One-Sample Hypothesis Tests for Means and Proportions

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What do the null and alternative hypotheses represent?

$H_0$ states the reference value or status quo, while $H_a$ states the difference or direction being investigated. The null includes equality.

How does the alternative hypothesis determine the test tail?

A less-than alternative is left-tailed, a greater-than alternative is right-tailed, and a not-equal alternative is two-tailed.

When should a one-sample z-test be used for a mean?

Use a z-test when the population standard deviation $\sigma$ is known and the distributional or sample-size conditions are reasonable.

What is the test statistic for a one-proportion z-test?

$z=(\hat{p}-p_0)/\sqrt{p_0(1-p_0)/n}$. The standard error uses the null proportion $p_0$.

What does failing to reject the null hypothesis mean?

The sample does not provide sufficient evidence against $H_0$ at the chosen significance level. It does not prove that $H_0$ is true.

Quiz questions

1. Which statement about the null hypothesis in a one-parameter hypothesis test is correct?Answer: It specifies the reference parameter value and includes equality.
2. A research question asks whether a population mean exceeds a reference value $\mu_0$. Which alternative and test direction fit that question?Answer: $H_a: \mu>\mu_0$, giving a right-tailed test.
3. For a one-sample t-test based on $n$ observations, what are the degrees of freedom?Answer: $n-1$
4. For a one-proportion z-test with null proportion $p_0$, which condition is a common check for sufficiently large expected counts?Answer: $np_0\ge10$ and $n(1-p_0)\ge10$
5. A hypothesis test has p-value 0.03 and significance level $\alpha=0.05$. What decision follows the usual p-value rule?Answer: Reject $H_0$ because the p-value is below $\alpha$.