91Ó°ÊÓ

A Michigan study concerning preference for outdoor activities used a questionnaire with a 6 -point Likert-type response in which 1 designated "not important" and 6 designated "extremely important." A random sample of \(n_{1}=46\) adults were asked about fishing as an outdoor activity. The mean response was \(\bar{x}_{1}=4.9 .\) Another random sample of \(n_{2}=51\) adults were asked about camping as an outdoor activity. For this group, the mean response was \(\bar{x}_{2}=4.3 .\) From previous studies, it is known that \(\sigma_{1}=1.5\) and \(\sigma_{2}=1.2 .\) Does this indicate a difference (either way) regarding preference for camping versus preference for fishing as an outdoor activity? Use a \(5 \%\) level of significance. Note: \(\mathrm{A}\) Likert scale usually has to do with approval of or agreement with a statement in a questionnaire. For example, respondents are asked to indicate whether they "strongly agree," "agree," "disagree," or "strongly disagree" with the statement.

Step by step solution

01

State the Hypotheses

We need to determine if there is a significant difference in preference for fishing versus camping. Therefore, we set up our null hypothesis as \[ H_0: \mu_1 = \mu_2 \]which states there is no difference in preference for camping and fishing. Our alternative hypothesis is \[ H_a: \mu_1 eq \mu_2 \]which indicates there is a difference.

02

Determine the Significance Level

The problem specifies a \(5\%\) level of significance, so we set \[ \alpha = 0.05. \]

03

Use the Z-Test for Two Means

Compute the test statistic to compare the two sample means using the formula for the Z-test for two independent samples: \[ z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}} \]Substitute the given values: \[ z = \frac{4.9 - 4.3}{\sqrt{\frac{1.5^2}{46} + \frac{1.2^2}{51}}} = \frac{0.6}{\sqrt{0.04891 + 0.02824}} = \frac{0.6}{\sqrt{0.07715}} = \frac{0.6}{0.27765} \approx 2.16. \]

04

Determine the Critical Value

For a two-tailed test at a \(5\%\) significance level, we find the critical Z-values using a standard normal distribution table. The critical values are \( \pm 1.96 \).

05

Make a Decision

Compare the calculated Z-value to the critical values. Since \(2.16 > 1.96\), the calculated Z-value falls in the rejection region. Therefore, we reject the null hypothesis.

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.

Understanding the Z-Test for Two Means

The Z-test for two means is a statistical method used to determine if there are significant differences between the means of two groups. It's particularly useful when you know the population standard deviations and are dealing with large sample sizes. In this exercise, we want to see if there's a difference in preferences for fishing compared to camping as outdoor activities among groups of adults. By using the Z-test, we're able to compare the average Likert scale scores between these two activities.

Here's how it works:

• Calculate the difference between the two sample means. This gives you a point of comparison.
• Determine the standard deviation of the difference by using the known population standard deviations and the sample sizes.
• Compute the Z-score, which tells you how far away, in standard deviations, your observed mean difference is from the hypothesized difference (which is zero in this case).

When the Z-score is greater than the critical value at your level of significance, you have evidence to suggest a difference exists.

Decoding the Significance Level

In hypothesis testing, the significance level, denoted by \( \alpha \), reflects the risk you're willing to take of making a Type I error, which is rejecting the null hypothesis when it's actually true. It's essentially the threshold you set for deciding whether to reject the null hypothesis. In this exercise, we've decided on a \(5\%\) significance level, or \( \alpha = 0.05 \).

The significance level affects the critical values in a Z-test. For instance:

• A \(5\%\) significance level corresponds to critical Z-values of \( \pm 1.96\) in a two-tailed test. This means the decision rule for rejecting the null hypothesis involves comparing the computed Z-score against these critical values.
• If the computed Z falls outside \( \pm 1.96\), it suggests the mean difference is statistically significant.

Understanding this concept helps determine how "sure" you need to be about the results before deciding that a real difference exists.

The Likert Scale Explained

A Likert scale is a common tool used in surveys and questionnaires to measure attitudes or preferences. It's named after psychologist Rensis Likert, who developed this scaling method. In our exercise, the Likert scale rates the perceived importance of certain outdoor activities on a scale from 1 to 6.

Here’s how a typical Likert scale operates:

• The lowest number indicates a negative response or low agreement—"not important" in this study.
• The highest number reflects strong agreement or high importance—"extremely important" in our questionnaire.
• The scale captures a range of attitudes rather than a simple yes-no dichotomy, allowing for more nuanced data analysis.

By using the scale, we translate qualitative feelings and opinions into quantitative data, which can then be statistically analyzed, as seen in the Z-test for two means.

What is a Null Hypothesis?

The null hypothesis (\(H_0\)) is a key concept in hypothesis testing that represents a statement of no effect or no difference. It is the default position that indicates no relationship between two measured phenomena. In statistical testing, you always start with the null hypothesis.

In our specific case regarding outdoor activity preferences:

• The null hypothesis posits that there is no difference in preference between fishing and camping. Mathematically, it can be represented as \( \mu_1 = \mu_2 \).
• This hypothesis is tested against the alternative hypothesis to determine if the observed data provides enough evidence to reject it.

Starting with a null hypothesis is crucial because it gives a baseline that alternative hypotheses are tested against, using data analysis to draw conclusions.

Understanding the Alternative Hypothesis

The alternative hypothesis (\(H_a\)) is a statement that contradicts the null hypothesis, suggesting that there is an effect, a difference, or a relationship between two variables. This hypothesis is what you hope to demonstrate is true in an experiment or study.

For our outdoor activity study:

• The alternative hypothesis posits that there is a difference in preference for fishing versus camping, expressed as \( \mu_1 eq \mu_2 \).
• If the statistical test results allow us to reject the null hypothesis, we accept the alternative hypothesis as the better explanation for the observed data.

The alternative hypothesis drives the research question and the direction of the testing. It represents the new claim or theory being evaluated by the data collected.