Calculate 90 Percent Confidence Interval For Μ Given $\bar{x}$ = 34, σ = 4, And N = 10.

Jun 18, 2025 by ADMIN 94 views

Calculating Confidence Intervals for μ: A Step-by-Step Guide

In statistical inference, estimating population parameters is a crucial task. When we want to estimate the population mean (μ), confidence intervals provide a range within which we believe the true mean lies, with a certain level of confidence. This article will walk you through the process of calculating a 90 percent confidence interval for μ, given sample data from a normal population. We'll use the provided sample information: sample mean ( $\overset{ˉ}{x}$ ) = 34, population standard deviation ( ${$ σ) = 4, and sample size (n) = 10.

Understanding Confidence Intervals

Before diving into the calculations, let's clarify what a confidence interval represents. A confidence interval is a range of values, calculated from sample data, that is likely to contain the true population parameter. The confidence level (e.g., 90 percent) indicates the probability that the interval will capture the true parameter if we repeat the sampling process many times. In other words, if we were to take many samples and construct confidence intervals for each, we would expect about 90 percent of those intervals to contain the true population mean.

Confidence intervals are essential tools in statistical analysis, offering a range within which the true population parameter is likely to fall. In many real-world scenarios, it's impractical or impossible to survey an entire population. Therefore, we rely on samples to estimate population characteristics. A confidence interval provides a measure of the uncertainty associated with this estimation, reflecting the inherent variability in sampling. The width of the interval is influenced by factors such as the sample size, the variability within the sample, and the desired level of confidence. A wider interval suggests a higher degree of uncertainty, whereas a narrower interval implies a more precise estimate. The confidence level, typically expressed as a percentage (e.g., 90%, 95%, 99%), reflects the proportion of times the interval is expected to contain the true population parameter if the sampling process were repeated multiple times. For instance, a 95% confidence interval suggests that if we were to draw numerous samples and compute intervals for each, approximately 95% of these intervals would capture the true parameter value. However, it's crucial to interpret confidence intervals correctly. A common misconception is that a 95% confidence interval implies a 95% probability that the true parameter falls within the calculated interval. Instead, the confidence level refers to the long-run frequency of capturing the true parameter across multiple samples. The true parameter is a fixed value, and it either lies within the calculated interval or it doesn't. The confidence interval provides a range of plausible values based on the available data, acknowledging the uncertainty inherent in statistical estimation.

The Formula for Confidence Intervals

When the population standard deviation ( ${}$ σ) is known and the population is normally distributed, or the sample size is large enough (typically n ≥ 30) that the sampling distribution of the sample mean is approximately normal (Central Limit Theorem), we can use the z-distribution to calculate the confidence interval. The formula for a confidence interval for the population mean (μ) is:

Confidence Interval = ${\bar{x}}$ ± z * ( ${}$ σ / √n)

Where:

${\bar{x}}$ is the sample mean.
z is the z-score corresponding to the desired confidence level.
${}$ σ is the population standard deviation.
n is the sample size.

Understanding the formula for confidence intervals is crucial for interpreting the results and applying them effectively in statistical analysis. This formula, specifically tailored for scenarios where the population standard deviation is known or the sample size is sufficiently large, allows us to estimate a range within which the true population mean is likely to reside. The formula is composed of several key components, each contributing to the precision and reliability of the estimated interval. At the heart of the formula lies the sample mean ( ${\bar{x}}$ ), which serves as our point estimate for the population mean. It represents the average value calculated from the sample data and forms the center of our confidence interval. However, since the sample mean is just one estimate from a potentially infinite number of samples, we need to account for the uncertainty associated with this estimation. This is where the standard error of the mean ( ${}$ σ / √n) comes into play. The standard error quantifies the variability of sample means around the true population mean. It reflects how much the sample means are likely to vary from the true population mean due to random sampling. A smaller standard error indicates that the sample means are clustered more closely around the true population mean, leading to a more precise estimate. The Z-score, denoted as 'z' in the formula, is a critical factor that determines the width of the confidence interval. It represents the number of standard deviations away from the mean in a standard normal distribution, corresponding to the desired level of confidence. For instance, a 95% confidence level corresponds to a Z-score of approximately 1.96, indicating that we are capturing the central 95% of the standard normal distribution. A larger Z-score results in a wider confidence interval, reflecting a higher level of confidence and a broader range of plausible values for the population mean. The sample size (n) also plays a significant role in the precision of the confidence interval. As the sample size increases, the standard error decreases, leading to a narrower confidence interval. This makes intuitive sense, as larger samples provide more information about the population, allowing for a more precise estimation of the population mean. In summary, the formula for confidence intervals elegantly combines the sample mean, standard error, Z-score, and sample size to provide a range of values that is likely to contain the true population mean. By understanding the interplay of these components, we can better interpret confidence intervals and make informed decisions based on statistical evidence.

