Hypothesis Testing

Null, Alternative, p-Value & Decision

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Hypothesis testing is a statistical method for using sample data to evaluate a claim about a population. It helps scientists, doctors, engineers, and social researchers decide whether an observed effect is likely real or could have happened by random chance alone. The process gives a structured way to compare evidence against a default assumption. This makes conclusions more consistent and transparent.

In a hypothesis test, you begin with a null hypothesis that represents no effect or no difference, and an alternative hypothesis that represents the claim you want to investigate. After choosing a significance level $\alpha$ , you calculate a test statistic from the sample and use it to find a $p$ -value. The $p$ -value measures how unusual the sample result would be if the null hypothesis were true. If the $p$ -value is small enough, you reject the null hypothesis; otherwise, you fail to reject it.

Key Facts

Null hypothesis $H_0$ usually states no difference, no effect, or a specific population value.
Alternative hypothesis $H_a$ states the competing claim, such as $\mu \neq \mu_0$ , $\mu > \mu_0$ , or $\mu < \mu_0$ .
Decision rule: reject $H_0$ if $p$ -value $\leq \alpha$ ; otherwise fail to reject $H_0$ .
A common significance level is $\alpha = 0.05$ , which is the probability of a Type I error.
For a $z$ test of a mean, $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$ .
For a one-sample $t$ test, $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ .

Vocabulary

Null hypothesis: The starting claim that there is no effect, no difference, or no change in the population.
Alternative hypothesis: The competing claim that says there is an effect, a difference, or a change.
p-value: The probability of getting a result at least as extreme as the sample result if the null hypothesis is true.
Significance level: The cutoff probability $\alpha$ used to decide when evidence is strong enough to reject the null hypothesis.
Test statistic: A standardized number computed from sample data that measures how far the sample result is from what the null hypothesis predicts.

Common Mistakes to Avoid

Saying the p-value is the probability that $H_0$ is true, which is wrong because the p-value assumes $H_0$ is true and measures the probability of the observed data or more extreme data.
Writing reject $H_0$ when p-value > $\alpha$ , which is wrong because results larger than $\alpha$ do not provide enough evidence against the null hypothesis.
Confusing fail to reject $H_0$ with proving $H_0$ true, which is wrong because the test may simply lack enough evidence or sample size to detect a real effect.
Choosing a one-tailed test after looking at the data, which is wrong because the direction of the test must be set before analysis to avoid biased conclusions.

Practice Questions

1 A factory claims the mean battery life is 20 hours. A sample of 36 batteries has mean 18.8 hours. Assume $\sigma = 3$ hours and test $H_0: \mu = 20$ versus $H_a: \mu < 20$ at $\alpha = 0.05$ . Compute the $z$ statistic and state the decision using the critical value method or p-value method.
2 A school tests whether a new tutoring program changes average test scores. For 25 students, the sample mean is 78, the hypothesized mean is 74, and the sample standard deviation is 10. Test $H_0: \mu = 74$ versus $H_a: \mu \neq 74$ at $\alpha = 0.05$ using a one-sample $t$ test. Compute the $t$ statistic and state whether to reject $H_0$ .
3 A study reports $p$ -value = 0.08 for a test conducted at $\alpha = 0.05$ . Explain what decision should be made and why this does not mean the null hypothesis has been proven true.

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