Mathematics 9709/61 — May/June 2022

Walkthrough

We are given summary statistics, not raw data. The best point estimate of the population mean $\mu$ is the sample mean $\bar{x} = \sum x / n$. Here that is $2520/200 = 12.6$.

To build a confidence interval we also need a measure of spread. Since we are estimating the population variance from a sample, we use the unbiased estimator

The term $n\bar{x}^2$ subtracts the part of $\sum x^2$ explained by the mean, leaving the sum of squared deviations. Dividing by $n-1$ rather than $n$ corrects the bias in the sample variance.

Here $200(12.6)^2 = 31752$, so $\sum x^2 - n\bar{x}^2 = 31852 - 31752 = 100$. Thus $s^2 = 100/199 \approx 0.5025$.

The standard error of the sample mean is $\sqrt{s^2/n}$. Because $n = 200$ is large, the sample mean is approximately normally distributed, so a 95% confidence interval uses $z = 1.96$. The interval is

This gives $12.6 \pm 0.09825$, i.e. from about 12.502 to 12.698, which is $12.5$ to $12.7$ to 3 significant figures.

Key Takeaways

• The sample mean is the point estimate of the population mean.
• Use the $n-1$ denominator for an unbiased estimate of population variance from a sample.
• For large samples, the normal distribution can be used for a confidence interval for the mean.
• A 95% confidence interval is of the form estimate $\pm$ (critical value) $\times$ standard error.

Common Mistakes

• Using the biased variance $\sum x^2/n - \bar{x}^2$ instead of the unbiased version with $n-1$. The mark scheme allows at most B1 M1 A0 B1 M1 A0 for this error.
• Forgetting to divide $s^2$ by $n$ when finding the standard error.
• Using the wrong critical value, such as 1.645 (for 90%) or 2.576 (for 99%), instead of 1.96 for 95%.
• Giving only the margin of error, or giving the interval as a single number, instead of two endpoints.

Things to Be Careful About

• The units are millimetres; the interval endpoints are in mm.
• The answer must be an interval, and to 3 significant figures it is $12.5$ to $12.7$.
• The mark scheme says 'CWO' (correct working only), so if an earlier variance estimate is wrong, the final interval may not receive the final A mark even if it appears numerically close.
• Since $n=200$, using $z$ is appropriate; do not use a $t$-table unless the course explicitly requires it.
• Keep enough decimal places in intermediate working so the final rounding is accurate.

Walkthrough

A 95% confidence interval is constructed so that, in repeated sampling, 95% of such intervals will contain the true population mean. Therefore, if Jean calculates 40 independent 95% confidence intervals, the expected number containing $\mu$ is $0.95 \times 40 = 38$.

This is an expected value, not a guarantee: the actual number could be different in any particular set of 40 samples.

Key Takeaways

• Confidence level is a long-run proportion: 95% of intervals contain the true mean.
• Expected count = confidence level × number of intervals.

Common Mistakes

• Saying 'exactly 38' as if it were certain; it is the expected number.
• Confusing the probability that a particular interval contains the mean with a probability statement about the mean itself: the population mean is fixed, not random.

Things to Be Careful About

• Use 0.95, not 0.05.
• The calculation is $0.95 \times 40$, giving 38.
• The mark scheme awards B1 for this early multiplication.