What is the first step to calculate the 90% confidence interval for the given sample?

The first step is to compute the sample mean and sample standard deviation of the data points: 7.9, 8.3, 8.4, 9.6, 7.7, and 8.1.

How do you calculate the sample mean for the sample data provided?

Add all the sample values together and divide by the number of observations: (7.9 + 8.3 + 8.4 + 9.6 + 7.7 + 8.1) / 6.

What is the formula for the sample standard deviation, and how is it applied here?

The sample standard deviation is calculated using s = sqrt[ Σ(xi - x̄)² / (n - 1) ], where xi are data points, x̄ is the mean, and n is the sample size.

How do you determine the t-value for a 90% confidence interval with 5 degrees of freedom?

Use a t-distribution table or calculator to find the t-value for a 90% confidence level with df = n - 1 = 5, which is approximately 2.015.

What is the formula for the confidence interval in this context?

The 90% confidence interval is given by: x̄ ± t (s / sqrt(n)), where x̄ is the mean, s is the standard deviation, n is the sample size, and t is the t-value.

Can you compute the sample mean for the sample data?

Yes. The mean x̄ = (7.9 + 8.3 + 8.4 + 9.6 + 7.7 + 8.1) / 6 = 49.0 / 6 ≈ 8.167.

What is the calculated standard deviation for the sample?

First, find each deviation squared, sum them, divide by 5, then take the square root: s ≈ 0.736.

What is the final 90% confidence interval based on these calculations?

Using the mean 8.167, standard deviation 0.736, t-value 2.015, and n=6: CI ≈ 8.167 ± 2.015 (0.736 / sqrt(6)) ≈ 8.167 ± 2.015 0.300 ≈ (7.557, 8.777).

Why is it important to use the t-distribution instead of the normal distribution in this calculation?

Because the sample size is small (n < 30), the t-distribution accounts for additional uncertainty in estimating the population mean, providing a more accurate confidence interval.

Calculate The 90% Confidence Interval For The Following Sample Sample: 7.9, 8.3, 8.4, 9.6, 7.7, 8.1,

Calculate The 90% Confidence Interval For The Following Sample Sample: 7.9, 8.3, 8.4, 9.6, 7.7, 8.1, is a fundamental statistical task that helps researchers and analysts understand the range within which a population parameter, such as the mean, is likely to fall with a specified level of confidence. Confidence intervals are essential tools in inferential statistics, providing a measure of uncertainty around a sample estimate. This article will guide you through the process of calculating the 90% confidence interval for a given sample, explain the underlying concepts, and illustrate each step with the provided data set.

Understanding Confidence Intervals

What Is a Confidence Interval?

A confidence interval (CI) is a range of values, derived from sample data, that is believed to contain the true population parameter (e.g., the mean) with a certain probability, known as the confidence level. In this case, a 90% confidence level indicates that if the same sampling process were repeated numerous times, approximately 90% of the calculated intervals would contain the true population mean.

Why Use Confidence Intervals?

Confidence intervals offer several advantages:

Quantify uncertainty in estimates

Provide a range that is likely to contain the true parameter

Assist in decision-making by understanding the precision of estimates

Given Data and Objective

The sample data provided is:

Our goal is to calculate the 90% confidence interval for the population mean based on this sample.

Step-by-Step Calculation of the 90% Confidence Interval

Step 1: Calculate the Sample Mean (Average)

The sample mean (\(\bar{x}\)) is the sum of all sample values divided by the number of observations.

Sum of sample values:

7.9 + 8.3 + 8.4 + 9.6 + 7.7 + 8.1 = 49.0

Number of observations (\(n\)):

Sample mean:

\[ \bar{x} = \frac{49.0}{6} \approx 8.1667 \]

Interpretation: The average of your sample data is approximately 8.17.

Step 2: Calculate the Sample Standard Deviation (s)

The sample standard deviation measures the variability or dispersion of the data points around the mean.

