Calculate 96% Confidence Interval For Light Bulb Lifetime
In the realm of statistical analysis, understanding confidence intervals is crucial for making informed decisions based on sample data. In this article, we delve into the process of calculating a confidence interval for the average lifetime of light bulbs manufactured by a firm. Our focus will be on a scenario where the lifetimes are approximately normally distributed, and we aim to determine the width of a 96% confidence interval. This exploration will not only enhance your understanding of confidence intervals but also provide practical insights into how they are applied in real-world scenarios.
Key Concepts: Confidence Intervals and Normal Distribution
Before we dive into the calculations, it's essential to grasp the fundamental concepts of confidence intervals and the normal distribution. A confidence interval is a range of values within which we are reasonably certain that the true population parameter lies. In simpler terms, it provides a plausible range for the true average lifetime of all light bulbs produced by the firm, based on the data collected from a sample. The level of confidence, in this case, 96%, represents the percentage of times that the confidence interval would contain the true population mean if we were to repeat the sampling process multiple times.
The normal distribution, often referred to as the Gaussian distribution or bell curve, is a probability distribution that is symmetrical around the mean. Many natural phenomena, including the lifetimes of light bulbs, tend to follow a normal distribution. This distribution is characterized by its mean (average) and standard deviation (a measure of the spread of the data). In our scenario, we are given that the light bulb lifetimes are approximately normally distributed, which allows us to utilize the properties of the normal distribution in our calculations. The standard deviation, a key parameter, quantifies the dispersion or variability within the dataset. A smaller standard deviation suggests the data points are clustered closer to the mean, whereas a larger standard deviation indicates a wider spread.
The interplay between confidence intervals and the normal distribution is pivotal in statistical inference. The normal distribution's well-defined properties, such as its symmetry and the known probabilities associated with specific standard deviations from the mean, enable us to construct confidence intervals with a specified level of confidence. For instance, we can use the standard normal distribution (a normal distribution with a mean of 0 and a standard deviation of 1) to determine the critical values needed for our 96% confidence interval. These critical values, often denoted as z-scores, represent the number of standard deviations away from the mean that correspond to the desired level of confidence. By understanding these concepts, we can proceed with the calculations, armed with the knowledge to interpret the results accurately and make informed decisions based on the data.
Problem Setup: Light Bulb Lifetimes and Sample Data
Let's lay the groundwork for our problem. We have a firm that manufactures light bulbs, and we're interested in understanding the typical lifespan of these bulbs. We know that the lifetimes of these bulbs, in terms of hours, tend to follow a normal distribution. This is a crucial piece of information because the normal distribution has well-understood statistical properties that we can leverage.
We're given the standard deviation of the light bulb lifetimes, which is 40 hours. The standard deviation tells us about the spread or variability in the lifetimes of the bulbs. A smaller standard deviation would mean the bulbs' lifetimes are more consistent, while a larger standard deviation indicates more variability. In this case, 40 hours gives us a measure of how much individual bulb lifetimes deviate from the average.
To estimate the average lifetime, the firm takes a sample of 25 light bulbs. The size of this sample is important because it affects the precision of our estimate. Larger samples generally lead to more precise estimates. For this sample, the average lifetime is found to be 780 hours. This sample mean is our best point estimate of the true average lifetime of all light bulbs the firm produces.
Our goal is to find a 96% confidence interval for the true average lifetime. This confidence interval will give us a range of values within which we can be 96% confident that the true population mean (the average lifetime of all bulbs) lies. The confidence level reflects the reliability of our estimation process. A higher confidence level, like 96%, means we want to be very sure that our interval contains the true mean.
To summarize, we have the following information:
- Population distribution: Approximately normal
- Population standard deviation: 40 hours
- Sample size: 25 light bulbs
- Sample mean: 780 hours
- Desired confidence level: 96%
With this information, we are well-equipped to calculate the confidence interval. The next step involves identifying the appropriate statistical formula and values to use, which we will delve into in the following sections.
