Hypothesis Testing and the One-Mean z-Test: Structured Study Notes

Hypothesis Testing in Statistics

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make inferences about population parameters based on sample data. It helps determine whether observed data provide sufficient evidence to support a specific claim about a population.

• Hypothesis Test: A procedure to decide if a statement about a population parameter is true, based on sample data.

• Applications: Testing means, proportions, variances, etc.

Null and Alternative Hypotheses

Every hypothesis test involves two competing hypotheses:

• Null Hypothesis (H0): The hypothesis to be tested. It usually states that there is no effect or no difference. Symbol: $H_0$

• Alternative Hypothesis (Ha): The hypothesis that is considered as an alternative to the null. It represents the claim to be tested. Symbol: $H_a$

Depending on the form of $H_a$, tests can be:

• Two-tailed test: $H_a: ar{x} eq ar{x}_0$

• Left-tailed test: $H_a: ar{x} < ar{x}_0$

• Right-tailed test: $H_a: ar{x} > ar{x}_0$

Hypothesis Test: The process of deciding whether to reject $H_0$ in favor of $H_a$.

Possible Conclusions for a Hypothesis Test

• If $H_0$ is rejected, we conclude that the data provide enough evidence to support $H_a$.

• If $H_0$ is not rejected, we conclude that the data do not provide enough evidence to support $H_a$.

• Results are often described as "statistically significant" if $H_0$ is rejected.

Types of Errors in Hypothesis Testing

Type I and Type II Errors

When making decisions in hypothesis testing, two types of errors can occur:

• Type I Error: Rejecting $H_0$ when it is actually true.

• Type II Error: Not rejecting $H_0$ when it is actually false.

	H0 True	H0 False
Reject H0	Type I Error	Correct Decision
Do Not Reject H0	Correct Decision	Type II Error

Significance Level ($\alpha$): The probability of making a Type I error (rejecting a true $H_0$).

Relationship: For a fixed sample size, increasing $\alpha$ increases the probability of a Type I error but decreases the probability of a Type II error.

Critical Value Approach to Hypothesis Testing

Terminology

• Test Statistic: A value calculated from sample data, used to decide whether to reject $H_0$.

• Rejection Region: The set of values for the test statistic that leads to rejection of $H_0$.

• Nonrejection (Acceptance) Region: The set of values for the test statistic that leads to non-rejection of $H_0$.

• Critical Value: The boundary value(s) that separate the rejection and nonrejection regions.

Test Type	Rejection Region	Do Not Reject Region
Two-tailed	$z < -z_{\alpha/2}$ or $z > z_{\alpha/2}$	$-z_{\alpha/2} \leq z \leq z_{\alpha/2}$
Left-tailed	$z < -z_{\alpha}$	$z \geq -z_{\alpha}$
Right-tailed	$z > z_{\alpha}$	$z \leq z_{\alpha}$

Obtaining Critical Values

At a 5% significance level ($\alpha = 0.05$):

• Two-tailed: $z_{\alpha/2} = z_{0.025} = 1.96$

• Left-tailed: $z_{\alpha} = z_{0.05} = -1.645$

• Right-tailed: $z_{\alpha} = z_{0.05} = 1.645$

When to Use the One-Sample z-Test

• Small samples (n < 15): Use only if the variable is normally distributed and there are no outliers.

• Moderate samples (15 ≤ n < 30): Use if data are roughly normal and outliers are absent.

• Large samples (n ≥ 30): Use without restriction, but check for outliers.

Outliers: If present, their effect should be checked by running the test with and without them.

One-Mean z-Test

Purpose and Steps

The one-mean z-test is used to test a hypothesis about the mean of a population when the population standard deviation is known.

• Step 1: State the null hypothesis $H_0: \mu = \mu_0$ and the alternative hypothesis $H_a$ (can be $\mu \neq \mu_0$, $\mu < \mu_0$, or $\mu > \mu_0$).

• Step 2: Decide on the significance level $\alpha$.

• Step 3: Compute the value of the test statistic:

$z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

• Step 4: Find the critical value(s) and rejection region(s) based on $\alpha$ and the type of test (left, right, or two-tailed).

• Step 5: Make a decision: If the test statistic falls in the rejection region, reject $H_0$; otherwise, do not reject $H_0$.

Example Application

Suppose a company wants to test if the mean price of books this year is different from last year. They set up $H_0: \mu = \mu_0$ and $H_a: \mu \neq \mu_0$, choose $\alpha = 0.05$, and calculate the test statistic and critical values as above.

Summary Table: Decision Making in Hypothesis Testing

Test Statistic	Critical Value(s)	Decision
$z$	Depends on $\alpha$ and test type	Reject $H_0$ if $z$ falls in rejection region

Key Points:

• Always check assumptions (normality, outliers) before applying the z-test.

• Interpret results in the context of the problem.

Additional info: These notes expand on brief points with definitions, formulas, and structured examples for clarity and completeness.