Margin of Error and Confidence Intervals Revision Notes for Leaving Cert Mathematics

Margin of Error and Confidence Intervals

Introduction to sampling and surveys

When conducting research, we often want to learn about a large group of people (called the population) by studying a smaller group (called the sample). This approach is used because surveying everyone in a population would be too expensive and time-consuming.

Opinion polls are a common example of sampling in action. Before elections, newspapers commission research companies to survey a sample of voters to predict voting intentions. A typical sample might include around 1000 people to represent the entire electorate.

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The key idea is that if we collect data properly from our sample, we can make reasonable estimates about the whole population. This fundamental principle underpins all survey research and statistical inference.

Sample proportion vs population proportion

Understanding the difference between sample and population proportions is crucial for interpreting survey results.

Sample proportion (p̂): This represents the fraction or percentage of people in your sample who have a particular characteristic. For example, if 40% of people in your sample support a political party, then p̂ = 0.40.

Population proportion (p): This represents the true fraction or percentage of people in the entire population who have that characteristic. This is usually unknown and is what we're trying to estimate.

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Since we cannot know the true population proportion, we use the sample proportion p̂ as our best estimator for the unknown population proportion p. This is a fundamental concept in statistics - we use sample data to make inferences about populations.

Understanding margin of error

Sample surveys are never 100% accurate because they only look at part of the population. There is always some uncertainty or error involved in our estimates.

Margin of error provides a way to express this uncertainty. It tells us how much our sample result might differ from the true population value.

For example, if a survey shows 40% support with a margin of error of 3%, this means:

The research company is confident the true population proportion lies between 37% and 43%
We can write this as: 40% ± 3%

Calculating margin of error

The margin of error (E) for opinion polls is calculated using the formula:

$E = \frac{1}{\sqrt{n}}$

where n is the sample size.

Let's see how this works with different sample sizes:

Sample size 400: $E = \frac{1}{\sqrt{400}} = \frac{1}{20} = 0.05$ (or 5%)
Sample size 1000: $E = \frac{1}{\sqrt{1000}} ≈ 0.03$ (or 3%)
Sample size 4000: $E = \frac{1}{\sqrt{4000}} ≈ 0.016$ (or 1.6%)

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Key insight: The margin of error decreases as the sample size increases. Larger samples give more accurate results. This relationship is not linear - to halve the margin of error, you need to quadruple the sample size!

Confidence intervals

A confidence interval is a range of values that likely contains the true population proportion. It combines our sample proportion with the margin of error to create this range.

If our sample proportion is p̂ and our margin of error is E, then our confidence interval runs from:

Lower bound: p̂ - E
Upper bound: p̂ + E

We can write this formally as: p̂ - E < p < p̂ + E

Understanding confidence levels

In this course, we use a 95% confidence level. This means:

Our method for creating confidence intervals works correctly 95% of the time
If we repeated our survey method 100 times, about 95 of those confidence intervals would contain the true population proportion
We can be quite confident (but not certain) that our interval contains the true value

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Think of confidence level as the reliability of your method, not the probability that any specific interval is correct. A 95% confidence level means that if you used this method many times, 95% of the intervals you create would capture the true population parameter.

Confidence interval formula

The 95% confidence interval for a population proportion is:

$\hat{p} \pm \frac{1}{\sqrt{n}}$

This can also be written as:

$\hat{p} - \frac{1}{\sqrt{n}} < p < \hat{p} + \frac{1}{\sqrt{n}}$

Worked examples

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Worked Example 1: Flu vaccine study

A random sample of 400 people received a flu vaccine and 136 of them experienced some discomfort.

Step 1: Find the sample size

Sample size n = 400

Step 2: Calculate the margin of error

$E = \frac{1}{\sqrt{n}} = \frac{1}{\sqrt{400}} = \frac{1}{20} = 0.05$

Step 3: Find the sample proportion

$\hat{p} = \frac{136}{400} = 0.34$

Step 4: Find the confidence interval

$\hat{p} \pm \frac{1}{\sqrt{n}} = 0.34 \pm 0.05$
Lower bound: 0.34 - 0.05 = 0.29
Upper bound: 0.34 + 0.05 = 0.39
Confidence interval: 0.29 < p < 0.39

Interpretation: We are 95% confident that between 29% and 39% of all people who receive this flu vaccine will experience some discomfort.

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Worked Example 2: Required sample sizes

What sample size would be needed to achieve specific margins of error?

Part (i): Margin of error of 0.05

Using $E = \frac{1}{\sqrt{n}}$ , we have $0.05 = \frac{1}{\sqrt{n}}$
Rearranging: $\sqrt{n} = \frac{1}{0.05} = 20$
Therefore: $n = 20^2 = 400$

Part (ii): Margin of error of 2.5% (or 0.025)

Using $0.025 = \frac{1}{\sqrt{n}}$
Rearranging: $\sqrt{n} = \frac{1}{0.025} = 40$
Therefore: $n = 40^2 = 1600$

Key insight: To halve the margin of error, you need to quadruple the sample size.

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Worked Example 3: Pet ownership survey

In a random sample of 500 households, 80 said they had at least one pet.

Step 1: Identify the sample size

n = 500

Step 2: Calculate margin of error

$E = \frac{1}{\sqrt{500}} ≈ 0.045$ (or 4.5%)

Step 3: Find sample proportion

$\hat{p} = \frac{80}{500} = 0.16$ (or 16%)

Step 4: Construct confidence interval

$\hat{p} \pm \frac{1}{\sqrt{n}} = 0.16 \pm 0.045$
Range: 0.115 to 0.205 (or 11.5% to 20.5%)

Interpretation: We are 95% confident that between 11.5% and 20.5% of all households in the population have at least one pet.

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Key Points to Remember:

Margin of error formula: $E = \frac{1}{\sqrt{n}}$ - larger samples give smaller margins of error
Sample proportion (p̂) estimates the unknown population proportion (p)
Confidence intervals give a range of likely values: $\hat{p} \pm \frac{1}{\sqrt{n}}$
95% confidence level means our method works correctly 95% of the time
To improve accuracy (reduce margin of error), you need a much larger sample size - the relationship is not linear

Margin of Error and Confidence Intervals (Leaving Cert Mathematics): Revision Notes