (50 hours), the formula for sample size is:
n = (z*σ / E)^2
With z = 1.645, σ = 50, and E = 15:
n = (1.645 * 50 / 15)^2 ≈ (1.645 * 3.333)^2 ≈ (5.482)^2 ≈ 30.07
Since sample size must be an integer, the required sample size is approximately 31. Increasing the sample size ensures the margin of error does not exceed 15 hours, enhancing the reliability of the mean estimate.
Conclusion
Confidence intervals are vital for making statistically informed decisions across various fields. The calculations demonstrated above provide meaningful insights into product durability, economic status, and public preferences, emphasizing the importance of choosing the correct confidence level, sample size, and statistical method based on known parameters. Properly applied, confidence interval estimation reduces uncertainty and supports data-driven decision-making processes in manufacturing, economics, and public opinion research.
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