A random sample of 802 supermarket shoppers determined that...

A random sample of 802 supermarket shoppers determined that 378 shoppers preferred generic-brand items. Test at the $10 \%$ level the null hypothesis that at least one-half of all shoppers preferred genericbrand items against the alternative that the population proportion is less than one-half. Find the power of a $10 \%$-level test if, in fact, $45 \%$ of the supermarket shoppers preferred generic brands.

Key Concepts

Power of a Test

The power of a test is the probability that the test correctly rejects a false null hypothesis, thereby avoiding a Type II error. It reflects the sensitivity of the test to detect an effect when one exists. High power is desirable in hypothesis testing because it means there is a higher likelihood of detecting a true deviation from the null hypothesis, which is particularly important in making informed decisions based on the test outcomes.

Hypothesis Testing

Hypothesis testing is a statistical procedure used to assess if there is enough evidence in a sample to infer that a certain condition holds for the entire population. It involves formulating a null hypothesis (a statement of no effect or the status quo) and an alternative hypothesis (an assertion that contradicts the null). The evidence from the sample is then compared against these hypotheses using a test statistic to decide whether to reject the null hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis is the claim that there is no significant difference or effect—in this case, that the proportion meets or exceeds a certain threshold—while the alternative hypothesis represents the claim to be tested and typically suggests a deviation from the null condition. The formulation of these hypotheses is crucial as it sets the framework for the statistical test design and interpretation of the results.

Significance Level and Type I Error

The significance level, often denoted by alpha (?), is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. It represents the threshold below which the observed test statistic must fall to reject the null hypothesis. The choice of ? reflects the level of risk a researcher is willing to take for this error, and it directly influences the stringency of the test.

Z-Test for Proportions

A Z-test for proportions is a statistical method used to determine whether the observed proportion in a sample is significantly different from a hypothesized population proportion. This test relies on the normal approximation to the sampling distribution of the sample proportion when the sample size is sufficiently large, allowing one to compute a Z-statistic that measures the distance between the observed proportion and the hypothesized value in standard error units.

Rejection Region

The rejection region is a set of values for the test statistic that leads to the rejection of the null hypothesis. It is determined based on the chosen significance level and the direction of the alternative hypothesis (one-tailed or two-tailed). If the calculated test statistic falls within this region, it provides sufficient evidence to conclude that the null hypothesis should be rejected.

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