Question

"Most Like It Hot" is the title of a press release issued by the Pew Research Center (March 18, 2009, www.pewsocialtrends. org). The press release states that "by an overwhelming margin, Americans want to live in a sunny place." This statement is based on data from a nationally representative sample of 2,260 adult Americans. Of those surveyed, 1,288 indicated that they would prefer to live in a hot climate rather than a cold climate. Suppose that you want to determine if there is convincing evidence that a majority of all adult Americans prefer a hot climate over a cold climate. a. What hypotheses should be tested in order to answer this question? b. The \(P\) -value for this test is 0.000001 . What conclusion would you reach if \(\alpha=0.01 ?\) For questions \(10.85-10.86,\) answer the following four key questions (introduced in Section 7.2 ) and indicate whether the method that you would consider would be a large-sample hypothesis test for a population proportion.

Short answer

a. Null Hypothesis: \(H_0: p \leq 0.5\), Alternative Hypothesis: \(H_A: p > 0.5\). b. Since the p-value (0.000001) is less than the significance level (\(\alpha=0.01\)), the null hypothesis is rejected favouring the alternative hypothesis. This indicates that a majority of adult Americans probably prefer a hot climate. The method used is a Large-Sample Hypothesis Test for a population proportion.

Step-by-step solution

Formulate Hypotheses

The Null Hypothesis \(H_0\) is that the proportion \(p\) of all adult Americans who prefer a hot climate is less or equal than 50%. In other words, \(H_0: p \leq 0.5\). The alternative hypothesis \(H_A\) must be the opposite, which is that more than 50% of all adult Americans prefer a hot climate, or \(H_A: p > 0.5\)

Evaluate the p-value

The p-value of 0.000001 obtained for the test is smaller than the significance level, \(\alpha=0.01 \). This indicates that the results are statistically significant and we reject the null hypothesis.

Conclusion

Since the p-value is less than the significance level, we can confidently reject the null hypothesis in favor of the alternative hypothesis. This implies that the evidence suggests that more than 50% of all adult Americans prefer a hot climate over a cold climate.

Determine the Method

The method used to answer this is indeed a large-sample hypothesis test for a population proportion since the sample size (2,260) is large and we are testing a single population proportion against a specific value (50%)

Key concepts

Population Proportion

Understanding population proportion is crucial in the realm of statistical analysis, particularly when determining preferences or characteristics within a whole population. The population proportion, denoted by the symbol \( p \), refers to the ratio of members in a population who possess a particular attribute to the total number of individuals in that population. For instance, if we consider our example with the preference for a hot climate, the population proportion would represent the percentage of all adult Americans who prefer living in warmer regions. When conducting a hypothesis test, such as the one in the exercise, this is the underlying parameter we wish to estimate and make inferences about, based on the sample data.

During this process, we calculate an estimate of the population proportion from the sample data, which in our scenario was determined from the 1,288 respondents preferring a hot climate out of 2,260 surveyed. This sample proportion is then used to measure against the hypothesized population proportion to determine if the observed data can provide sufficient evidence for or against the claim being tested.

Null Hypothesis

The null hypothesis, symbolized as \( H_0 \), serves as a statement that there is no effect or no difference, and it will hold true until evidence indicates otherwise. It's a default position that suggests the suspected phenomenon—such as the majority preference for a hot climate—is not present. In our example, the null hypothesis posits that the true proportion of all adult Americans preferring a hot climate is 50% or less (\(H_0: p \leq 0.5\)).

This hypothesis sets the stage for statistical testing, where the objective is to either reject or fail to reject the null based on the data from our sample. The null hypothesis is framed so that it denotes no change or no difference, making it a critical component of hypothesis testing because it defines the statement that we are seeking to challenge with sample evidence.

P-Value

The p-value is a fundamental concept in hypothesis testing that measures the strength of evidence against the null hypothesis. It represents the probability of observing a sample outcome as extreme as, or more extreme than, the actual observed results, assuming that the null hypothesis is true. A small p-value indicates that such an extreme observed outcome would be very unlikely under the null hypothesis. Therefore, it suggests strong evidence against \( H_0 \) and in favor of the alternative hypothesis \( H_A \).

In our example with the preference for a hot climate, the p-value calculated from the sample data was 0.000001. Such a low p-value suggests that, assuming the null hypothesis is true, the chance of randomly selecting a sample where 1,288 out of 2,260 preferring hot climate would be virtually nonexistent. Therefore, the evidence weighs heavily against the null hypothesis.

Significance Level

The significance level, often denoted by \( \alpha \), is a threshold chosen by the researcher to decide whether to reject the null hypothesis. It reflects the researcher's tolerance for making a Type I error, which is the mistake of rejecting the null hypothesis when it is actually true. Common significance levels include 0.05, 0.01, and 0.10, with a smaller value indicating a lower probability of making a Type I error.

In our exercise, a significance level of 0.01 (\( \alpha = 0.01 \) was chosen. When we compare this significance level to our calculated p-value of 0.000001, we find that the p-value is much smaller than \( \alpha \), thus according to the rules of hypothesis testing, we reject the null hypothesis. This decision indicates that there is sufficient statistical evidence that more than 50% of all adult Americans prefer a hot climate, with a less than 1% chance of committing a Type I error.

Alternative Hypothesis

In contrast to the null hypothesis, the alternative hypothesis (\( H_A \) or \( H_1 \) suggests that there is a significant effect or difference that contradicts \( H_0 \). It is essentially what we suspect might be true instead of the null hypothesis. One essential aspect of the alternative hypothesis is that it is directional or non-directional, indicating whether the research is testing for a specific direction of the effect or for any change at all.

For the climate preference example, the alternative hypothesis was that more than 50% of adult Americans prefer a hot climate, denoted as \( H_A: p > 0.5 \). The directionality here is clear; we're looking for proof of a majority preference for a warmer climate. This conclusion typically requires strong evidence from the sample, which is evaluated through the p-value. As the evidence from the p-value was compellingly against the null hypothesis, we accept the alternative hypothesis, thus supporting the claim that a majority of adult Americans prefer hotter climates.