91Ó°ÊÓ

Please provide the following information for Problems. (a) What is the level of significance? State the null and alternate hypotheses. (b) Check Requirements What sampling distribution will you use? What assumptions are you making? Compute the sample test statistic and corresponding \(z\) or \(t\) value as appropriate. (c) Find (or estimate) the \(P\) -value. Sketch the sampling distribution and show the area corresponding to the \(P\) -value. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level \(\alpha ?\) (e) Interpret your conclusion in the context of the application. Note: For degrees of freedom \(d . f .\) not in the Student's \(t\) table, use the closest \(d . f .\) that is smaller. In some situations, this choice of \(d . f .\) may increase the \(P\) -value a small amount and therefore produce a slightly more "conservative" answer. Sociology: Trusting People Generally speaking, would you say that most people can be trusted? A random sample of \(n_{1}=250\) people in Chicago ages \(18-25\) showed that \(r_{1}=45\) said yes. Another random sample of \(n_{2}=280\) people in Chicago ages \(35-45\) showed that \(r_{2}=71\) said yes (based on information from the National Opinion Research Center, University of Chicago). Does this indicate that the population proportion of trusting people in Chicago is higher for the older group? Use \(\alpha=0.05\)

Step by step solution

01

Level of Significance and Hypotheses

The level of significance given is \( \alpha = 0.05 \). The null hypothesis \( H_0 \) is that the population proportion of trusting people is the same between the two age groups: \( p_1 = p_2 \). The alternative hypothesis \( H_a \) is that the population proportion of trusting people is higher for the older age group: \( p_1 < p_2 \).

02

Check Requirements and Compute Test Statistic

For a hypothesis test for two proportions, we use a standard normal distribution (\( z \)-test) if the sampling distribution conditions are met. The assumptions are that both samples are random and independent, and both \( n_1 \times \hat{p}_1 \), \( n_1 \times (1-\hat{p}_1) \), \( n_2 \times \hat{p}_2 \), and \( n_2 \times (1-\hat{p}_2) \) are all \( \ge 5 \). Calculate the sample proportions: \( \hat{p}_1 = \frac{45}{250} \) and \( \hat{p}_2 = \frac{71}{280} \). Compute the pooled sample proportion: \( \hat{p} = \frac{r_1 + r_2}{n_1 + n_2} = \frac{45 + 71}{250 + 280} \). Use the formula for the test statistic: \[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \].

03

Calculate P-value and Sketch the Distribution

Calculate the \( z \)-value using the proportions from Step 2. Find the corresponding \( P \)-value for this \( z \)-value using the standard normal distribution table. Sketch a bell curve marking the critical region based on the \( \alpha = 0.05 \) level and the \( z \)-value obtained to show the area corresponding to the \( P \)-value.

04

Decision Regarding Null Hypothesis

Compare the \( P \)-value with \( \alpha = 0.05 \). If \( P \)-value < \( \alpha \), reject the null hypothesis; otherwise, fail to reject the null hypothesis. Determine if the data are statistically significant at the \( 0.05 \) level based on this comparison.

05

Interpretation

If the null hypothesis is rejected, conclude that there is significant evidence to support that the proportion of trusting people is higher in the older age group in Chicago. If not rejected, conclude that there is not enough evidence to support a difference in trust proportions between the age groups.

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Key Concepts

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

Level of Significance

The level of significance, often denoted by \( \alpha \), is crucial in hypothesis testing. It represents the threshold for making a decision about the null hypothesis. If the \( P \)-value is less than or equal to \( \alpha \), there's enough evidence to reject the null hypothesis. In the original exercise, \( \alpha = 0.05 \). This means there is a 5% risk of concluding that the older age group in Chicago is more trusting when that might not be the case. It's akin to setting your tolerance for making a wrong decision. Choosing the right level of significance depends on the context, often striking a balance between being too strict (which might ignore a true effect) and too lenient (which might accept a false effect). Most common choices are \( \alpha = 0.05 \) or \( \alpha = 0.01 \), depending on how much certainty is needed in the results.

Null and Alternate Hypotheses

In any hypothesis testing, you start with the null hypothesis \( H_0 \), which is essentially the status quo. It declares that no statistical difference or effect exists. For the exercise, \( H_0 \) was that the proportion of trusting people is the same in both age groups. Mathematically, it's expressed as \( p_1 = p_2 \). Then comes the alternate hypothesis \( H_a \), which represents the effect or difference you suspect exists. Here, the \( H_a \) proposes that the older age group is actually more trusting: \( p_1 < p_2 \).Formulating these hypotheses correctly is fundamental since they define what you're testing. They guide the statistical analysis and decision-making process, referring back to what you initially aimed to prove or disprove.

P-value

The \( P \)-value helps us decide whether to reject the null hypothesis. It measures the probability of observing data as extreme as the current sample, assuming the null hypothesis is true.In simpler terms, if this \( P \)-value is really low, it suggests that the observed data is very unlikely under the null hypothesis, prompting us to reject it.For instance, in the provided solution, we calculate the \( P \)-value corresponding to the \( z \)-value from our hypothesis test. We use this value to see if it's smaller than our level of significance \( \alpha \) (0.05). If so, it indicates significant results, suggesting a difference in trust levels between age groups.

Sampling Distribution

Understanding the sampling distribution is crucial in determining the proper test statistic to use. When you conduct hypothesis testing for sample proportions, you often rely on the standard normal distribution, also known as the \( z \)-distribution.Here, we test whether the proportions differ between two samples (ages 18-25 and ages 35-45 in Chicago). For this, you first calculate the sample proportions: \( \hat{p}_1 \) and \( \hat{p}_2 \). Then, you find the pooled sample proportion \( \hat{p} \) which combines both samples to provide a more stable estimate.The assumptions for using the \( z \)-test involve randomness, independence of samples, and sufficient sample size so that \( n \cdot \hat{p} \) and \( n \cdot (1-\hat{p}) \) for each sample are at least 5. These criteria ensure that the sampling distribution approximates a normal distribution, letting you apply the \( z \)-test correctly.