AP Statistics 4.3-4.4: Tests for a Population Mean

Rare event in hypothesis testing

A rare event is a sample result that would be very unlikely if the null hypothesis were true. Such an outcome provides evidence against the null hypothesis, although it does not logically prove the alternative.

How does a hypothesis test use sample data to evaluate a population claim?

Assume the null hypothesis is true, then determine how likely the observed sample result—or a result more extreme—would be under that assumption. If the result is sufficiently unlikely, reject the null hypothesis.

Null hypothesis $H_0$

The null hypothesis is the baseline claim about a population parameter, often representing no effect, no difference, or a specified value. Statistical testing evaluates whether sample evidence is strong enough to reject it.

Alternative hypothesis $H_a$

The alternative hypothesis states the effect, difference, or direction that the investigation seeks evidence for. It must contradict the null hypothesis.

How should the phrase “the population mean is greater than a specified value” be written as hypotheses?

Use $H_a: \mu > \mu_0$ and a contradictory null hypothesis such as $H_0: \mu \leq \mu_0$. The inequality direction belongs in the alternative hypothesis.

How are hypotheses set up for a two-sided test about a population mean?

For a claim that the population mean differs from $\mu_0$, use $H_0: \mu = \mu_0$ and $H_a: \mu \ne \mu_0$.

How are hypotheses set up for a left-tailed test about a population mean?

For evidence that the population mean is less than $\mu_0$, use $H_a: \mu < \mu_0$ and a contradictory null hypothesis such as $H_0: \mu \geq \mu_0$.

p-value

The p-value is the probability, assuming $H_0$ is true, of obtaining the observed test result or a result at least as extreme. A smaller p-value indicates stronger evidence against $H_0$.

What does a large p-value indicate?

A large p-value means the observed result is reasonably plausible when $H_0$ is true. Therefore, the data do not provide sufficient evidence to reject $H_0$; this does not prove that $H_0$ is true.

What does a very small p-value indicate?

A very small p-value means the observed result would be unlikely under $H_0$. This is evidence against the null hypothesis and may justify rejecting it at the chosen significance level.

Significance level $\alpha$

The significance level is a preset threshold used to decide whether a p-value is small enough to reject $H_0$. It is also the probability of a Type I error under the testing procedure.

How is a p-value compared with $\alpha$ to make a hypothesis-test decision?

Reject $H_0$ when $p < \alpha$; otherwise, do not reject $H_0$. The results are called statistically significant when the null hypothesis is rejected.

What does “do not reject the null hypothesis” mean?

It means the sample did not provide sufficiently strong evidence against $H_0$ at the chosen $\alpha$. It does not mean that $H_0$ has been proven true.

Type I error

A Type I error occurs when a test rejects $H_0$ even though $H_0$ is true. The probability assigned to this error by the testing procedure is $\alpha$.

How should a hypothesis-test conclusion be written?

State the decision about $H_0$, then interpret it in the original scientific or practical context. For example, explain whether the data provide sufficient evidence for the claimed increase, decrease, difference, or effect.

Why is a graph useful when calculating a p-value?

A graph of the null distribution shows the observed test statistic and the tail area corresponding to results at least as extreme. The relevant one-sided or two-sided tail area is the p-value.

Student's $t$-test

A Student's $t$-test evaluates a population mean when the population standard deviation is unknown and is estimated from sample data. Its test statistic follows a $t$ distribution under appropriate assumptions.

One-sample $t$ statistic for testing a population mean

For testing $H_0:\mu=\mu_0$, use $t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}$, where $s$ is the sample standard deviation. The statistic measures how many estimated standard errors the sample mean is from the null value.

Degrees of freedom for a one-sample $t$-test

The degrees of freedom are $df=n-1$. One degree of freedom is lost because the sample mean is estimated from the same observations used to calculate $s$.

How is the p-value found for a one-sample $t$-test?

Use the calculated $t$ statistic and $df=n-1$ with the appropriate Student's $t$ distribution. The p-value is the area in the alternative-hypothesis tail(s) beyond the observed statistic.

What conditions support a one-sample $t$-test for a population mean?

The data should come from a random sample or randomized experiment, observations should be independent, and the population should be approximately normal. For moderately large samples, the Central Limit Theorem often supports approximate normality of the sample mean.

What is the 10% condition for a one-sample mean test?

When sampling without replacement from a finite population, the sample size should be no more than 10% of the population: $n \leq 0.10N$. This helps justify treating observations as approximately independent.

Why is the random condition required for a test about a population mean?

A random sample or randomized experiment helps ensure that the sample is representative and that probability-based conclusions about the population mean are valid. Voluntary or convenience samples can create bias.

Why does a $t$ distribution replace the standard normal distribution when $\sigma$ is unknown?

Replacing $\sigma$ with the variable estimate $s$ adds uncertainty to the standardized statistic. The $t$ distribution has heavier tails than the normal distribution, especially for small samples, and approaches the normal distribution as sample size increases.

How does sample size affect the sampling distribution of the mean?

By the Central Limit Theorem, the distribution of sample means tends toward normality as $n$ increases, provided observations are independent and have finite variance. Its spread is the standard error, which decreases as $1/\sqrt{n}$.

What assumptions make the theoretical one-sample $t$ distribution exact?

The observations are typically assumed to come from a normal population, and the sample mean and sample standard deviation satisfy the conditions underlying the $t$ distribution. With large samples, the test is often approximately valid under weaker conditions.

What conclusion follows from a p-value of $0.0013$ when testing $H_a:\mu>12$?

At common significance levels such as $\alpha=0.05$ or $0.01$, $0.0013<\alpha$, so reject $H_0$. The sample provides strong evidence that the population mean is greater than $12$.

What is the relationship between a $t$-test and a $z$-test for large samples?

As sample size grows, the sample standard deviation estimates the population standard deviation more reliably and the $t$ distribution approaches the standard normal distribution. Consequently, $t$- and $z$-test results become very similar.