Statistics is used to interpret data and draw conclusions about populations based on sample data. Hypothesis testing involves evaluating two statements (the null and alternative hypotheses) about a population using sample data. A hypothesis test determines which statement is best supported.
The key steps in hypothesis testing are to formulate the hypotheses, select an appropriate statistical test, choose a significance level, collect and analyze sample data to calculate a test statistic, determine the probability or critical value associated with the test statistic, and make a decision to reject or fail to reject the null hypothesis based on comparing the probability or test statistic to the significance level and critical value.
An example tests whether the proportion of internet users who shop online is greater than 40% using
Dowhy: An end-to-end library for causal inferenceAmit Sharma
In addition to efficient statistical estimators of a treatment's effect, successful application of causal inference requires specifying assumptions about the mechanisms underlying observed data and testing whether they are valid, and to what extent. However, most libraries for causal inference focus only on the task of providing powerful statistical estimators. We describe DoWhy, an open-source Python library that is built with causal assumptions as its first-class citizens, based on the formal framework of causal graphs to specify and test causal assumptions. DoWhy presents an API for the four steps common to any causal analysis---1) modeling the data using a causal graph and structural assumptions, 2) identifying whether the desired effect is estimable under the causal model, 3) estimating the effect using statistical estimators, and finally 4) refuting the obtained estimate through robustness checks and sensitivity analyses. In particular, DoWhy implements a number of robustness checks including placebo tests, bootstrap tests, and tests for unoberved confounding. DoWhy is an extensible library that supports interoperability with other implementations, such as EconML and CausalML for the the estimation step.
Dowhy: An end-to-end library for causal inferenceAmit Sharma
In addition to efficient statistical estimators of a treatment's effect, successful application of causal inference requires specifying assumptions about the mechanisms underlying observed data and testing whether they are valid, and to what extent. However, most libraries for causal inference focus only on the task of providing powerful statistical estimators. We describe DoWhy, an open-source Python library that is built with causal assumptions as its first-class citizens, based on the formal framework of causal graphs to specify and test causal assumptions. DoWhy presents an API for the four steps common to any causal analysis---1) modeling the data using a causal graph and structural assumptions, 2) identifying whether the desired effect is estimable under the causal model, 3) estimating the effect using statistical estimators, and finally 4) refuting the obtained estimate through robustness checks and sensitivity analyses. In particular, DoWhy implements a number of robustness checks including placebo tests, bootstrap tests, and tests for unoberved confounding. DoWhy is an extensible library that supports interoperability with other implementations, such as EconML and CausalML for the the estimation step.
Hypothesis Testing: Central Tendency – Normal (Compare 1:1)Matt Hansen
An extension on a series about hypothesis testing, this lesson reviews the 2 Sample T & Paired T tests as central tendency measurements for normal distributions.
A hypothesis is a testable statement about the relationship between two or more variables and errors reveal about the rejection and acceptance of the statement.
Please like, comment and share
The number that divides the normal distribution into region where we will reject the null hypothesis and the region where we fail to reject the null hypothesis. For normal distribution Z at 5% level of significance (z= plus-minus 1.96) is often referred to as the critical value (or critical region).
Page 266LEARNING OBJECTIVES· Explain how researchers use inf.docxkarlhennesey
Page 266
LEARNING OBJECTIVES
· Explain how researchers use inferential statistics to evaluate sample data.
· Distinguish between the null hypothesis and the research hypothesis.
· Discuss probability in statistical inference, including the meaning of statistical significance.
· Describe the t test and explain the difference between one-tailed and two-tailed tests.
· Describe the F test, including systematic variance and error variance.
· Describe what a confidence interval tells you about your data.
· Distinguish between Type I and Type II errors.
· Discuss the factors that influence the probability of a Type II error.
· Discuss the reasons a researcher may obtain nonsignificant results.
· Define power of a statistical test.
· Describe the criteria for selecting an appropriate statistical test.
Page 267IN THE PREVIOUS CHAPTER, WE EXAMINED WAYS OF DESCRIBING THE RESULTS OF A STUDY USING DESCRIPTIVE STATISTICS AND A VARIETY OF GRAPHING TECHNIQUES. In addition to descriptive statistics, researchers use inferential statistics to draw more general conclusions about their data. In short, inferential statistics allow researchers to (a) assess just how confident they are that their results reflect what is true in the larger population and (b) assess the likelihood that their findings would still occur if their study was repeated over and over. In this chapter, we examine methods for doing so.
