91Ó°ÊÓ

In a 2009 nonscientific poll on the Web site of the Daily Gazette of Schenectady, New York, readers were asked the following question: "Are you less inclined to buy a General Motors or Chrysler vehicle now that they have filed for bankruptcy?" Of the respondents, \(56.1 \%\) answered "Yes" (http://www. dailygazette.com/polls/2009/jun/Bankruptcy/). In a recent survey of 1200 adult Americans who were asked the same question, 615 answered "Yes." Can you reject the null hypothesis at the \(1 \%\) significance level in favor of the alternative that the percentage of all adult Americans who are less inclined to buy a General Motors or Chrysler vehicle since the companies filed for bankruptcy is different from \(56.1 \%\) ? Use both the \(p\) -value and the critical-value approaches.

Step by step solution

01

Formulating Hypotheses

In this case, the null hypothesis (\(H_0\)) is that the proportion of all adult Americans less inclined to buy a General Motors or Chrysler vehicle is still \(56.1\%\) i.e., \(p = 0.561\). The alternative hypothesis (\(H_1\)) is that the proportion is different from \(56.1\%\) i.e., \(p\neq0.561\).

02

Calculating Test Statistic

The test statistic for comparing two proportions is calculated using the formula: \(z = \frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}\), where \(\hat{p}\) is the sample proportion, \(p_0\) is the population proportion, and \(n\) is the sample size. Substituting the given numbers, the test statistic is \(z = \frac{615/1200-0.561}{\sqrt{0.561(1-0.561)/1200}}\).

03

Calculating p-value

The p-value is the probability that you would obtain the observed statistic given that the null hypothesis is true. It is calculated by looking up the test statistic in a standard normal (Z) table or using software that can compute it.

04

Determining Critical Values

For a two-tailed test at \(1\%\) significance level, the critical z-values are \(z = -2.58\) and \(z = 2.58\). Any test statistic that lies beyond these critical values falls in the rejection region.

05

Conclusion

If the p-value is less than the significance level (\(0.01\)), or if the test statistic falls in the rejection region, then we reject the null hypothesis in favor of the alternative. Otherwise, we do not have enough evidence to reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Null Hypothesis

In hypothesis testing, the **Null Hypothesis**, often denoted as \(H_0\), is a statement that indicates no effect or no difference, and it serves as a starting point for statistical testing. It is what we assume to be true unless there is substantial evidence against it. For example, in the scenario where we analyze American opinions on buying General Motors or Chrysler vehicles post-bankruptcy, the null hypothesis states that the true proportion of individuals with reduced inclination to buy these vehicles is \(56.1\%\), or mathematically \(p = 0.561\).

The key points to understand about the null hypothesis are:

• It represents the status quo or commonly accepted fact until proven otherwise.
• We aim to find evidence against \(H_0\) in order to consider an alternative point of view.

By setting this hypothesis, researchers have a clear benchmark to compare against using statistical evidence.

Alternative Hypothesis

The **Alternative Hypothesis**, denoted as \(H_1\) or \(H_a\), suggests that there is a statistically significant effect or a difference from what is stated in the null hypothesis. In our example, the alternative hypothesis states that the proportion of people less inclined to purchase a vehicle from General Motors or Chrysler is different from \(56.1\%\), which can be expressed as \(p eq 0.561\).

Here are some notable aspects about the alternative hypothesis:

• It represents the outcome that the researcher aims to support.
• The ultimate goal of hypothesis testing is to determine if there is enough evidence to reject the null hypothesis in favor of this alternative.
• Depending on the research question, the alternative hypothesis may be one-sided or two-sided. In this case, it's two-sided because we're interested in any difference from \(56.1\%\).

Identifying the alternative hypothesis helps to frame the direction and type of analysis necessary in the study.

p-value

The **p-value** is a vital concept in hypothesis testing. It indicates the probability of obtaining the observed results when the null hypothesis is true. Essentially, it helps to measure the strength of the evidence against the null hypothesis.

For instance, if you calculate a test statistic and use this value to find a p-value, this will tell you:

• If the p-value is low (usually below \(0.01\), \(0.05\), or other predetermined significance levels), there is enough evidence to reject the null hypothesis.
• If the p-value is high, it suggests that the null hypothesis cannot be rejected based on the evidence at hand.

The p-value serves as a critical decision-making tool, helping researchers determine the statistical significance of their findings. The smaller the p-value, the stronger the evidence against \(H_0\).

Critical Value

The **Critical Value** is a threshold used in hypothesis testing to define the boundaries of the acceptance region for the null hypothesis. Depending on the significance level, it helps decide whether the test statistic falls within the range of values that would lead to rejecting the null hypothesis.

In the context of our problem, with a significance level of \(1\%\), we use a critical value corresponding to this confidence level:

• For a two-tailed test, the critical z-values are \(-2.58\) and \(2.58\).
• If the calculated test statistic is lower than \(-2.58\) or higher than \(2.58\), it falls into the rejection region, providing enough grounds to reject \(H_0\).

By using critical values, you can make concrete decisions about the null hypothesis and understand the range of normal vs. exceptional outcomes.

Test Statistic

The **Test Statistic** is a standardized value computed from sample data. It underpins the decision-making process in hypothesis testing, indicating how far the sample statistic is from the null hypothesis value. In our example, the test statistic is calculated to compare the proportions of respondents who were less inclined to buy the vehicles.

The formula used here is:\[ z = \frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)/n}} \]where:

• \(\hat{p}\) is the sample proportion, calculated as \(\frac{615}{1200}\).
• \(p_0\) is the hypothesized population proportion, \(0.561\).
• \(n\) is the sample size, \(1200\).

The test statistic, often represented as a 'z-score' in this context, helps determine how unusual the sample results are under the null hypothesis. This metric is essential in assessing whether to reject \(H_0\) by comparing it to the critical values.