Consider the following hypothesis test: H0: μ≤12 Ha: μ>12 A sample of 25 provided a

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Consider the following hypothesis test: H0: μ≤12 ...

Consider the following hypothesis test: $$ \begin{aligned} & H_0: \mu \leq 12 \\ & H_{\mathrm{a}}: \mu>12 \end{aligned} $$ A sample of 25 provided a sample mean $\bar{x}-14$ and a sample standard deviation $s=4.32$. a. Compute the value of the test statistic. b. Use the $t$ distribution table (Table 2 in Appendix B) to compute a range for the $p$-value. c. At $\alpha=.05$, what is your conclusion? d. What is the rejection rule using the critical value? What is your conclusion?

Consider the following hypothesis test:
$$
\begin{aligned}
& H_0: \mu \leq 12 \\
& H_{\mathrm{a}}: \mu>12
\end{aligned}
$$
A sample of 25 provided a sample mean $\bar{x}-14$ and a sample standard deviation $s=4.32$.
a. Compute the value of the test statistic.
b. Use the $t$ distribution table (Table 2 in Appendix B) to compute a range for the $p$-value.
c. At $\alpha=.05$, what is your conclusion?
d. What is the rejection rule using the critical value? What is your conclusion?

Key Concepts

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to decide whether there is enough evidence in a sample to infer that a certain condition is true for the entire population. It involves setting up two competing hypotheses and using sample data to decide between them.

Null and Alternative Hypotheses

The null hypothesis is a statement of no effect or no difference and serves as the default assumption until evidence suggests otherwise. The alternative hypothesis claims that there is an effect or a difference, and the goal of the test is to determine if there is sufficient evidence to reject the null hypothesis in favor of the alternative.

Test Statistic

A test statistic is a standardized quantity calculated from sample data during a hypothesis test. It measures how far the sample statistic deviates from the null hypothesis, taking into account the variability in the data, and is used to decide whether to reject the null hypothesis.

t-Distribution

The t-distribution is a probability distribution used in statistical analyses when the sample size is small and the population standard deviation is unknown. It adjusts for the extra uncertainty by having heavier tails compared to the normal distribution, and is commonly used in the context of t-tests.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities which can be assigned to a statistical distribution. In the context of t-tests, it is typically calculated as the sample size minus one, and it influences the shape of the t-distribution.

P-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It provides a measure of the strength of the evidence against the null hypothesis; a smaller p-value indicates stronger evidence in favor of the alternative hypothesis.

Significance Level

The significance level, often denoted by alpha, is the threshold for rejecting the null hypothesis. It represents the probability of committing a Type I error, which is the error of rejecting a true null hypothesis, and is typically set at common levels like 0.05.

Critical Value/Rejection Rule

The critical value is the cutoff point on the test statistic distribution that defines the boundary of the rejection region for the null hypothesis. If the test statistic exceeds this value, the null hypothesis is rejected. The rejection rule specifies the conditions under which the null hypothesis should be rejected based on the comparison of the test statistic to the critical value.

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