Calculate each data point's deviation from the mean and square it:

| Data Point | Deviation (\(xi - \bar{x}\)) | Squared Deviation \((xi - \bar{x})^2\) | |--------------|------------------------------|-------------------------------------------| | 7.9 | 7.9 - 8.1667 = -0.2667 | 0.0711 | | 8.3 | 8.3 - 8.1667 = 0.1333 | 0.0178 | | 8.4 | 8.4 - 8.1667 = 0.2333 | 0.0544 | | 9.6 | 9.6 - 8.1667 = 1.4333 | 2.0560 | | 7.7 | 7.7 - 8.1667 = -0.4667 | 0.2178 | | 8.1 | 8.1 - 8.1667 = -0.0667 | 0.0044 |

Sum of squared deviations:

\[ 0.0711 + 0.0178 + 0.0544 + 2.0560 + 0.2178 + 0.0044 = 2.4215 \]

Sample variance (\(s^2\)):

\[ s^2 = \frac{\text{Sum of squared deviations}}{n - 1} = \frac{2.4215}{5} \approx 0.4843 \]

Standard deviation (\(s\)):

\[ s = \sqrt{0.4843} \approx 0.6960 \]

Interpretation: The variability of the data points around the mean is approximately 0.696.

Step 3: Determine the Appropriate Critical t-Value

Since the sample size is small (\(n=6\)), and the population standard deviation is unknown, we use the Student's t-distribution.

Degrees of freedom (\(df\)):

\[ df = n - 1 = 5 \]

Critical t-value for a 90% confidence level:

Using a t-distribution table or calculator, for \(df = 5\) and a 90% confidence interval (which corresponds to \(\alpha=0.10\), and thus 0.05 in each tail):

\[
t_{0.95, 5} \approx 2.015
\]

Note: The t-value varies slightly depending on the table, but 2.015 is a common approximation.

Step 4: Calculate the Margin of Error (ME)

The margin of error quantifies the extent of the interval on either side of the sample mean.

\[
ME = t_{\alpha/2, df} \times \frac{s}{\sqrt{n}}
\]

Standard error (SE):

\[
SE = \frac{s}{\sqrt{n}} = \frac{0.6960}{\sqrt{6}} \approx \frac{0.6960}{2.449} \approx 0.284
\]

Margin of error:

\[
ME = 2.015 \times 0.284 \approx 0.573
\]

Interpretation: The margin of error is approximately 0.573.

Final Step: Calculate the Confidence Interval

Lower bound:

\[ \bar{x} - ME = 8.1667 - 0.573 \approx 7.593 \]

Upper bound:

\[ \bar{x} + ME = 8.1667 + 0.573 \approx 8.740 \]

Conclusion: The 90% confidence interval for the population mean based on this sample is approximately (7.593, 8.740).

Interpreting the Confidence Interval

This interval suggests that we can be 90% confident that the true population mean falls between 7.593 and 8.740. It provides a range of plausible values for the mean, accounting for sampling variability.

Importance of Confidence Intervals in Statistical Analysis

Confidence intervals are invaluable in various fields such as medical research, economics, manufacturing, and social sciences. They help in:

Assessing the reliability of sample estimates

Making informed decisions based on data

Communicating uncertainty effectively

Summary of the Calculation Process

To recap, calculating a confidence interval involves the following steps:

Compute the sample mean (\(\bar{x}\))

Calculate the sample standard deviation (s)

Identify the appropriate critical t-value based on degrees of freedom and confidence level

Calculate the standard error (SE)

Determine the margin of error (ME)

Construct the interval by adding and subtracting ME from the sample mean

By following these steps, you can accurately estimate the interval within which the true population parameter is likely to lie with a specified confidence level.

Final Thoughts

Calculating the 90% confidence interval for a sample like 7.9, 8.3, 8.4, 9.6, 7.7, 8.1 provides valuable insights into the data's underlying population. Whether you're conducting scientific research, quality control, or business analysis, understanding how to compute and interpret confidence intervals enhances the reliability and clarity of your conclusions. Remember, the confidence interval is not a guarantee but a probabilistic estimate, reflecting the inherent variability of sampling processes.

If you want to learn more about confidence intervals, statistical inference, or data analysis techniques, consider exploring additional resources or consulting with a statistician to deepen your understanding of these essential tools.