# Null hypothesis and alternative hypothesis

##### Intros

###### Lessons

- Hypothesis Testing is the method of testing whether or not a claim is valid

Two types of claims:

• Proportions: Data given by percentages, %

• Means: given by data measurements, $\mu$

Null Hypothesis $(H_0)$:

The result that is hoped to be proven false. It is a single parameter.

Given by: "$=$"

Alternative Hypothesis $(H_1)$:

The result that is hoped to be true. It is a wide range of parameters, where the truth of this hypothesis is tested based off the verity of the Null Hypothesis.

Given by: "< , >, $\neq$"

##### Examples

###### Lessons

**Intuitively Judging Validity of Claims**

You meet a woman on the street who says she has Extra Sensory Perception (ESP) and can predict the probability of dice rolls with 70% probability. To test this you roll the die 90 times, and see how many times she "guesses" correctly.**Determining Claims**

What were the Null Hypothesis and the Alternative Hypothesis from the previous question?

- Which claims are Null Hypotheses and which claims are Alternative Hypotheses?
- For each of the following claims:

i) State whether each claim refers to proportions or means.

ii) Identify which claims are $H_0$ and which claims are $H_1$.

iii) Form the following claims into mathematical statements.

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###### Topic Notes

## Introduction to Hypothesis Testing

Welcome to our exploration of hypothesis testing, a fundamental concept in statistics! As we begin this journey, it's crucial to understand two key components: the null hypothesis and the alternative hypothesis. The null hypothesis, often denoted as H, represents the status quo or the assumption of no effect. In contrast, the alternative hypothesis, symbolized as H or H, suggests a significant difference or relationship. Our introduction video, which I highly recommend watching, provides a clear and engaging overview of these concepts. It's an excellent starting point for grasping the essence of hypothesis testing. As we delve deeper, you'll see how these hypotheses form the foundation for making informed decisions based on data. Remember, hypothesis testing is like being a detective in the world of statistics we're looking for evidence to support or refute our claims. Let's embark on this exciting statistical adventure together!

## Understanding Claims in Hypothesis Testing

In the world of hypothesis testing, two primary types of claims take center stage: proportions and means. Understanding these claim types is crucial for anyone delving into statistical analysis. Let's explore each type and see how they differ in their application and interpretation.

Proportions, as the name suggests, deal with percentages or parts of a whole. These claims are often used when we're interested in the frequency of occurrence within a population. For example, a researcher might claim that "60% of adults prefer coffee over tea." This is a proportion claim because it's expressing a percentage of the population.

On the other hand, means focus on average measurements or quantities. These claims involve numerical data that can be measured on a continuous scale. For instance, a manufacturer might assert that "the average lifespan of their light bulbs is 1000 hours." This is a mean claim as it's dealing with an average measurement.

Identifying whether a claim deals with proportions or means is crucial because it determines the type of statistical test we'll use. For proportion claims, we might employ a z-test for proportions, while for mean claims, we might use a t-test or z-test for means, depending on the sample size and other factors.

Let's look at more examples to solidify our understanding. A political pollster might claim that "52% of voters support a particular candidate." This is clearly a proportion claim. In contrast, a nutritionist stating that "the average daily calorie intake for adults is 2500 calories" is making a mean claim.

It's important to note that sometimes claims can be phrased in ways that might obscure whether they're about proportions or means. For instance, "3 out of 5 dentists recommend this toothpaste" is actually a proportion claim (60%), even though it's not expressed as a percentage.

In the realm of business, we often encounter both types of claims. A marketing manager might say, "75% of our customers are satisfied with our service" (a proportion claim), while a finance director might state, "Our average transaction value is $150" (a mean claim).

When conducting hypothesis testing, the nature of the claim guides our entire approach. For proportion claims, we're often interested in whether a stated percentage is accurate or if there's a significant difference between two proportions. With mean claims, we might be testing if an average differs from a specified value or comparing averages between groups.

