Mathematics 9709/73 — May/June 2019

Walkthrough

We are estimating the unknown probability $p$ that a coin shows heads. The best point estimate from the data is the sample proportion $\hat{p} = 60/100 = 0.6$. Because the sample size is 100, the distribution of $\hat{p}$ is approximately normal, so we can use the standard confidence interval formula for a proportion.

For a 98% confidence interval, the total tail probability is $1 - 0.98 = 0.02$, so each tail has probability $0.01$. The corresponding critical value is $z = 2.326$. This is the value that leaves 1% in each tail of the standard normal distribution.

The standard error of $\hat{p}$ is $\sqrt{\hat{p}(1-\hat{p})/n}$. Substituting $\hat{p} = 0.6$ and $n = 100$ gives $\sqrt{0.6 \times 0.4 / 100} = \sqrt{0.0024} \approx 0.04899$. Multiplying by $z = 2.326$ gives the margin of error, about $0.11395$. Adding and subtracting this from $0.6$ gives the endpoints $0.486$ and $0.714$ to 3 significant figures.

Key Takeaways

• A confidence interval for a population proportion is based on the sample proportion and its standard error.
• The critical value depends on the confidence level; for 98% use $z = 2.326$.
• The interval is written as point estimate $\pm$ margin of error.

Common Mistakes

• Using $z = 1.96$ (for 95%) instead of $z = 2.326$ for 98% confidence.
• Forgetting to divide by $n$ inside the square root.
• Using $p = 0.5$ instead of the sample proportion $0.6$ in the standard error.
• Giving only the point estimate or only the margin of error, not the interval.

Things to Be Careful About

• The mark scheme accepts $z$ from 2.326 to 2.329, depending on the table used.
• The final answer must be an interval; here $0.486$ to $0.714$ to 3 significant figures.
• Since this is an approximate interval, it is valid because the sample size is large enough for the normal approximation to apply.