Testing Claims Using P-Values (a) identify the claim and...

Testing Claims Using P-Values $(a)$ identify the claim and state $H_{0}$ and $H_{a}$, (b) use technology to find the P-value, (c) decide whether to reject or fail to reject the null hypothesis, and (d) interpret the decision in the context of the original claim. Assume the population is normally distributed. Class Size You receive a brochure from a large university. The brochure indicates that the mean class size for full-time faculty is fewer than 32 students. You want to test this claim. You randomly select 18 classes taught by full-time faculty and determine the class size of each. The results are shown in the table at the left. At $\alpha=0.05,$ can you support the university's claim?

Testing Claims Using P-Values $(a)$ identify the claim and state $H_{0}$ and $H_{a}$, (b) use technology to find the P-value, (c) decide whether to reject or fail to reject the null hypothesis, and (d) interpret the decision in the context of the original claim. Assume the population is normally distributed.
Class Size You receive a brochure from a large university. The brochure indicates that the mean class size for full-time faculty is fewer than
32 students. You want to test this claim. You randomly select 18 classes taught by full-time faculty and determine the class size of each. The results are shown in the table at the left. At $\alpha=0.05,$ can you support the university's claim?

Key Concepts

Significance Level

The significance level, denoted as alpha (?), is the threshold used to decide whether the p-value indicates sufficient evidence to reject the null hypothesis. It represents the maximum acceptable probability of making a Type I error (incorrectly rejecting a true null hypothesis). Commonly used significance levels include 0.05, and decisions in hypothesis testing are made by comparing the p-value with ?.

t-Test

When the sample size is small and the population standard deviation is unknown, a t-test is used to perform hypothesis testing. The t-test uses the t-distribution, which adjusts for the sample size via degrees of freedom. It is appropriate for testing claims about a population mean under the assumption that the population is normally distributed.

Decision Making and Interpretation

After calculating the p-value using a t-test or appropriate statistical software, the next step is to compare it to the significance level. If the p-value is less than the chosen ?, the null hypothesis is rejected in favor of the alternative hypothesis. The final step involves interpreting this decision in the context of the original claim, i.e., determining whether the evidence supports or refutes the claim being tested.

p-value

The p-value is the probability, under the assumption that the null hypothesis is true, of obtaining a test statistic as extreme as, or more extreme than, the one observed. It quantifies the evidence against the null hypothesis. A small p-value indicates that the observed result would be very unlikely if the null hypothesis were true, thereby providing support for rejecting the null hypothesis.

Null Hypothesis

The null hypothesis (H?) is a formal statement asserting that there is no effect or no difference, or that the parameter equals a specific value. In the context of testing claims with p-values, H? serves as the default assumption and is typically set up to reflect the scenario of 'no change' or 'no difference' from a stated value. It provides a framework against which the alternative hypothesis is tested.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or decisions about a population parameter based on sample data. It involves defining a null hypothesis representing a baseline assumption and an alternative hypothesis representing a claim to be tested. The process uses sample data, test statistics, and p-values to evaluate whether the evidence is strong enough to reject the null hypothesis in favor of the alternative hypothesis.

Alternative Hypothesis

The alternative hypothesis (H?) is a formal statement that contradicts the null hypothesis and represents the claim or effect that the researcher is interested in proving. In testing claims with p-values, the alternative hypothesis specifies the direction of the effect (for example, that a population mean is less than a stated value) and it is accepted only when there is sufficient evidence against the null hypothesis.

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