# Confidence Interval

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Confidence intervals are a fundamental concept in statistics and data analysis. They provide a way to estimate the range within which a population parameter, such as a population mean or proportion, is likely to fall.

### What is a Confidence Interval?

A confidence interval is a range of values that is constructed around a sample statistic to estimate an unknown population parameter. It provides a level of confidence that the true parameter value falls within the specified range. In other words, it quantifies the uncertainty associated with our estimate.

or in other words:

A confidence interval is a statistical concept that provides a range of values within which the true population parameter is likely to fall with a certain level of confidence. It is calculated using a sample from the population and employs statistical procedures to determine the interval.

## Assumptions/Conditions

To construct a confidence interval, certain conditions must be met. These include:

- a random sample that is representative of the population,

- a normal distribution of sampling means (for the mean), and

- a large enough sample size.

These conditions are necessary for the validity of the confidence interval and to ensure accurate estimation.

## The width of the confidence interval

The width of the confidence interval is influenced by three key factors: the desired degree of confidence.,  the variability within the sample and the sample size.

The degree of confidence, often expressed as a percentage (such as 90% or 95%), indicates how certain we are that the true population parameter falls within the interval. A higher degree of confidence results in a wider interval.

The variability in the sample is measured by the standard deviation. The samples with higher standard deviations have wider confidence intervals.

The sample size is an important factor in the construction of a confidence interval. Generally, larger sample sizes produce narrower intervals and the degree of precision increases as the sample size increases.

## Interpretation of the result

Once you have calculated the confidence interval, you can interpret it as follows: there is a 95% (or whatever confidence level you choose) chance that the true population mean lies within the calculated interval. The wider the confidence interval, the less precise the estimate of the mean.

## Calculating Confidence Intervals

The formula for calculating a confidence interval depends on the type of data and the parameter being estimated. Here, we'll cover two common cases: confidence intervals for the population mean and confidence intervals for the population proportion.

#### Confidence Interval for the Population Mean

When estimating the population mean (μ) with a confidence interval, you typically use the t-distribution (for small samples) or the z-distribution (for large samples) along with the standard error (SE) of the sample mean.

The formula for a confidence interval for the population mean is:

$$\bar{x} \pm Z\left(\frac{s}{\sqrt{n}}\right)$$

Where:

•  $$\bar{x}$$ is the sample mean.
• Z is the critical value from the standard normal distribution (or t-distribution for smaller samples), often chosen to correspond to a specific confidence level (e.g., 1.96 for a 95% confidence interval).
• s is the sample standard deviation.
• n is the sample size.

#### Confidence Interval for the Population Proportion

When estimating the population proportion (p), you use the standard normal distribution (z-distribution) along with the standard error of the sample proportion (SE).

The formula for a confidence interval for the population proportion is:

$$\hat{p} \pm Z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$$

Where:

•  $$\hat{p}$$ is the sample proportion.
• Z is the critical value from the standard normal distribution, chosen based on the desired confidence level.
• n is the sample size.

## Confidence Interval Practice Problems

### Practice Problem 1: Confidence Interval for a Mean with Known Population Standard Deviation

Scenario: A researcher wants to estimate the average height of adult males in a particular city. A random sample of 100 adult males is selected from the population. The population standard deviation is known to be 2.5 inches. Calculate a 95% confidence interval.

Solution:

Step 1: Obtain a random sample and collect the necessary information. Sample size (n) = 100, population standard deviation (σ) = 2.5 inches, confidence level (1 - α) = 0.95.

Step 2: Calculate the standard error (SE) using the formula SE = σ / √n. In this case, SE = 2.5 / √100 = 0.25.

Step 3: Determine the margin of error (ME) by multiplying the critical value (Z) corresponding to the desired confidence level with the standard error. For a 95% confidence level, the Z value is approximately 1.96. So, ME = 1.96 * 0.25 = 0.49.

Step 4: Construct the confidence interval by subtracting and adding the margin of error to the sample mean. The mean of the sample is denoted as x̄. Confidence interval = x̄ ± ME. Hence, the confidence interval is x̄ ± 0.49.

