Making Decisions from Data

This topic brings together key ideas in statistics that help us make informed decisions using data. You will explore how data can be described, summarised, and used to draw conclusions with confidence.

The Empirical Rule — understanding the spread of data in a normal distribution using the 68–95% rule.
Margin of Error — measuring how accurate a sample result is, using the rule \( E = \frac{1}{\sqrt{n}} \).
Confidence Intervals — estimating a range within which the true population proportion is likely to fall.
Hypothesis Testing — deciding whether to accept or reject a claim about a population proportion.

Each section below includes short explanations, worked examples, and interactive quizzes to help you practise applying these ideas.

The Empirical Rule (68%–95%)

Many sets of data — like exam marks or heights — follow a normal distribution: a symmetric, bell-shaped curve centred at the mean \(\bar{x}\). The standard deviation \(\sigma\) measures how spread out the data are.

Empirical Rule showing 68% and 95% shaded bell curves

The Rule

68% of values lie within \(\bar{x}\pm\sigma\) (between \(-1\sigma\) and \(+1\sigma\)).
95% of values lie within \(\bar{x}\pm2\sigma\) (between \(-2\sigma\) and \(+2\sigma\)).

Example 1: Test marks: \(\bar{x}=60,\ \sigma=10.\)
68% of marks: \(60\pm10\Rightarrow [50,70]\).
95% of marks: \(60\pm20\Rightarrow [40,80]\).

Example 2: Heights: \(\bar{x}=170\text{ cm},\ \sigma=8\text{ cm}.\)
68%: \([162,178]\text{ cm}\).
95%: \([154,186]\text{ cm}\).

Empirical Rule Quiz

Labels: a = mean − 2σ, b = mean − 1σ, c = mean, d = mean + 1σ, e = mean + 2σ.

Use the diagram for every question.

1) Mean = 100, σ = 12. What is the value at b (mean − 1σ)?

A) 92

B) 88

C) 112

D) 76

2) Mean = 60, σ = 7. What is the value at e (mean + 2σ)?

A) 67

B) 70

C) 74

D) 81

3) Approximately what percentage of data lies between b and d?

A) 34%

B) 68%

C) 95%

D) 99%

4) Approximately what percentage lies between a and e?

A) 68%

B) 90%

C) 95%

D) 99%

5) Approximately what percentage lies above e (greater than +2σ)?

A) 2.5%

B) 5%

C) 16%

D) 34%

6) 150 students sit a test. How many would you expect to score between b and d?

A) 95

B) 100

C) 102

D) 68

Margin of Error for a Population Proportion

At a 5% level of significance, the margin of error is given by:

\[ E = \frac{1}{\sqrt{n}} \] where \(n\) is the sample size.

Example

Find the margin of error for a sample of \(n = 400\).

\[ E = \frac{1}{\sqrt{400}} = \frac{1}{20} = 0.05 = 5.0\% \]

Practice Question

Find the margin of error when \(n = 900\).

\[ E = \frac{1}{\sqrt{900}} = \frac{1}{30} \approx 0.0333 = 3.3\% \]

Population Proportion \( \hat{p} \)

The sample proportion \( \hat{p} \) represents the proportion of individuals in a sample who have a certain characteristic.

\[ \hat{p} = \frac{x}{n} \] where:

\(x\) = number in the sample with the characteristic
\(n\) = total sample size

It can be written as a fraction, decimal or percentage.

Example

In a survey of 200 people, 90 said they travel to school by bus. Find \( \hat{p} \).

\[ \hat{p} = \frac{90}{200} = 0.45 = 45\% \] So 45% of the sample travel by bus.

Quiz – Finding \( \hat{p} \)

Score: 0 / 0

Find the sample proportion \( \hat{p} = \dfrac{x}{n} \) and choose the correct percentage.

Result

95% Confidence Interval for a Proportion

To construct a 95% confidence interval for a population proportion:

Compute the sample proportion \( \hat{p}=\dfrac{x}{n} \) (write as a percentage).
Find the margin of error \( E=\dfrac{1}{\sqrt{n}} \) (convert to a percentage).
Write the interval \( [\,\hat{p}-E,\;\hat{p}+E\,] \) (in percentages).

Sample Question

In a survey, \(n=400\) and \(x=116\) said “Yes”. Construct a 95% confidence interval for the true percentage who would say “Yes”.

\[ \hat{p}=\frac{116}{400}=0.29=29\%,\qquad E=\frac{1}{\sqrt{400}}=\frac{1}{20}=0.05=5\%. \] \[ \text{CI}:\ [\,29\%-5\%,\ 29\%+5\%\,]=[\,24\%,\ 34\%\,]. \] We are 95% confident that the true percentage lies between \(24\%\) and \(34\%\).

Quiz — Choose the Correct 95% Confidence Interval

Score: 0 / 0

For each question, compute \( \hat{p}=\dfrac{x}{n} \) and \( E=\dfrac{1}{\sqrt{n}} \), then pick the correct interval \( [\,\hat{p}-E,\ \hat{p}+E\,] \) (in percentages).

Result

Hypothesis Test using the Margin of Error

This method tests a claim about a population proportion using the Margin of Error (E).

Steps:

Find the sample proportion: \( \hat{p} = \dfrac{x}{n} \)
Find the margin of error: \( E = \dfrac{1}{\sqrt{n}} \)
Construct the 95% confidence interval: \( [\,\hat{p} - E,\;\hat{p} + E\,] \)
Check if the hypothesised value \( p_0 \) is inside or outside the interval.
Conclude:
- If \( p_0 \) is inside → Do not reject the claim.
- If \( p_0 \) is outside → Reject the claim.

Example

In a survey of \(n = 400\) students, \(220\) said they prefer digital textbooks. Test, at the 5% level, whether the true proportion of students who prefer digital textbooks is 50%.

Step 1: Sample proportion

\[ \hat{p} = \frac{220}{400} = 0.55 = 55\% \]

Step 2: Margin of error

\[ E = \frac{1}{\sqrt{400}} = \frac{1}{20} = 0.05 = 5\% \]

Step 3: 95% Confidence interval

\[ [\,55\% - 5\%,\; 55\% + 5\%\,] = [\,50\%,\;60\%\,] \]

Step 4: The hypothesised value \(p_0 = 50\%\) is inside the interval.

Step 5: Do not reject the claim. There is no evidence that the proportion differs from 50%.

Practice Question

A survey of \(n = 900\) adults finds that \(432\) support a new local policy. Test, at the 5% level, whether the true proportion of support is 45%.

Step 1: Sample proportion

\[ \hat{p} = \frac{432}{900} = 0.48 = 48\% \]

Step 2: Margin of error

\[ E = \frac{1}{\sqrt{900}} = \frac{1}{30} \approx 0.0333 = 3.3\% \]

Step 3: 95% Confidence interval

\[ [\,48\% - 3.3\%,\; 48\% + 3.3\%\,] = [\,44.7\%,\;51.3\%\,] \]

Step 4: The hypothesised value \(p_0 = 45\%\) is inside the interval.

Step 5: Do not reject the claim. The data are consistent with 45% support.

Hypothesis Test – Accept or Reject?

Score: 0 / 0

Decide whether to accept or reject the claim based on the 95% confidence interval.

Result