It's worth mentioning that while proportions and means are the most common types of claims in hypothesis testing, there are other types as well, such as claims about variances or correlations. However, mastering proportions and means provides a solid foundation for most statistical analyses.

In conclusion, recognizing whether a claim deals with percentages (proportions) or measurements (means) is a critical first step in hypothesis testing. This distinction shapes the entire testing process, from choosing the appropriate test statistic to interpreting the results. By understanding these two fundamental types of claims, you'll be well-equipped to tackle a wide range of statistical problems and make informed decisions based on data.

## Null Hypothesis: Definition and Examples

The null hypothesis, denoted as H, is a fundamental concept in statistical hypothesis testing. It represents the default position or the status quo that researchers aim to challenge or disprove. Understanding how to find the null hypothesis is crucial for conducting accurate statistical hypothesis testing and drawing valid conclusions from data.

In essence, the null hypothesis states that there is no significant difference, relationship, or effect between variables being studied. It serves as a starting point for statistical investigations and is typically formulated to be rejected, allowing researchers to support an alternative hypothesis if sufficient evidence is found.

To determine the null hypothesis, one must first identify the claim or statement being tested. The null hypothesis is often the opposite of this claim or represents no change from the current situation. For example, if a researcher claims that a new medication is more effective than a placebo, the null hypothesis would state that there is no difference in effectiveness between the medication and the placebo.

When dealing with proportion claims, a null hypothesis example might be: "The proportion of left-handed people in a population is 10%." In this case, H would be expressed as p = 0.10, where p represents the population proportion. For mean claims, a null hypothesis could be: "The average height of adult males is 170 cm." Here, H would be written as μ = 170, where μ represents the population mean.

It's important to note that the null hypothesis always includes an equality sign (=, , or ). For instance, if a claim states that more than 60% of students prefer online learning, the null hypothesis would be H: p 0.60, indicating that the proportion is less than or equal to 60%.

To identify the null hypothesis from a given claim, follow these steps:

- Determine the parameter of interest (proportion, mean, etc.).
- Identify the claim being made about the parameter.
- Write the opposite of the claim or express it as no change from the current situation.
- Ensure the null hypothesis includes an equality sign.

For example, if the claim is "The average time spent on social media exceeds 2 hours per day," the null hypothesis would be H: μ 2, where μ represents the population mean time spent on social media.

Understanding how to determine the null hypothesis is crucial because it forms the basis for statistical testing. Researchers collect data and perform analyses to gather evidence against the null hypothesis. If sufficient evidence is found, the null hypothesis can be rejected in favor of an alternative hypothesis.

The significance of the null hypothesis in hypothesis testing cannot be overstated. It provides a clear, objective starting point for statistical analysis and helps researchers avoid bias in their investigations. By formulating a null hypothesis, researchers can quantify the likelihood of observing their data if the null hypothesis were true, leading to more robust and reliable conclusions.

In practice, null hypotheses are used across various fields, including medicine, psychology, economics, and social sciences. For instance, in medical research, a null hypothesis might state that a new drug has no effect on patient recovery times compared to existing treatments. In marketing, a null hypothesis could propose that there is no difference in consumer preferences between two product designs.

It's worth noting that failing to reject the null hypothesis does not necessarily mean it is true. It simply means that there isn't enough evidence to support the alternative hypothesis. This nuance is crucial in interpreting statistical results and avoiding common misconceptions in hypothesis testing.

In conclusion, the null hypothesis is a cornerstone of statistical inference. By learning how to find the null hypothesis, researchers can design effective studies, analyze data accurately, and draw meaningful conclusions. Whether dealing with proportions, means, or other statistical parameters, the ability to formulate and test null hypotheses is an essential skill for anyone engaged in data-driven research or decision-making.

## Alternative Hypothesis: Definition and Examples

The alternative hypothesis, denoted by the symbol H, is a crucial component in statistical hypothesis testing process. It represents the claim or statement that the researcher aims to support or prove, in contrast to the null hypothesis. Understanding the alternative hypothesis and its relationship to the null hypothesis is essential for conducting effective statistical analyses and drawing meaningful conclusions from data.

