Inferential Statistics

The One Thing

Inferential statistics uses a sample to draw qualified conclusions about a population, while probability and sampling distributions quantify the uncertainty in those conclusions. Confidence intervals estimate population parameters, and hypothesis tests evaluate claims about them.

Definitions and Results

• Population: The complete group of individuals or observations about which a conclusion is required.
• Sample: A smaller group selected from the population for study.
• Parameter: A numerical measure describing a population, such as the population mean or proportion.
• Statistic: A numerical measure calculated from a sample, such as the sample mean or sample proportion.
• Random Sample: A sample selected so that each population member has a known and fair chance of being chosen.
• Sampling Distribution: The probability distribution of a statistic obtained from all possible samples of a fixed size.
• Standard Error: The standard deviation of a sampling distribution; it measures the typical variation of a sample statistic.
• Point Estimate: A single sample statistic used to estimate a population parameter.
• Confidence Interval: An interval calculated from sample data that is expected to contain the population parameter at a stated confidence level.
• Confidence Level: The long-run percentage of similarly constructed intervals that would contain the true population parameter, such as 90%, 95%, or 99%.
• Null Hypothesis: The initial claim that there is no significant difference, effect, or relationship.
• Alternative Hypothesis: The claim that contradicts the null hypothesis and suggests a difference, effect, or relationship.
• Level of Significance: The maximum probability of rejecting a true null hypothesis, usually denoted by alpha.
• Test Statistic: A standardized value calculated from sample data to assess a hypothesis.
• Critical Region: The set of values of the test statistic for which the null hypothesis is rejected.
• P-value: The probability of obtaining a result at least as extreme as the observed result, assuming the null hypothesis is true.
• Type I Error: Rejecting the null hypothesis when it is actually true.
• Type II Error: Failing to reject the null hypothesis when it is actually false.
• One-Tailed Test: A hypothesis test in which the rejection region lies in one tail of the sampling distribution.
• Two-Tailed Test: A hypothesis test in which the rejection regions lie in both tails of the sampling distribution.
• Statistical Significance: A result is statistically significant when it provides sufficient evidence to reject the null hypothesis at the chosen significance level.

Important results and conditions include:

• For a sample mean,

$$SE(\bar{x})=\frac{\sigma }{\sqrt{n}}$$ when the population standard deviation is known, and $$SE(\bar{x})=\frac{s}{\sqrt{n}}$$ when it is estimated from the sample.

• For a sample proportion,

$$SE(\hat{p})=\sqrt{\frac{p(1-p)}{n}}.$$

• By the Central Limit Theorem, the sampling distribution of the sample mean is approximately normal for a sufficiently large sample size, even when the population is not normal.
• A general confidence interval has the form

$$\text{estimate}\pm \text{critical value}\times \text{standard error}.$$

• A confidence interval does not mean that the fixed population parameter has a stated probability of lying in one particular calculated interval. The confidence level describes the long-run reliability of the method.
• A statistically significant result provides evidence against the null hypothesis, not absolute proof that the alternative hypothesis is true.
• Failure to reject the null hypothesis does not prove that the null hypothesis is true; it means that the available evidence is insufficient.
• Practical importance and statistical significance are different. A very small effect may be statistically significant in a large sample, whereas an important effect may fail to reach significance in a small sample.
• Correlation or a statistically significant result does not by itself prove a cause-and-effect relationship.
• A smaller significance level makes rejection more difficult and reduces the risk of a Type I error, but it may increase the risk of a Type II error.
• Increasing the sample size generally decreases the standard error and makes estimates more precise.
• A sample can provide useful information about a population only when it is selected appropriately and is reasonably representative.

Worked Methods

Constructing a confidence interval for a population mean when the standard deviation is known

• Identify the sample mean $\bar{x}$, known population standard deviation $\sigma$, sample size $n$, and confidence level.
• Select the appropriate standard normal critical value $z_{\alpha /2}$.
• Calculate the standard error:

$$SE(\bar{x})=\frac{\sigma }{\sqrt{n}}.$$

• Use:

$$\bar{x}\pm z_{\alpha /2}\frac{\sigma }{\sqrt{n}}.$$

• Interpret the interval in relation to the population parameter.

Common standard normal critical values are:

• $z=1.645$ for 90% confidence;
• $z=1.96$ for 95% confidence;
• $z=2.576$ for 99% confidence.

Constructing a confidence interval for a population mean when the standard deviation is unknown

• Identify $\bar{x}$, the sample standard deviation $s$, the sample size $n$, and the confidence level.
• Use the $t$-distribution with $n-1$ degrees of freedom.
• Calculate the standard error:

$$SE(\bar{x})=\frac{s}{\sqrt{n}}.$$

• Use the $t$-interval:

$$\bar{x}\pm t_{\alpha /2,n-1}\frac{s}{\sqrt{n}}.$$

• State the resulting interval and its contextual interpretation.