Calculating the Confidence Interval: Step-by-Step
Now, let's break down the calculation of the 96% confidence interval step-by-step. This process involves using the given information and applying the appropriate statistical formulas. Here’s how we'll proceed:
1. Determine the Critical Value (Z-score)
The first step is to find the critical value, often denoted as a Z-score, that corresponds to our desired confidence level of 96%. The Z-score represents the number of standard deviations away from the mean in a standard normal distribution that captures the desired level of confidence. Since we want a 96% confidence interval, this means we want 96% of the area under the standard normal curve to lie within our interval, with the remaining 4% split equally between the two tails (2% in each tail).
To find the Z-score, we need to look up the value that corresponds to 1 - (0.04 / 2) = 0.98 in a standard normal distribution table or use a statistical calculator. This value is approximately 2.05. So, our critical value (Z-score) is 2.05. This Z-score tells us how many standard errors away from the sample mean we need to go to capture the middle 96% of the distribution.
2. Calculate the Standard Error
The standard error is a measure of how much the sample mean is likely to vary from the true population mean. It's calculated by dividing the population standard deviation by the square root of the sample size. In our case, the population standard deviation is 40 hours, and the sample size is 25 light bulbs. So, the standard error (SE) is:
SE = Population Standard Deviation / √Sample Size
SE = 40 / √25
SE = 40 / 5
SE = 8 hours
The standard error of 8 hours gives us an estimate of the variability we would expect in sample means if we were to take many samples of 25 light bulbs.
3. Determine the Margin of Error
The margin of error is the range of values above and below the sample mean within which the true population mean is likely to fall. It is calculated by multiplying the critical value (Z-score) by the standard error. In our case, the Z-score is 2.05, and the standard error is 8 hours. So, the margin of error (ME) is:
ME = Z-score * Standard Error
ME = 2.05 * 8
ME = 16.4 hours
The margin of error of 16.4 hours tells us how much we need to add to and subtract from the sample mean to create our confidence interval.
4. Calculate the Confidence Interval
Finally, we can calculate the confidence interval by adding and subtracting the margin of error from the sample mean. Our sample mean is 780 hours, and our margin of error is 16.4 hours. So, the confidence interval (CI) is:
CI = Sample Mean ± Margin of Error
CI = 780 ± 16.4
The lower limit of the interval is:
780 - 16.4 = 763.6 hours
And the upper limit of the interval is:
780 + 16.4 = 796.4 hours
Thus, our 96% confidence interval for the average lifetime of the light bulbs is (763.6 hours, 796.4 hours).
Interpreting the Results: What Does the Confidence Interval Tell Us?
Now that we've calculated the 96% confidence interval for the average lifetime of the light bulbs, it's crucial to understand what this interval actually tells us. The interval is (763.6 hours, 796.4 hours). But what does this range of values signify in practical terms?
A confidence interval is a range of values that we are reasonably confident contains the true population parameter. In our case, the population parameter is the true average lifetime of all light bulbs produced by the firm. We don't know this true average, but the confidence interval gives us a plausible range for it.
Our 96% confidence interval means that if we were to take many samples of 25 light bulbs and calculate a confidence interval for each sample, we would expect 96% of those intervals to contain the true average lifetime. This doesn't mean there's a 96% chance that the true average falls within our specific interval (763.6 hours, 796.4 hours). Instead, it means that our method of constructing the interval is reliable 96% of the time.
So, we can say that we are 96% confident that the true average lifetime of all light bulbs produced by the firm lies between 763.6 hours and 796.4 hours. This is a more nuanced interpretation than simply saying there's a 96% chance the true average is within this range.
It's also important to consider the width of the confidence interval. The width of the interval gives us a sense of the precision of our estimate. A narrower interval suggests a more precise estimate, while a wider interval indicates more uncertainty. In our case, the width of the interval is:
Width = Upper Limit - Lower Limit
Width = 796.4 hours - 763.6 hours
Width = 32.8 hours
The width of 32.8 hours gives us a sense of the range within which the true average might lie. If we wanted a narrower interval (a more precise estimate), we could increase the sample size or decrease the confidence level. However, these changes come with trade-offs. Increasing the sample size costs more time and resources, while decreasing the confidence level reduces our certainty that the interval contains the true average.