### Practice Problem 2: Confidence Interval for a Proportion

Scenario: A survey was conducted to estimate the proportion of adults who own a smartphone. A random sample of 500 adults was selected, and 320 of them owned a smartphone. Calculate a 90% confidence interval for the proportion of adults who own a smartphone.

Solution:

Step 1: Obtain a random sample and collect the necessary information. Sample size (n) = 500, number of successes (x) = 320, confidence level (1 - α) = 0.90.

Step 2: Calculate the sample proportion (p) by dividing the number of successes by the sample size. In this case, p = 320 / 500 = 0.64.

Step 3: Determine the standard error (SE) using the formula SE = √(p(1-p)/n). SE = √((0.64 * (1-0.64)) / 500) ≈ 0.021.

Step 4: Determine the margin of error (ME) by multiplying the critical value (Z) corresponding to the desired confidence level with the standard error. For a 90% confidence level, the Z value is approximately 1.645. So, ME = 1.645 * 0.021 ≈ 0.0353.

Step 5: Construct the confidence interval by subtracting and adding the margin of error to the sample proportion. Confidence interval = p ± ME. Hence, the confidence interval is 0.64 ± 0.0353.

These practice problems demonstrate how to construct confidence intervals for both means and proportions. By following the step-by-step instructions and understanding the rationale behind each calculation, you can confidently estimate population parameters based on sample data.

## Confidence Interval Calculation App:

Confidence Interval Calculator

Conditions: Samples are random, with a normal distribution. For sample size greater than 30 normal distribution and for sample size less than 30 Student's t-distribution is assumed.

 function changeInputFields() { const ciType = document.getElementById('ciTypeSelector').value; document.getElementById('meanFields').style.display = ciType === 'mean' ? 'block' : 'none'; document.getElementById('proportionFields').style.display = ciType === 'proportion' ? 'block' : 'none'; } function normalDistribution(x, mean, stdDev) { return Math.exp(-0.5 * Math.pow((x - mean) / stdDev, 2)) / (stdDev * Math.sqrt(2 * Math.PI)); } function computeCI() { const ciType = document.getElementById('ciTypeSelector').value; const n = parseInt(document.getElementById('sampleSize').value); const alpha = 1 - parseFloat(document.getElementById('confidenceLevel').value); let resultContainer = document.getElementById('result'); let ci, mean, stdDev; if (isNaN(n) || isNaN(alpha)) { resultContainer.innerHTML = "Please fill all fields correctly."; return; } let zOrTValue; if (n < 30) { zOrTValue = jStat.studentt.inv(1 - alpha / 2, n - 1); } else { zOrTValue = jStat.normal.inv(1 - alpha / 2, 0, 1); } if (ciType === 'mean') { mean = parseFloat(document.getElementById('mean').value); stdDev = parseFloat(document.getElementById('stdDev').value); if (isNaN(mean) || isNaN(stdDev)) { resultContainer.innerHTML = "Please fill all fields correctly."; return; } ci = zOrTValue * (stdDev / Math.sqrt(n)); } else { const successCount = parseInt(document.getElementById('successCount').value); if (isNaN(successCount)) { resultContainer.innerHTML = "Please fill all fields correctly."; return; } mean = successCount / n; stdDev = Math.sqrt((mean * (1 - mean)) / n); ci = zOrTValue * stdDev; } resultContainer.innerHTML = <span class='bold-text'>Confidence Interval: [${(mean - ci).toFixed(4)},${(mean + ci).toFixed(4)}]</span>; // Plotting const x = Array.from({ length: 500 }, (_, i) => mean - 4*stdDev + i * 8 * stdDev / 500); const y = x.map(val => normalDistribution(val, mean, stdDev)); const trace1 = { x: x, y: y, mode: 'lines', type: 'scatter', name: 'Normal Distribution' }; const xShaded = x.filter(val => val >= mean - ci && val <= mean + ci); const yShaded = xShaded.map(val => normalDistribution(val, mean, stdDev)); const trace2 = { x: xShaded, y: yShaded, fill: 'tozeroy', type: 'scatter', name: 'Confidence Interval' }; const layout = { title: 'Confidence Interval', xaxis: { title: 'X' }, yaxis: { title: 'Density' } }; Plotly.newPlot('plotDiv', [trace1, trace2], layout); } 

Posted on September 23, 2023 by  Quality Gurus

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