91Ó°ÊÓ

The press release referenced in the previous exercise also included data from independent surveys of teenage drivers and parents of teenage drivers. In response to a question asking if they approved of laws banning the use of cell phones and texting while driving, \(74 \%\) of the teens surveyed and \(95 \%\) of the parents surveyed said they approved. The sample sizes were not given in the press release, but for purposes of this exercise, suppose that 600 teens and 400 parents of teens responded to the surveys and that it is reasonable to regard these samples as representative of the two populations. Do the data provide convincing evidence that the proportion of teens that approve of cell- phone and texting bans while driving is less than the proportion of parents of teens who approve? 'Test the relevant hypotheses using a significance level of \(.05\).

Step by step solution

01

State the hypotheses

The null hypothesis, denoted \(H_0\), assumes that the proportions are equal, i.e., \(p_{teens} = p_{parents}\).\n\nThe alternative hypothesis, denoted \(H_a\), assumes that the proportion of teens' approval is less than parents', i.e., \(p_{teens} < p_{parents}\).\n\nHere, \(p_{teens}\) is the proportion of teens who approve the use of cell phones and texting while driving, and \(p_{parents}\) is the proportion of parents of teens who approve.

02

Calculate pooled proportion

First, we need to calculate the combined proportion of teens and parents who approve.\n\nThis is done with the formula \(p = \frac{x_1 + x_2}{n_1 + n_2}\), where \(x_1\) and \(x_2\) are the number of successes in each sample, and \(n_1\) and \(n_2\) are the sample sizes. \n\nGiven that \(74 \%\) of 600 teens and \(95 \%\) of 400 parents approved, that translates to \(444\) teens and \(380\) parents. Thus the pooled proportion is \(p = \frac{444 + 380}{600 + 400}\).

03

Calculate z-value

Next, calculate the z-value. The formula for this is \[ z = \frac{p_{teens} - p_{parents}}{\sqrt{p*(1-p)*[\frac{1}{n_{teens}} + \frac{1}{n_{parents}}] }} \]\n\nWe just calculated \(p = \frac{824}{1000}\). \n\nSince \(p_{teens} = \frac{444}{600}\) and \(p_{parents} = \frac{380}{400}\), we can now calculate the z-value.

04

Decision

Now with the resultant z-value we compare this to the z-critical value at significance level \(.05\). If the calculated z-value falls into the rejection region (z < -z critical value) we reject the null hypothesis, otherwise we do not. The z-critical value at \(.05\) significance level for a one-tailed test is approximately -1.645.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Null Hypothesis

In hypothesis testing, the null hypothesis is an initial statement that suggests no effect or no difference in the situation being analyzed. It's represented as \(H_0\). For example, in our scenario involving drivers, the null hypothesis assumes that both teens and parents have an equal proportion of approval regarding banning cell phones and texting while driving. Essentially, this means \(p_{teens} = p_{parents}\).
The null hypothesis is important because it establishes a baseline that the data can be tested against. It is typically assumed to be true until statistical evidence indicates otherwise. If evidence suggests the null hypothesis is unlikely, researchers may reject it in favor of an alternative hypothesis.

Alternative Hypothesis

The alternative hypothesis, denoted as \(H_a\), counters the null hypothesis. It suggests a different outcome than the null hypothesis, implying a change or difference exists. In the case of our study on cell phone use approval, it indicates that the proportion of teens approving is less than that of parents, denoted as \(p_{teens} < p_{parents}\).
Choosing the correct alternative hypothesis is vital because it signals what the research aims to prove. Typically, it's formulated based on prior studies or theories suggesting a certain trend or effect. If the results of the hypothesis test show significant evidence supporting the alternative hypothesis, researchers may conclude that the alternative hypothesis is more plausible than the null hypothesis.

Pooled Proportion

The pooled proportion is a critical concept when comparing two groups. When you have data from two samples, like our teens and parents approving cell and text bans, the pooled proportion combines their data to create one overall proportion of successes. It is calculated with the formula:

• \( p = \frac{x_1 + x_2}{n_1 + n_2} \)

Here, \(x_1\) and \(x_2\) represent the number of approvals in each group, while \(n_1\) and \(n_2\) are the total sample sizes. In the example given, \(444\) teens approve, along with \(380\) parents. Thus, the pooled proportion becomes \( p = \frac{444 + 380}{600 + 400} \).
The pooled proportion helps in standardizing the data, which makes it easier to determine the significance of any differences between groups. This is crucial in hypothesis testing, allowing for accurate comparisons.

Significance Level: 0.05

The significance level, often noted as \( \alpha \), is a threshold used in hypothesis testing to determine when to reject the null hypothesis. A common choice for \( \alpha \) is 0.05, which indicates a 5% risk of concluding a finding is statistically significant when it is not.
Selecting a 0.05 significance level comprises a common practice across many fields, striking a balance between too easily accepting or rejecting hypotheses. In this test, if the calculated test statistic, like the z-value, falls beyond the critical value associated with \( \alpha = 0.05 \), the results are considered statistically significant, implying real differences exist beyond mere chance.
Keep in mind that lowering \( \alpha \) reduces the risk of a false positive, but increases the risk of negatively impacting findings that might actually exist.