Consider the following three tests for a fair coin: (i) H0:...

Consider the following three tests for a fair coin: (i) $H_0: p=0.5$ vs. $H_1: p \neq 0.5$ (ii) $H_0: p=0.5$ vs. $H_1: p>0.5$ (iii) $H_0: p=0.5$ vs. $H_1: p<0.5$. Assume $n=100$ coin tosses in each test and that the rejection regions for the above tests are selected for $\alpha=1 \%$. (a) Find the power curves for the three tests as a function of $p$. (b) Explain the power curve of the two-sided test in comparison to those of the onesided tests.

Consider the following three tests for a fair coin:
(i) $H_0: p=0.5$ vs. $H_1: p \neq 0.5$
(ii) $H_0: p=0.5$ vs. $H_1: p>0.5$
(iii) $H_0: p=0.5$ vs. $H_1: p<0.5$.

Assume $n=100$ coin tosses in each test and that the rejection regions for the above tests are selected for $\alpha=1 \%$.
(a) Find the power curves for the three tests as a function of $p$.
(b) Explain the power curve of the two-sided test in comparison to those of the onesided tests.

Key Concepts

Significance Level

The significance level, typically denoted by alpha, is the probability of making a Type I error — that is, rejecting the null hypothesis when it is actually true. In hypothesis testing, setting a small alpha (such as 1%) controls the likelihood of a false positive, ensuring that only strong evidence against the null hypothesis leads to its rejection.

Rejection Region

The rejection region is the set of outcomes, defined by critical values based on the significance level, for which the null hypothesis is rejected. This region is determined by the distribution of the test statistic under the null hypothesis and dictates the conditions under which the test concludes that the alternative hypothesis is more plausible.

Type I and Type II Errors

A Type I error occurs when the null hypothesis is erroneously rejected (a false positive), while a Type II error happens when the null hypothesis is not rejected even though it is false (a false negative). The trade-off between these errors is a key consideration in test design, as reducing the probability of one type of error typically increases the risk of the other, and the power of the test is directly related to minimizing Type II errors.

Power Function

The power function of a test indicates the probability of correctly rejecting the null hypothesis for each possible true value of the parameter. It is a function that shows how effective the test is at detecting departures from the null hypothesis. The power at any particular parameter value is the probability that the test statistic will fall into the pre-determined rejection region under that true parameter value.

Hypothesis Testing

Hypothesis testing is a statistical inference procedure that starts with a null hypothesis, representing a baseline or presumed condition (such as a coin being fair), and an alternative hypothesis, which reflects a deviation from that baseline. The objective is to use sample data to decide whether there is enough evidence to reject the null hypothesis in favor of the alternative.

One-Sided vs. Two-Sided Tests

One-sided tests are designed to detect deviations in one specific direction (for example, testing if the probability is greater than 0.5 or less than 0.5), while two-sided tests assess deviations in both directions (checking whether the probability is simply different from 0.5). Because one-sided tests allocate the entire significance level to one tail of the distribution, they generally have higher power in the chosen direction compared to two-sided tests, where the significance level is split between both tails.

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