SAMPLES AND POPULATIONS
Inferential statistics are necessary because the results of a given study are based only on data obtained from a single sample of research participants. Researchers rarely, if ever, study entire populations; their findings are based on sample data. In addition to describing the sample data, we want to make statements about populations. Would the results hold up if the experiment were conducted repeatedly, each time with a new sample?
In the hypothetical experiment described in Chapter 12 (see Table 12.1), mean aggression scores were obtained in model and no-model conditions. These means are different: Children who observe an aggressive model subsequently behave more aggressively than children who do not see the model. Inferential statistics are used to determine whether the results match what would happen if we were to conduct the experiment again and again with multiple samples. In essence, we are asking whether we can infer that the difference in the sample means shown in Table 12.1 reflects a true difference in the population means.
Recall our discussion of this issue in Chapter 7 on the topic of survey data. A sample of people in your state might tell you that 57% prefer the Democratic candidate for an office and that 43% favor the Republican candidate. The report then says that these results are accurate to within 3 percentage points, with a 95% confidence level. This means that the researchers are very (95%) confident that, if they were able to study the entire population rather than a sample, the actual percentage who preferred th ...
Hypothesis Testing: Central Tendency – Normal (Compare 1:1)Matt Hansen
An extension on a series about hypothesis testing, this lesson reviews the 2 Sample T & Paired T tests as central tendency measurements for normal distributions.
A hypothesis is a testable statement about the relationship between two or more variables and errors reveal about the rejection and acceptance of the statement.
Please like, comment and share
The number that divides the normal distribution into region where we will reject the null hypothesis and the region where we fail to reject the null hypothesis. For normal distribution Z at 5% level of significance (z= plus-minus 1.96) is often referred to as the critical value (or critical region).
Page 266LEARNING OBJECTIVES· Explain how researchers use inf.docxkarlhennesey
Page 266
LEARNING OBJECTIVES
· Explain how researchers use inferential statistics to evaluate sample data.
· Distinguish between the null hypothesis and the research hypothesis.
· Discuss probability in statistical inference, including the meaning of statistical significance.
· Describe the t test and explain the difference between one-tailed and two-tailed tests.
· Describe the F test, including systematic variance and error variance.
· Describe what a confidence interval tells you about your data.
· Distinguish between Type I and Type II errors.
· Discuss the factors that influence the probability of a Type II error.
· Discuss the reasons a researcher may obtain nonsignificant results.
· Define power of a statistical test.
· Describe the criteria for selecting an appropriate statistical test.
Page 267IN THE PREVIOUS CHAPTER, WE EXAMINED WAYS OF DESCRIBING THE RESULTS OF A STUDY USING DESCRIPTIVE STATISTICS AND A VARIETY OF GRAPHING TECHNIQUES. In addition to descriptive statistics, researchers use inferential statistics to draw more general conclusions about their data. In short, inferential statistics allow researchers to (a) assess just how confident they are that their results reflect what is true in the larger population and (b) assess the likelihood that their findings would still occur if their study was repeated over and over. In this chapter, we examine methods for doing so.
SAMPLES AND POPULATIONS
Inferential statistics are necessary because the results of a given study are based only on data obtained from a single sample of research participants. Researchers rarely, if ever, study entire populations; their findings are based on sample data. In addition to describing the sample data, we want to make statements about populations. Would the results hold up if the experiment were conducted repeatedly, each time with a new sample?
In the hypothetical experiment described in Chapter 12 (see Table 12.1), mean aggression scores were obtained in model and no-model conditions. These means are different: Children who observe an aggressive model subsequently behave more aggressively than children who do not see the model. Inferential statistics are used to determine whether the results match what would happen if we were to conduct the experiment again and again with multiple samples. In essence, we are asking whether we can infer that the difference in the sample means shown in Table 12.1 reflects a true difference in the population means.
Recall our discussion of this issue in Chapter 7 on the topic of survey data. A sample of people in your state might tell you that 57% prefer the Democratic candidate for an office and that 43% favor the Republican candidate. The report then says that these results are accurate to within 3 percentage points, with a 95% confidence level. This means that the researchers are very (95%) confident that, if they were able to study the entire population rather than a sample, the actual percentage who preferred th ...