Determining the Width of the Confidence Interval
The primary goal of this exercise was to find the width of the 96% confidence interval. We've already calculated the interval to be (763.6 hours, 796.4 hours), and we interpreted what this interval means. Now, let's explicitly state the width of the interval and discuss its significance.
As we calculated in the previous section, the width of the confidence interval is the difference between the upper and lower limits:
Width = Upper Limit - Lower Limit
Width = 796.4 hours - 763.6 hours
Width = 32.8 hours
So, the width of the 96% confidence interval is 32.8 hours. This value is crucial because it gives us a measure of the precision of our estimate. A narrower width suggests a more precise estimate of the true average lifetime, while a wider width indicates more uncertainty.
In this context, a width of 32.8 hours means that our estimate of the true average lifetime has a margin of error of 16.4 hours on either side of the sample mean (780 hours). This can be quite informative for the firm manufacturing the light bulbs. For example, if the firm has a target average lifetime and the lower limit of the confidence interval is below that target, they might need to investigate their manufacturing process to ensure the bulbs meet the desired standard.
Factors Affecting the Width of the Confidence Interval
Several factors influence the width of a confidence interval. Understanding these factors can help us design studies and interpret results more effectively:
- Sample Size: As the sample size increases, the standard error decreases, leading to a narrower confidence interval. Larger samples provide more information about the population, resulting in a more precise estimate.
- Standard Deviation: A larger population standard deviation leads to a larger standard error and, consequently, a wider confidence interval. More variability in the population makes it harder to estimate the true mean precisely.
- Confidence Level: A higher confidence level (e.g., 99% instead of 96%) requires a larger critical value (Z-score), which results in a wider confidence interval. To be more confident that the interval contains the true mean, we need to allow for a wider range of values.
In our example, the width of 32.8 hours reflects the interplay of these factors. The sample size of 25, the standard deviation of 40 hours, and the confidence level of 96% collectively determine the precision of our estimate. By understanding how these factors influence the width of the confidence interval, we can make informed decisions about data collection and interpretation.
Conclusion: The Importance of Confidence Intervals in Quality Control
In conclusion, we've successfully calculated the 96% confidence interval for the average lifetime of light bulbs manufactured by a firm. We found the interval to be (763.6 hours, 796.4 hours), with a width of 32.8 hours. This interval provides a range of values within which we can be 96% confident that the true average lifetime of all light bulbs produced by the firm lies.
Confidence intervals are invaluable tools in statistical analysis, particularly in quality control and manufacturing. They allow us to make informed decisions based on sample data, even when we can't measure the entire population. In the context of light bulb manufacturing, the confidence interval can help the firm assess the consistency and reliability of their products. If the interval falls within acceptable limits, the firm can be confident that their manufacturing process is producing bulbs with a consistent lifespan. If the interval is too wide or falls below desired thresholds, it may indicate issues that need to be addressed.
The width of the confidence interval is a key indicator of the precision of our estimate. A narrower interval suggests a more precise estimate, while a wider interval indicates more uncertainty. The width is influenced by several factors, including sample size, standard deviation, and confidence level. Understanding these factors allows us to design studies and interpret results more effectively.
The steps we took to calculate the confidence interval include:
- Determining the critical value (Z-score) corresponding to the desired confidence level.
- Calculating the standard error, which measures the variability of sample means.
- Determining the margin of error by multiplying the critical value by the standard error.
- Calculating the confidence interval by adding and subtracting the margin of error from the sample mean.
By following these steps, we can construct confidence intervals for various parameters in a wide range of applications. Whether it's estimating the average lifetime of light bulbs, the average weight of a product, or the proportion of defective items in a batch, confidence intervals provide a framework for making data-driven decisions.
In summary, confidence intervals are essential for quality control because they:
- Provide a range of plausible values for the true population parameter.
- Quantify the uncertainty associated with our estimate.
- Help us assess the consistency and reliability of products or processes.
- Inform decisions about process improvement and quality assurance.
By mastering the concepts and calculations involved in confidence intervals, we can enhance our ability to analyze data, draw meaningful conclusions, and make informed decisions in a variety of contexts.