Constructing an approximate confidence interval for a population proportion

• Identify the sample proportion $\hat{p}$, sample size $n$, and confidence level.
• Ensure that the sample is sufficiently large for the approximation.
• Calculate the estimated standard error:

$$\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.$$

• Use:

$$\hat{p}\pm z_{\alpha /2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.$$

• Interpret the interval as an estimate of the population proportion.

Testing a population mean when the standard deviation is known

• Define the population and parameter.
• State the hypotheses before examining or analysing the data. The null hypothesis generally contains equality, for example:

$$H_0:\mu ={\mu }_0.$$

• State the alternative hypothesis and determine whether the test is one-tailed or two-tailed.
• Choose the significance level $\alpha$.
• Calculate the $z$-test statistic:

$$z=\frac{\bar{x}-{\mu }_0}{\sigma /\sqrt{n}}.$$

• Find the critical value or calculate the p-value.
• Using the critical-value method, reject $H_0$ if the test statistic lies in the critical region; otherwise, do not reject $H_0$.
• Using the p-value method, reject $H_0$ when the p-value is less than or equal to $\alpha$; otherwise, do not reject $H_0$.
• State the conclusion in the context of the original population and variable.

Testing a population mean when the standard deviation is unknown

• Define the population and parameter.
• State the hypotheses before performing the test.
• Choose $\alpha$ and identify the relevant degrees of freedom, $n-1$.
• Calculate the $t$-test statistic:

$$t=\frac{\bar{x}-{\mu }_0}{s/\sqrt{n}},$$ with $n-1$ degrees of freedom.

• Find the critical value or p-value from the $t$-distribution.
• Apply the appropriate decision rule.
• Interpret the result in context.

Applying one-tailed and two-tailed decision rules

• Base the choice of test on the research question and make it before examining the data.
• For a two-tailed test at significance level $\alpha$, place the rejection regions below

$$-z_{\alpha /2}$$ and above $$z_{\alpha /2}.$$

• For a right-tailed test, reject $H_0$ when the test statistic exceeds the upper critical value.
• For a left-tailed test, reject $H_0$ when the test statistic is below the lower critical value.
• Report whether the result is statistically significant at the chosen significance level.

Complete inferential procedure

• Define the population and parameter.
• Select a suitable sample.
• State the null and alternative hypotheses.
• Choose $\alpha$.
• Calculate the relevant statistic.
• Find the critical value or p-value.
• Make the decision to reject or not reject $H_0$.
• Interpret the result in context, including the population, variable, confidence level or significance level, and limitations of the sample.

Where It Goes Wrong

• Treating a statistic as if it were a parameter: a statistic describes the sample, whereas a parameter describes the population.
• Forgetting that the standard error for a sample mean uses $\sigma /\sqrt{n}$ when the population standard deviation is known, but $s/\sqrt{n}$ when it is estimated from the sample.
• Using the normal critical value rather than the $t$-distribution when the population standard deviation is unknown.
• Stating hypotheses after examining the data, rather than before performing the test; the choice between a one-tailed and two-tailed test must be based on the research question in advance.
• Interpreting failure to reject $H_0$ as proof that $H_0$ is true, or interpreting statistical significance as absolute proof of the alternative hypothesis.
• Forgetting that correlation or statistical significance does not by itself establish a cause-and-effect relationship, and that conclusions must account for sampling limitations.

What Gets Asked

This material supports questions requiring students to:

• distinguish populations, samples, parameters, statistics, random samples, and sampling distributions;
• calculate standard errors for sample means and sample proportions;
• explain the Central Limit Theorem and the conditions under which the sample mean is approximately normally distributed;
• construct confidence intervals for a population mean with known or unknown standard deviation;
• use the standard normal critical values $1.645$, $1.96$, and $2.576$ for 90%, 95%, and 99% confidence;
• construct an approximate confidence interval for a population proportion when the sample is sufficiently large;
• formulate null and alternative hypotheses, including hypotheses such as $H_0:\mu ={\mu }_0$;
• calculate and interpret $z$- and $t$-test statistics;
• identify one-tailed and two-tailed tests and locate their critical regions;
• make decisions using either critical values or p-values;
• distinguish Type I and Type II errors;
• explain how changing $\alpha$ or increasing $n$ affects error risk, standard error, and precision;
• distinguish statistical significance from practical importance;
• interpret confidence intervals and hypothesis-test conclusions in the context of the original population and variable;
• explain why inferential conclusions remain uncertain and why statistically significant findings do not necessarily establish causation.