The alternative hypothesis is typically the opposite of the null hypothesis, which is denoted as H. While the null hypothesis usually represents no effect, no difference, or no relationship, the alternative hypothesis suggests that there is a significant effect, difference, or relationship between variables. In essence, the alternative hypothesis challenges the status quo represented by the null hypothesis.

To identify the alternative hypothesis from a given claim, researchers need to carefully examine the research question or statement being investigated. The alternative hypothesis is often the statement that the researcher hopes to support with evidence from the data. It's important to note that the alternative hypothesis can take different forms depending on the nature of the research question and the type of statistical test being conducted.

For proportion claims, an example of an alternative hypothesis might be: "The proportion of customers who prefer product A is greater than 50%." In this case, the null hypothesis would be that the proportion is equal to or less than 50%. The alternative hypothesis symbol (H) for this example could be written as H: p > 0.50, where p represents the population proportion.

When dealing with mean claims, an alternative hypothesis could be: "The average time to complete a task using the new software is less than 10 minutes." Here, the null hypothesis would state that the average time is equal to or greater than 10 minutes. The alternative hypothesis symbol for this scenario might be expressed as H: μ < 10, where μ represents the population mean.

It's crucial to understand that the alternative hypothesis can be directional or non-directional. A directional alternative hypothesis specifies the direction of the expected difference or relationship (e.g., greater than, less than), while a non-directional alternative hypothesis only indicates that there is a difference or relationship without specifying the direction.

To find the alternative hypothesis, researchers should follow these steps:

- Clearly state the research question or claim being investigated.
- Identify the parameter of interest (e.g., proportion, mean, difference between means).
- Determine whether the claim suggests a directional or non-directional hypothesis.
- Formulate the alternative hypothesis as the opposite of the null hypothesis, ensuring it aligns with the research question.

The relationship between the null and alternative hypotheses is complementary. They are mutually exclusive and exhaustive, meaning that if one is true, the other must be false, and together they cover all possible outcomes. This relationship is fundamental to the process of hypothesis testing process, where statistical evidence is used to decide whether to reject the null hypothesis in favor of the alternative hypothesis.

In practice, researchers often focus on disproving the null hypothesis rather than directly proving the alternative hypothesis. This approach is based on the principle that while it's challenging to prove something with absolute certainty, it's often possible to gather evidence that contradicts the null hypothesis, thereby lending support to the alternative hypothesis.

Understanding how to formulate and interpret the alternative hypothesis is crucial for researchers across various fields, including social sciences, natural sciences, and business analytics. By correctly identifying and stating the alternative hypothesis, researchers can design appropriate statistical tests, interpret results accurately, and draw valid conclusions from their data.

In conclusion, the alternative hypothesis plays a vital role in statistical inference and hypothesis testing. It represents the researcher's claim or expectation about a population parameter, contrasting with the null hypothesis. By understanding how to find the alternative hypothesis, interpret its symbol (H), and recognize its relationship to the null hypothesis, researchers can conduct more robust and meaningful statistical analyses, ultimately contributing to the advancement of knowledge in their respective fields.

## Identifying Null and Alternative Hypotheses in Practice

Welcome to our guide on how to determine null and alternative hypotheses! This crucial skill is essential for anyone diving into the world of hypothesis testing. Let's break it down step-by-step and explore some examples to help you master this concept.

### Step 1: Understand the Claim

The first step in identifying null and alternative hypotheses is to carefully read and understand the claim being made. Is it about a population proportion or a population mean? What exactly is being asserted?

### Step 2: Identify the Parameter of Interest

Determine whether you're dealing with a proportion (p) or a mean (μ). This will guide how you formulate your hypotheses.

### Step 3: Formulate the Null Hypothesis (H)

The null hypothesis always represents the status quo or no change. It often includes phrases like "equal to," "no difference," or "no effect." Remember, we assume the null hypothesis is true unless we have strong evidence to reject it.

