/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 List the five parts of a statist... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 1

List the five parts of a statistical test.

Short Answer

Expert verified
Question: Describe and explain the five parts of a statistical test. Answer: The five parts of a statistical test are: 1. Null Hypothesis (H鈧�): A statement about the population parameter, which we assume to be true until proven otherwise, usually stating that there is no effect or relationship between variables being tested. 2. Alternative Hypothesis (H鈧� or H_A): A statement that contradicts the null hypothesis, representing the claim we wish to test or what we believe to be true. 3. Test Statistic: A value calculated from the sample data that helps to decide whether to accept or reject the null hypothesis, such as the t-value, z-value, or Chi-square value. 4. Critical Value and Rejection Region: The critical value divides the region where the null hypothesis is accepted from the region where it is rejected. If the test statistic falls within the rejection region, we reject the null hypothesis. 5. Conclusion and Interpretation: Based on the test statistic and the rejection region, we either reject or fail to reject the null hypothesis. We must also interpret the result in the context of the research question and consider factors such as practical significance, effect size, and study limitations.

Step by step solution

01

1. Null Hypothesis

The Null Hypothesis, denoted as H鈧�, is a statement about the population parameter, which we assume to be true until it is proven otherwise. Typically, the null hypothesis states that there is no effect (e.g., no difference between two population means) or only a chance relationship between the variables being tested.
02

2. Alternative Hypothesis

The Alternative Hypothesis, denoted as H鈧� or H_A, is a statement that is contradictory to the null hypothesis. It represents the claim we wish to test or what we believe to be true (e.g., a difference between two population means). In the test, if we reject the null hypothesis, we accept the alternative hypothesis.
03

3. Test Statistic

The Test Statistic is a value calculated from the sample data, which helps to decide whether to accept or reject the null hypothesis. It establishes the connection between the null hypothesis and the sample data and compares the observed data to what is expected under the null hypothesis. Some common test statistics include the t-value, z-value, and Chi-square value.
04

4. Critical Value and Rejection Region

The Critical Value is the value that divides the region where the null hypothesis is accepted from the region where it is rejected (the Rejection Region). If the test statistic falls within the rejection region, we reject the null hypothesis and accept the alternative hypothesis. The critical value depends on the level of significance (伪) and the type of statistical test conducted (one-tailed or two-tailed).
05

5. Conclusion and Interpretation

In the final step, we make a conclusion based on the test statistic and the rejection region. If the test statistic falls within the rejection region, we reject the null hypothesis, and if it falls outside the rejection region, we fail to reject the null hypothesis. We must also interpret the result in the context of the research question, considering factors such as practical significance, effect size, and the limitations of the study.

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 91影视!

Key Concepts

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

Null Hypothesis

The null hypothesis, symbolized as H鈧�, forms the backbone of any statistical test. It is a default position that indicates no relationship or difference between the variables tested. Think of the null hypothesis as the skeptic's stance鈥攊t assumes that whatever you're trying to prove hasn't happened (like no change in a treatment effect or no correlation between two factors).

In a clinical trial, for example, the null hypothesis might state that a new drug has the same effect as the current standard medication. It's the starting point of hypothesis testing, and the goal is to gather evidence that challenges this assumption. The theory that there's no effect will hold unless data provides strong evidence to the contrary.

Improving Comprehension:

  • Simplify terminology: The null hypothesis is the initial belief that there's no effect or difference.
  • Use everyday analogies: It's like assuming a light switch won't affect a bulb's state鈥攐ff is off unless proven otherwise.
Alternative Hypothesis

The alternative hypothesis, denoted as H鈧� or H_A, is the assertion we're trying to support with evidence. Contrary to the null hypothesis, the alternative hypothesis suggests that there is indeed an effect or a relationship present between the variables we're examining.

Using our clinical trial example, the alternative hypothesis might propose that the new drug has a different effect than that of the current medication鈥攅ither better or worse. If evidence (i.e., data) suggests this is the case, we may have grounds to reject the null hypothesis in favor of this alternative stance.

Effective Educational Approach:

  • Contrast with null hypothesis: Highlight that the alternative hypothesis is essentially the opposite claim of the null hypothesis.
  • Relate to real-world scenarios: Explain the alternative hypothesis as the 'case for change' or 'evidence for a new effect' in a scientific setting.
Test Statistic

The test statistic is a key number calculated from the data, allowing us to decide whether the data supports the null hypothesis or not. This statistic is derived from your sample and is used to measure how far your sample statistic is from the null hypothesis's proposed parameter.

For example, in testing whether a coin is biased, the test statistic could be the number of heads from a series of coin flips compared to what we'd expect if the coin were fair. Different types of tests have different test statistics, such as t-statistics for t-tests, z-statistics for z-tests, and chi-square statistics for chi-square tests.

