Question

According to a Washington Post-ABC News poll, 331 of 502 randomly selected U.S. adults interviewed said they would not be bothered if the National Security Agency collected records of personal telephone calls they had made. Is there sufficient evidence to conclude that a majority of U.S. adults feel this way? Test the appropriate hypotheses using a .01 significance level.

Short answer

The null hypothesis is rejected, so there is sufficient evidence at the 0.01 significance level to conclude that a majority of U.S. adults would not be bothered if the NSA collected records of personal calls.

Step-by-step solution

Set up the hypotheses

The null hypothesis \(H_0\): \(p = 0.5\) suggests that half of the U.S. adults would not be bothered if the NSA collected records of personal phone calls they made. The alternative hypothesis \(H_1\): \(p > 0.5\) suggests that more than 50% of U.S. adults would not be bothered.

Calculate the sample proportion

The sample proportion \(\hat{p}\) is given by the number of successes (those wouldn't be bothered) divided by the sample size, i.e., \(\hat{p}=331/502=0.659\).

Perform the test statistic calculation

The test statistic for a hypothesis test for a population proportion is \(Z = (\hat{p} - p_0)/\sqrt{p_0(1 - p_0)/n}\), where \(p_0\) is the proportion under the null hypothesis, \(n\) is the sample size. Substituting in the values gives \(Z = (0.659 - 0.5)/\sqrt{0.5(1 - 0.5)/502} = 7.45.\)

Determine the p-value and make a decision

The p-value for a Z-value of 7.45 is nearly 0, much less than the significance level of 0.01. Therefore, reject the null hypothesis. There is sufficient evidence at the 0.01 significance level to conclude that a majority of U.S. adults would not be bothered if the NSA collected records of personal calls.

Key concepts

Statistical Significance

Statistical significance is a term used to determine if the results observed in a study are unlikely to have occurred by chance alone. When conducting hypothesis tests, like in the provided exercise, researchers set a significance level, often denoted by \( \alpha \), which serves as a threshold for deciding when to reject the null hypothesis. In our example, the significance level is set at 0.01, indicating there is a 1% risk of concluding that a difference exists when there is no actual difference. If the p-value calculated from the test statistic is less than \( \alpha \), we say that the results are statistically significant and the null hypothesis is rejected. Crucially, this does not mean that the result is practically significant or important, only that it is statistically unlikely to be due to random variation in the sample.

In the context of the poll, finding statistical significance means we have enough evidence to reasonably conclude that the observed proportion represents the population's behavior rather than being a fluke of this specific sample.

Population Proportion

Population proportion, denoted as \( p \), is a measure that represents the fraction of the population that has a particular characteristic. In the exercise, we're concerned with the proportion of U.S. adults who wouldn't be bothered by NSA surveillance of personal calls. It’s important to distinguish between the actual population proportion and the sample proportion (denoted as \(\textasciicircum p\)). The latter is an estimate derived from a sample and is used to make inferences about the true population proportion.

The accuracy and reliability of these inferences depend on the size and randomness of the sample. If the sample size is too small, or if it's not representative of the population, our conclusions about the population proportion may be flawed. In the given solution, the sample proportion calculated from the poll data served as the foundation for testing hypotheses about the general sentiment of U.S. adults.

p-value

The p-value is a pivotal concept in hypothesis testing. It is a probability that expresses how extreme the observed data are, assuming the null hypothesis is true. In other words, it tells us how likely it is to obtain a result at least as extreme as the one observed if there were actually no effect in the population. A small p-value suggests that the observed data are unlikely under the null hypothesis and hence provide evidence against it.

In our exercise, a p-value nearly 0 indicated that obtaining such a high sample proportion of U.S. adults who aren't bothered by NSA's records collection would be extremely unlikely if the true proportion was really 50% or less. As a result, we reject the null hypothesis in favor of the alternative hypothesis because the p-value is smaller than the pre-set alpha level of 0.01.

Null Hypothesis

The null hypothesis, represented as \( H_0 \), is a default statement used in statistical hypothesis testing that there is no effect or no difference. It serves as a starting point for statistical significance testing and is presumed true until evidence suggests otherwise. In hypothesis testing, we never prove the null hypothesis; we can only provide enough evidence to reject it.

For the poll regarding NSA surveillance, the null hypothesis states that the proportion of U.S. adults who wouldn't be bothered (\( p \)) is equal to 0.5 - implying no majority either way. The exercise's question revolves around whether we can reject this hypothesis and assert with confidence that the true sentiment of the population deviates from this stated proportion.

Alternative Hypothesis

Contrasting the null hypothesis is the alternative hypothesis, labeled as \( H_1 \) or \( H_a \). It proposes what we suspect might be true instead and is the claim we are trying to find evidence for. The alternative hypothesis is only accepted if the data provides sufficient evidence to reject the null hypothesis.

In the case of our exercise, the alternative hypothesis declares that the proportion of people who would not be bothered by NSA call record collection (\( p \)) is greater than 0.5. When the test statistic led to a nearly 0 p-value, it indicated strong evidence in support of this alternative hypothesis, suggesting that it is likely that a majority of the population is indeed unconcerned by the surveillance in question.