### Step 4: Formulate the Alternative Hypothesis (H or H)

The alternative hypothesis is what we're testing for. It represents the claim or the change from the status quo. It's often the opposite of the null hypothesis.

### Examples with Proportions

**Example 1:** A company claims that 60% of customers prefer their new product.

- Null Hypothesis (H): p = 0.60
- Alternative Hypothesis (H): p 0.60

**Example 2:** A researcher believes that less than 30% of students in a school are left-handed.

- Null Hypothesis (H): p 0.30
- Alternative Hypothesis (H): p < 0.30

### Examples with Means

**Example 3:** A nutritionist claims that the average daily calorie intake of adults is more than 2000 calories.

- Null Hypothesis (H): μ 2000
- Alternative Hypothesis (H): μ > 2000

**Example 4:** A manufacturer states that the mean battery life of their new phones is 24 hours.

- Null Hypothesis (H): μ = 24
- Alternative Hypothesis (H): μ 24

### Key Points to Remember

- The null hypothesis always includes the equality sign (=, , or ).
- The alternative hypothesis never includes the equality sign.
- If the claim uses words like "at least" or "no more than," these typically go in the null hypothesis.
- Words like "increased," "decreased," or "changed" usually indicate the alternative hypothesis.

Correctly identifying null and alternative hypotheses is crucial because it sets the foundation for your entire hypothesis testing. A mistake at this stage can lead to incorrect conclusions, no matter how well you perform the rest of the analysis.

Remember, practice makes perfect! Try formulating hypotheses for various claims you encounter in daily life or in your studies. With time, you'll find it becomes second nature to identify the null and alternative hypotheses quickly and accurately.

By mastering this skill, you're taking a significant step towards becoming proficient in statistical analysis and hypothesis testing. Keep up the great work, and don't hesitate to review these steps

## Common Mistakes and Tips for Hypothesis Formulation

Formulating null and alternative hypotheses is a crucial step in statistical analysis, but it's one where students often stumble. Let's explore common errors and provide strategies to help you master this essential skill. Understanding how to calculate null hypothesis and distinguishing between null and alternative hypotheses are key to successful statistical testing.

One frequent mistake is confusing the null hypothesis with the alternative hypothesis. Remember, the null hypothesis (H0) typically represents no effect or no difference, while the alternative hypothesis (H1 or Ha) suggests a significant effect or difference. For example, a correct null hypothesis might be "There is no difference in test scores between two teaching methods," while the alternative would be "There is a difference in test scores between two teaching methods."

Another common error is formulating hypotheses that are not testable or measurable. Always ensure your hypotheses can be evaluated using statistical methods. For instance, instead of "The new medication improves patient well-being," a better formulation would be "Patients taking the new medication show a statistically significant increase in quality of life scores compared to the control group."

Students often struggle with directional hypotheses. When you have a specific prediction about the direction of an effect, it's important to reflect this in your alternative hypothesis. For example, if you believe Method A is more effective than Method B, your alternative hypothesis should state "Method A is more effective than Method B" rather than just "There is a difference between Method A and Method B."

To avoid these mistakes, follow these tips:

- Always start with the null hypothesis, which assumes no effect or difference.
- Ensure your hypotheses are clear, concise, and testable.
- Use precise language and avoid vague terms.
- Include relevant variables and specify the relationship you're testing.
- Double-check that your null and alternative hypotheses are mutually exclusive and exhaustive.

When learning how to calculate null hypothesis, remember that you don't actually calculate it; rather, you test it against your data. The process involves collecting data, choosing an appropriate statistical test, and then determining whether to reject or fail to reject the null hypothesis based on the results.

It's important to note that we test the null hypothesis, not the alternative hypothesis. The alternative hypothesis is what we hope to support if we reject the null. This is a key concept in understanding do you test the null or alternative hypothesis.