Steps to Calculate the 90 Percent Confidence Interval

Let's apply the formula to our given data.

Identify the Given Values:
- Sample mean ( ${\bar{x}}$ ) = 34
- Population standard deviation ( ${}$ σ) = 4
- Sample size (n) = 10
- Confidence level = 90 percent
Determine the Z-score:

For a 90 percent confidence interval, we need to find the z-score that corresponds to the middle 90 percent of the standard normal distribution. This leaves 5 percent in each tail (100% - 90% = 10%, divided by 2 for two tails). We can use a z-table or a calculator to find the z-score corresponding to 0.95 (0.90 + 0.05) or 0.05. The z-score for a 90 percent confidence level is approximately 1.645.

Determining the Z-score is a pivotal step in constructing confidence intervals, as it directly influences the width of the interval and, consequently, the precision of our estimate. The Z-score, also known as the standard score, represents the number of standard deviations a particular data point is away from the mean in a standard normal distribution. In the context of confidence intervals, the Z-score serves as a critical link between the desired level of confidence and the boundaries of the interval. The confidence level, typically expressed as a percentage, reflects the proportion of times we expect the interval to contain the true population parameter if we were to repeat the sampling process multiple times. For instance, a 95% confidence level implies that if we drew numerous samples and calculated intervals for each, approximately 95% of those intervals would capture the true parameter value. To determine the appropriate Z-score for a given confidence level, we need to consider the distribution of sample means. According to the Central Limit Theorem, the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the underlying population distribution. This theorem allows us to leverage the properties of the standard normal distribution, which has a mean of 0 and a standard deviation of 1, to find the Z-score. The Z-score corresponds to the cumulative probability associated with the desired confidence level. For a two-tailed confidence interval, we need to divide the significance level (1 - confidence level) by 2 to account for the area in each tail of the distribution. This is because we are interested in capturing the central portion of the distribution, leaving an equal amount of probability in both tails. For example, to construct a 95% confidence interval, we have a significance level of 0.05, and we divide it by 2 to get 0.025. This means we need to find the Z-score that corresponds to a cumulative probability of 0.975 (1 - 0.025) in the standard normal distribution. We can utilize a Z-table or a statistical calculator to find this Z-score. A Z-table provides the cumulative probabilities for various Z-scores in the standard normal distribution. By looking up the value closest to 0.975 in the table, we can find the corresponding Z-score, which is approximately 1.96. Alternatively, a statistical calculator can directly compute the Z-score for a given cumulative probability. In summary, determining the Z-score is a crucial step in constructing confidence intervals. It allows us to translate the desired level of confidence into a specific number of standard deviations from the mean in the standard normal distribution, which ultimately determines the width of the interval. By utilizing Z-tables or statistical calculators, we can accurately find the Z-score and ensure the reliability of our confidence interval estimate.
Calculate the Margin of Error:

Margin of Error = z * ( ${}$ σ / √n) Margin of Error = 1.645 * (4 / √10) Margin of Error ≈ 1.645 * (4 / 3.162) Margin of Error ≈ 1.645 * 1.265 Margin of Error ≈ 2.081

