Which of the following specify legitimate pairs of null and...

Which of the following specify legitimate pairs of null and alternative hypotheses? a. $H_{0}: p=0.25 \quad H_{i}: p>0.25$ b. $H_{0}: p<0.40 \quad H_{i}: p>0.40$ c. $H_{0}: p=0.40 \quad H: p<0.65$ d. $H_{0}: p \neq 0.50 \quad H_{i}: p=0.50$ e. $H_{\mathrm{n}}: p=0.50 \quad H_{i}: p>0.50$ f. $H_{0}: \hat{p}=0.25 \quad H_{e^{\prime}} \hat{p}>0.25$

Which of the following specify legitimate pairs of null and alternative hypotheses?
a. $H_{0}: p=0.25 \quad H_{i}: p>0.25$
b. $H_{0}: p<0.40 \quad H_{i}: p>0.40$
c. $H_{0}: p=0.40 \quad H: p<0.65$
d. $H_{0}: p \neq 0.50 \quad H_{i}: p=0.50$
e. $H_{\mathrm{n}}: p=0.50 \quad H_{i}: p>0.50$
f. $H_{0}: \hat{p}=0.25 \quad H_{e^{\prime}} \hat{p}>0.25$

Key Concepts

Null Hypothesis

The null hypothesis is a formal statement that assumes no effect or no difference in the population parameter. It is usually defined as an equality (e.g., p = a specified value) which serves as a baseline for comparison against the alternative hypothesis. This equality requirement makes the null hypothesis a clear reference for testing statistical claims.

Alternative Hypothesis

The alternative hypothesis represents what the researcher wants to test for and is typically formulated as a directional or non-directional statement. It includes inequality signs (e.g., p > value, p ? value) indicating that the parameter differs from the value stated in the null hypothesis, thus capturing the effect or difference expected in the study.

One-Tailed vs Two-Tailed Tests

One-tailed and two-tailed tests refer to the directionality of the alternative hypothesis. A one-tailed test is used when the research hypothesis predicts that the parameter will either be greater than or less than a specified value, while a two-tailed test is applied when the alternative hypothesis does not specify a direction, only that there is a difference from the specified value.

Parameter vs Statistic Clarification

In hypothesis testing, it is important to distinguish between population parameters and sample statistics. Hypotheses must be formulated in terms of the parameter (e.g., the true proportion p) and not the sample statistic (e.g., sample proportion p?). This ensures that the inference pertains to the population rather than just the sample observed.

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