With practice and attention to detail, you'll become proficient in formulating clear, testable hypotheses. Remember, mastering this skill is fundamental to conducting robust statistical analyses and drawing valid conclusions from your research. Keep refining your approach, and don't hesitate to seek guidance when needed. Your ability to craft precise hypotheses will greatly enhance the quality and reliability of your statistical work.

## Conclusion

Understanding null and alternative hypotheses is crucial in hypothesis testing. The null hypothesis assumes no effect or relationship, while the alternative hypothesis proposes a significant effect or relationship. The introduction video provides essential insights into these concepts, helping students grasp their fundamental roles in statistical analysis. It's vital for learners to practice identifying and formulating both types of hypotheses across various scenarios. This skill enhances critical thinking and prepares students for real-world data analysis. Remember, the null hypothesis is always the default position, and the alternative hypothesis challenges this stance. By regularly engaging with these concepts, students can develop a strong foundation in statistical reasoning. We encourage you to revisit the video, apply these concepts to different examples, and discuss with peers to solidify your understanding. For further exploration, consider researching advanced hypothesis testing methods and their applications in your field of study.

### Example:

**Intuitively Judging Validity of Claims**

You meet a woman on the street who says she has Extra Sensory Perception (ESP) and can predict the probability of dice rolls with 70% probability. To test this you roll the die 90 times, and see how many times she "guesses" correctly.

If she guesses 23 rolls correctly is it likely that she has ESP?

#### Step 1: Understanding the Claim

The first step is to understand the claim being made. The woman claims she has ESP and can predict the outcome of a dice roll with a 70% probability. This is significantly higher than the probability of a random guess, which is 1/6 or approximately 16.67%. The goal is to test this claim by rolling a die 90 times and recording how many times she guesses correctly.

#### Step 2: Establishing the Null and Alternative Hypotheses

In hypothesis testing, we start by establishing the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis typically represents the status quo or a statement of no effect. In this case, the null hypothesis (H0) would be that the woman does not have ESP and is guessing randomly, with a probability of 1/6. The alternative hypothesis (H1) would be that the woman does have ESP and can guess correctly with a probability of 70%.

#### Step 3: Calculating the Expected Number of Correct Guesses Under the Null Hypothesis

Next, we calculate the expected number of correct guesses if the woman is guessing randomly. This is a binomial experiment where the number of trials (n) is 90 and the probability of success (p) is 1/6. The expected number of correct guesses (E) can be calculated using the formula E = n * p. Substituting the values, we get E = 90 * (1/6) = 15. Therefore, if the woman is guessing randomly, we would expect her to guess correctly about 15 times out of 90.

#### Step 4: Comparing the Observed Number of Correct Guesses to the Expected Number

In the experiment, the woman guessed correctly 23 times. We need to compare this observed number to the expected number of 15. If the woman truly has ESP and can guess with 70% accuracy, we would expect her to guess correctly 90 * 0.7 = 63 times on average. The observed number of 23 is much closer to the expected number under the null hypothesis (15) than it is to the expected number under the alternative hypothesis (63).

#### Step 5: Intuitive Judgment

Based on the comparison, we can make an intuitive judgment about the validity of the woman's claim. The observed number of correct guesses (23) is closer to what we would expect if she were guessing randomly (15) rather than if she had ESP (63). Therefore, it seems unlikely that she has ESP based on this experiment. However, this is an intuitive judgment, and a more rigorous statistical test would be needed to make a definitive conclusion.

#### Step 6: Considering Variability and Statistical Significance

While our intuitive judgment suggests that the woman does not have ESP, it's important to consider the variability in the number of correct guesses. In a binomial distribution, there is natural variability, and the number of correct guesses can fluctuate. To make a more precise judgment, we would need to calculate the standard deviation and use a statistical test, such as a z-test or a chi-square test, to determine if the observed number of correct guesses is significantly different from what we would expect under the null hypothesis.

