Confidence Intervals for the Population Mean

In statistical inference, we use sample data to make estimates about unknown population parameters. While a point estimate (such as the sample mean $\bar{x}$) provides a single value, a confidence interval provides a range of values within which the true population parameter is expected to lie, with a certain level of confidence.

1. Fundamental Concepts

Population Mean ($\mu$): The true average of the entire population (usually unknown).
Sample Mean ($\bar{x}$): The average calculated from a sample, used as a point estimate for $\mu$.
Standard Deviation ($\sigma$): The population standard deviation. If unknown and the sample size is large ($n \ge 30$), the sample standard deviation $s$ can be used as an approximation.
Margin of Error ($M$): The distance from the sample mean to the endpoints of the confidence interval.

KEY TAKEAWAY: A confidence interval is an interval estimate. It provides more information than a point estimate because it accounts for the variability inherent in sampling.

2. Construction of Confidence Intervals

For a population with a known standard deviation $\sigma$, or for a large sample where $\sigma$ is approximated by $s$, the $C\%$ confidence interval for the population mean $\mu$ is given by:

\[\left( \bar{x} - z \frac{\sigma}{\sqrt{n}}, \bar{x} + z \frac{\sigma}{\sqrt{n}} \right)\]

Where:
* $\bar{x}$ is the sample mean.
* $n$ is the sample size.
* $z$ (or $z^*$) is the critical value for the desired level of confidence.

Common Critical $z$-values

The value of $z$ is determined such that the area under the standard normal curve $Z \sim N(0,1)$ between $-z$ and $z$ is equal to the confidence level $C$.

Confidence Level	$C$ (decimal)	Critical Value ($z$)
90%	0.90	1.645
95%	0.95	1.960
99%	0.99	2.576

EXAM TIP: If an unusual confidence level is requested (e.g., 92%), use the CAS command invNorm to find the $z$-score. For a $C\%$ interval, the area to the left of the upper $z$-score is $C + \frac{1-C}{2}$.

3. The Margin of Error and Interval Width

The Margin of Error ($M$) is defined as:
\$$M = z \frac{\sigma}{\sqrt{n}}$\$

The total Width ($W$) of the confidence interval is:
\$$W = 2M = 2z \frac{\sigma}{\sqrt{n}}$\$

Factors Affecting Width

The precision of a confidence interval is determined by its width. A narrower interval is more precise.

Action	Effect on Width	Reason
Increase Confidence Level	Increases Width	Larger $z$-value required to be more certain.
Increase Sample Size ($n$)	Decreases Width	Standard error $\frac{\sigma}{\sqrt{n}}$ decreases.
Increase Standard Deviation ($\sigma$)	Increases Width	More variability in the population leads to less precision.

The Square Root Law

The width of a confidence interval is inversely proportional to the square root of the sample size:
\$$W \propto \frac{1}{\sqrt{n}}$\$

To decrease the width by a factor of $k$, the sample size must be increased by a factor of $k^2$.

COMMON MISTAKE: Students often think doubling the sample size halves the width. In reality, to halve the width ($k=2$), you must quadruple the sample size ($2^2 = 4$).

4. Interpretation of Confidence Intervals

The interpretation of a confidence interval is a common source of theoretical questions in VCE exams.

Correct Interpretation: If we were to take many random samples of the same size and calculate a 95% confidence interval for each, approximately 95% of these intervals would contain the true population mean $\mu$.
Incorrect Interpretation: “There is a 95% probability that the true mean $\mu$ lies within this specific interval.” (Once the interval is calculated, the mean is either in it or it isn’t; there is no “probability” for a fixed constant).

VCAA FOCUS: VCAA frequently tests the definition of the confidence level. Remember: the interval is the random variable, not the population parameter $\mu$. The parameter $\mu$ is a fixed (though unknown) constant.

5. Solving for Sample Size $n$

To determine the minimum sample size required to achieve a specific margin of error $M$ at a certain confidence level:

Set up the equation: $M = z \frac{\sigma}{\sqrt{n}}$
Rearrange for $n$: $\sqrt{n} = \frac{z\sigma}{M} \implies n = \left( \frac{z\sigma}{M} \right)^2$
Always round up to the nearest whole number to ensure the margin of error is not exceeded.

Example: Changing Width

If a researcher wants to reduce the width of a 95% confidence interval to one-third of its original size:
* New Width = $\frac{1}{3} \times$ Old Width
* Since $W \propto \frac{1}{\sqrt{n}}$, we need $\frac{1}{\sqrt{n_{new}}} = \frac{1}{3\sqrt{n_{old}}}$
* $\sqrt{n_{new}} = 3\sqrt{n_{old}}$
* $n_{new} = 9 \times n_{old}$
* The sample size must be increased by a factor of 9.

STUDY HINT: Practice rearranging the width formula $W = \frac{2z\sigma}{\sqrt{n}}$ quickly. Many multiple-choice questions require finding the ratio between two sample sizes or two widths.

Confidence Intervals for the Population Mean

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