Calculating the margin of error is a crucial step in constructing confidence intervals, as it quantifies the uncertainty associated with our estimate of the population parameter. The margin of error represents the range within which the true population parameter is likely to fall around our sample statistic. It is influenced by several factors, including the sample size, the variability within the sample, and the desired level of confidence. In essence, the margin of error provides a buffer around our point estimate, acknowledging the inherent limitations of sampling and the possibility of random variation. A smaller margin of error indicates a more precise estimate, while a larger margin of error suggests greater uncertainty. The margin of error is calculated using a specific formula that depends on the type of confidence interval being constructed. For instance, when estimating the population mean with a known population standard deviation, the margin of error is calculated as the product of the critical value (Z-score or t-score) and the standard error of the mean. The critical value is determined by the desired level of confidence and the distribution of the sample statistic. The standard error of the mean, on the other hand, reflects the variability of sample means around the true population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. The formula for the margin of error elegantly captures the interplay of these factors. The critical value reflects the desired level of confidence, with larger values corresponding to wider intervals and higher confidence levels. The standard error quantifies the variability of the sample means, with smaller values indicating more precise estimates. The sample size also plays a crucial role, as larger samples generally lead to smaller margins of error. Understanding the components of the margin of error is essential for interpreting confidence intervals effectively. A narrow margin of error suggests that our sample statistic is a good representation of the population parameter, while a wide margin of error indicates greater uncertainty and the need for caution in drawing conclusions. When reporting confidence intervals, it is customary to present both the point estimate and the margin of error. For example, a 95% confidence interval for the population mean might be reported as 34 ± 2.081, indicating that we are 95% confident that the true population mean falls within the range of 31.919 to 36.081. In summary, calculating the margin of error is a critical step in constructing confidence intervals, providing a measure of the uncertainty associated with our estimate of the population parameter. By understanding the factors that influence the margin of error, we can better interpret confidence intervals and make informed decisions based on statistical evidence.
Calculate the Confidence Interval:

Confidence Interval = ${\bar{x}}$ ± Margin of Error Confidence Interval = 34 ± 2.081

Lower Limit = 34 - 2.081 = 31.919 Upper Limit = 34 + 2.081 = 36.081

Therefore, the 90 percent confidence interval for μ is approximately (31.919, 36.081).

Calculating the confidence interval is the culminating step in statistical estimation, providing a range of plausible values for the population parameter based on sample data. The confidence interval is constructed around a point estimate, which is the best single-value estimate for the population parameter. However, since point estimates are subject to sampling variability, confidence intervals provide a more informative approach by acknowledging the uncertainty associated with the estimation process. The confidence interval is defined by two limits: the lower limit and the upper limit. These limits are calculated by adding and subtracting the margin of error from the point estimate. The margin of error, as discussed earlier, quantifies the uncertainty associated with the estimation process and is influenced by factors such as sample size, variability within the sample, and the desired level of confidence. The width of the confidence interval, determined by the margin of error, reflects the precision of our estimate. A narrow interval suggests a more precise estimate, while a wider interval indicates greater uncertainty. The confidence level, typically expressed as a percentage (e.g., 90%, 95%, 99%), reflects the proportion of times we expect the interval to contain the true population parameter if we were to repeat the sampling process multiple times. For instance, a 95% confidence interval suggests that if we drew numerous samples and calculated intervals for each, approximately 95% of those intervals would capture the true parameter value. It's crucial to interpret confidence intervals correctly. A common misconception is that a 95% confidence interval implies a 95% probability that the true parameter falls within the calculated interval. Instead, the confidence level refers to the long-run frequency of capturing the true parameter across multiple samples. The true parameter is a fixed value, and it either lies within the calculated interval or it doesn't. The confidence interval provides a range of plausible values based on the available data, acknowledging the uncertainty inherent in statistical estimation. When reporting confidence intervals, it is essential to clearly state the confidence level and the limits of the interval. For example, a 95% confidence interval for the population mean might be reported as (31.919, 36.081), indicating that we are 95% confident that the true population mean falls within this range. In summary, calculating the confidence interval is the final step in statistical estimation, providing a range of plausible values for the population parameter. By understanding the components of the confidence interval and interpreting it correctly, we can make informed decisions based on statistical evidence.

Interpretation

We are 90 percent confident that the true population mean (μ) lies within the interval (31.919, 36.081). This means that if we were to take many samples and construct 90 percent confidence intervals for each, about 90 percent of those intervals would contain the true population mean.

Rounding the Answers

As requested, we have rounded the answers to 4 decimal places.

Conclusion

In this article, we have demonstrated how to calculate a 90 percent confidence interval for the population mean (μ) when the population standard deviation is known and the population is normally distributed. By following these steps, you can effectively estimate population parameters and make informed decisions based on sample data. Understanding confidence intervals is essential for anyone working with statistical data, as it provides a valuable tool for quantifying the uncertainty associated with estimations.