Clarification for Students:

  • Focus on utility: The test statistic helps us make the jump from sample data to a conclusion about the entire population.
  • Contextual examples: Illustrate test statistics using examples relevant to the student's field of study for better understanding.
Critical Value

The critical value is a threshold that determines the boundary between the rejection and non-rejection areas in a statistical test. It aligns with your chosen level of significance (commonly 0.05 for a 5% significance level), which reflects your tolerance for error.

If the test statistic exceeds the critical value, it lands in the rejection zone, which indicates that the null hypothesis is unlikely given the data. This step is akin to a 'boundary marker'鈥攊t tells us how extreme the test statistic needs to be for us to consider the results statistically significant and to reject the null hypothesis.

Demystifying Concepts:

  • Relate to decision-making: The critical value helps us make a 'yes or no' decision about the null hypothesis.
  • Use clear language: Explain the critical value as the 'line in the sand' which separates the outcome into 'normal' or 'unusual' in terms of the data.
Hypothesis Testing Interpretation

Interpretation is the culminating part of hypothesis testing where we give meaning to the results. It's about translating the numbers into a coherent understanding of what the data tells us about our question. Depending on where the test statistic falls in relation to the critical value, we either reject or fail to reject the null hypothesis.

But our job doesn't end with this binary decision. Understanding the practical significance of the outcomes, evaluating the effect size, and acknowledging the test's limitations are all crucial parts of a well-rounded interpretation. The point is to connect the statistical results back to the real-world question we started with.

Interpretation Guidance:

  • Go beyond 'reject' or 'do not reject': Encourage discussion about what the statistical result means in the broader research or real-world context.
  • Importance of effect size: Emphasize that even if we reject the null hypothesis, understanding the magnitude of the difference or relationship (effect size) is essential.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Analyses of drinking water samples for 100 homes in each of two different sections of a city gave the following information on lead levels (in parts per million): $$ \begin{array}{lcc} \hline & \text { Section 1 } & \text { Section 2 } \\ \hline \text { Sample Size } & 100 & 100 \\ \text { Mean } & 34.1 & 36.0 \\ \text { Standard Deviation } & 5.9 & 6.0 \end{array} $$ a. Calculate the test statistic and its \(p\) -value to test for a difference in the two population means. Use the \(p\) -value to evaluate the significance of the results at the \(5 \%\) level. b. Use a \(95 \%\) confidence interval to estimate the difference in the mean lead levels for the two sections of the city. c. Suppose that the city environmental engineers will be concerned only if they detect a difference of more than 5 parts per million in the two sections of the city. Based on your confidence interval in part b, is the statistical significance in part a of practical significance to the city engineers? Explain.

Ground beef is packaged in small trays, intended to hold 1 pound of meat. A random sample of 35 packages in the small tray produced weight measurements with an average of 1.01 pounds and a standard deviation of .18 pound. a. If you were the quality control manager and wanted to make sure that the average amount of ground beef was indeed 1 pound, what hypotheses would you test? b. Find the \(p\) -value for the test and use it to perform the test in part a. c. How would you, as the quality control manager, report the results of your study to a consumer interest group?

As a group, students majoring in the engineering disciplines have the highest salary expectations, followed by those studying the computer science fields, according to a Michigan State University study. \({ }^{8}\) To compare the starting salaries of college graduates majoring in electrical engineering and computer science, random samples of 50 recent college graduates in each major were selected and the following information was obtained: $$ \begin{array}{lcc} \hline \text { Major } & \text { Mean (\$) } & \text { SD } \\ \hline \text { Electrical Engineering } & 62,428 & 12,500 \\ \text { Computer Science } & 57,762 & 13,330 \\ \hline \end{array} $$ a. Do the data provide sufficient evidence to indicate a difference in average starting salaries for college graduates who majored in electrical engineering and computer science? Test using \(\alpha=.05\). b. Calculate a \(95 \%\) confidence interval for the difference in the two population means. Does this confirm your conclusion in part a? Explain.

A random sample of \(n=35\) observations from a quantitative population produced a mean \(\bar{x}=2.4\) and a standard deviation of \(s=.29 .\) Your research objective is to show that the population mean \(\mu\) exceeds 2.3. Use this information to answer the questions. Locate the rejection region for the test using a \(5 \%\) significance level.

Although there is a big difference in costs between a Tesla \(S\) and a Chevrolet Bolt, is there a difference in the range of miles for these vehicles between charges? Suppose that the results of driving tests are as follows: \({ }^{-10}\) $$ \begin{array}{lccc} \hline & & \text { Standard } & \text { Sample } \\ \text { Car } & \text { Mean } & \text { Deviation } & \text { Size } \\ \hline \text { Tesla S } & 230.2 & 14.3 & 40 \\ \text { Bolt } & 236.8 & 18.8 & 40 \\ \hline \end{array} $$ Is there sufficient evidence to indicate a difference in the average driving ranges for these two vehicles? Test using \(\alpha=.01\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.