#### Step 7: Conclusion

In conclusion, while the observed number of correct guesses (23) is closer to the expected number under the null hypothesis (15), a more rigorous statistical analysis is needed to make a definitive conclusion. Intuitively, it seems unlikely that the woman has ESP based on this experiment, but further statistical testing would provide a more accurate assessment.

### FAQs

**Q1: What is the difference between null and alternative hypotheses?**

A: The null hypothesis (H) represents the status quo or no effect, while the alternative hypothesis (H or H) suggests a significant difference or relationship. For example, if testing a new drug, H might state "The drug has no effect," while H might state "The drug has a significant effect."

**Q2: How do you denote alternative hypothesis?**

A: The alternative hypothesis is typically denoted as H or H. It's written using inequality symbols (, >, or <) to show a difference from the null hypothesis. For instance, if testing whether a population mean is greater than 10, the alternative hypothesis would be written as H: μ > 10.

**Q3: What is an example of a null hypothesis and alternative hypothesis?**

A: Consider testing whether the average height of adult males is 170 cm. The null hypothesis would be H: μ = 170 cm (the average height is 170 cm). The alternative hypothesis could be H: μ 170 cm (the average height is not 170 cm).

**Q4: How do you determine the null hypothesis?**

A: To determine the null hypothesis, start with the claim being tested and formulate its opposite or the "no effect" version. It should include an equality sign (=, , or ). For example, if the claim is "The new teaching method improves test scores," the null hypothesis would be "The new teaching method does not improve test scores" or H: μnew μold.

**Q5: Why do you test the null hypothesis?**

A: We test the null hypothesis because it's easier to disprove than to prove the alternative hypothesis directly. By gathering evidence against the null hypothesis, we can indirectly support the alternative hypothesis. This approach allows for more rigorous statistical analysis and helps control for Type I errors (false positives) in research.

### Prerequisite Topics

Understanding the foundation of statistical hypothesis testing is crucial when delving into the concepts of null hypothesis and alternative hypothesis. One key prerequisite topic that plays a significant role in this understanding is Chi-Squared hypothesis testing. This fundamental concept serves as a stepping stone to grasping the intricacies of null and alternative hypotheses.

Chi-Squared hypothesis testing is an essential component of statistical analysis that helps researchers evaluate the relationship between categorical variables. By mastering this concept, students gain valuable insights into the process of formulating and testing hypotheses, which directly relates to the broader topic of null and alternative hypotheses.

The null hypothesis and alternative hypothesis are central to the scientific method and statistical inference. To fully appreciate their importance, it's crucial to have a solid grasp of statistical hypothesis testing techniques, such as the Chi-Squared test. This prerequisite knowledge provides the necessary context for understanding how researchers formulate competing explanations for observed phenomena.

When studying null and alternative hypotheses, students will find that their prior knowledge of Chi-Squared testing enhances their ability to interpret results and make informed decisions. The Chi-Squared test, for instance, often involves setting up null and alternative hypotheses about the independence or association between variables. This direct application of hypothesis formulation reinforces the importance of mastering prerequisite topics.

Moreover, the concepts learned in Chi-Squared hypothesis testing lay the groundwork for understanding more complex statistical procedures. Students who are well-versed in this prerequisite topic will find it easier to grasp the nuances of type I and type II errors, p-values, and significance levels all of which are integral to working with null and alternative hypotheses.

By building a strong foundation in Chi-Squared testing, students develop critical thinking skills that are essential for formulating meaningful null and alternative hypotheses. They learn to question assumptions, consider various possibilities, and design experiments that can effectively test their hypotheses. This prerequisite knowledge empowers students to approach hypothesis testing with confidence and precision.

In conclusion, the journey to mastering null and alternative hypotheses is greatly enhanced by a solid understanding of prerequisite topics like Chi-Squared hypothesis testing. This foundational knowledge not only facilitates a smoother learning experience but also equips students with the tools to become proficient in statistical analysis and scientific inquiry. As students progress in their studies, they'll find that their grasp of these prerequisite concepts becomes an invaluable asset in tackling more advanced statistical challenges.

###### Basic Concepts

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