Topic Learning TeamNumber of Pages 2 (Double Spaced)Num.docxAASTHA76
Topic: Learning Team
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Correlation
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Correlation
A researcher is interested in investigating the relationship between viewing time (in seconds) and ratings of aesthetic appreciation. Participants are asked to view a painting for as long as they like. Time (in seconds) is measured. After the viewing time, the researcher asks the participants to provide a ‘preference rating’ for the painting on a scale ranging from 1-10. Create a scatter plot depicting the following data:
Viewing Time in Seconds
Preference Rating
10
3
12
4
24
7
5
3
16
6
3
4
11
4
5
2
21
8
23
9
9
5
3
3
17
5
14
6
What does the scatter plot suggest about the relationship between viewing time and aesthetic preference? Is it accurate to state that longer viewing times are the result of greater preference for paintings? Explain. Submit your scatter plot and your answers to the questions to your instructor.
LEARNING OBJECTIVES
· Explain how researchers use inferential statistics to evaluate sample data.
· Distinguish between the null hypothesis and the research hypothesis.
· Discuss probability in statistical inference, including the meaning of statistical significance.
· Describe the t test and explain the difference between one-tailed and two-tailed tests.
· Describe the F test, including systematic variance and error variance.
· Describe what a confidence interval tells you about your data.
· Distinguish between Type I and Type II errors.
· Discuss the factors that influence the probability of a Type II error.
· Discuss the reasons a researcher may obtain nonsignificant results.
· Define power of a statistical test.
· Describe the criteria for selecting an appropriate statistical test.
Page 267IN THE PREVIOUS CHAPTER, WE EXAMINED WAYS OF DESCRIBING THE RESULTS OF A STUDY USING DESCRIPTIVE STATISTICS AND A VARIETY OF GRAPHING TECHNIQUES.In addition to descriptive statistics, researchers use inferential statistics to draw more general conclusions about their data. In short, inferential statistics allow researchers to (a) assess just how confident they are that their results reflect what is true in the larger population and (b) assess the likelihood that their findings would still occur if their study was repeated over and over. In this chapter, we examine methods for doing so.
SAMPLES AND POPULATIONS
Inferential statistics are necessary because the results of a given study are based only on data obtained from a single sample of research participants. Researchers rarely, if ever, study entire populations; their findings are based on sample data. In addition to describing the sample data, we want to make statements ab.
Hypothesis Testing Definitions A statistical hypothesi.docxwilcockiris
Hypothesis Testing
Definitions:
A statistical hypothesis is a guess about a population parameter. The guess may or not be
true.
The null hypothesis, written H0, is a statistical hypothesis that states that there is no
difference between a parameter and a specific value, or that there is no difference between
two parameters.
The alternative hypothesis, written H1 or HA, is a statistical hypothesis that specifies a
specific difference between a parameter and a specific value, or that there is a difference
between two parameters.
Example 1:
A medical researcher is interested in finding out whether a new medication will have
undesirable side effects. She is particularly concerned with the pulse rate of patients who
take the medication. The research question is, will the pulse rate increase, decrease, or
remain the same after a patient takes the medication?
Since the researcher knows that the mean pulse rate for the population under study is 82
beats per minute, the hypotheses for this study are:
H0: µ = 82
HA: µ ≠ 82
The null hypothesis specifies that the mean will remain unchanged and the alternative
hypothesis states that it will be different. This test is called a two-tailed test since the
possible side effects could be to raise or lower the pulse rate. Notice that this is a non
directional hypothesis. The rejection region lies in both tails. We divide the alpha in two
and place half in each tail.
Example 2:
An entrepreneur invents an additive to increase the life of an automobile battery. If the
mean lifetime of the automobile battery is 36 months, then his hypotheses are:
H0: µ ≤ 36
HA: µ > 36
Here, the entrepreneur is only interested in increasing the lifetime of the batteries, so his
alternative hypothesis is that the mean is greater than 36 months. The null hypothesis is
that the mean is less than or equal to 36 months. This test is one-tailed since the interest
is only in an increased lifetime. Notice that the direction of the inequality in the alternate
hypothesis points to the right, same as the area of the curve that forms the rejection
region.
Example 3:
A landlord who wants to lower heating bills in a large apartment complex is considering
using a new type of insulation. If the current average of the monthly heating bills is $78,
his hypotheses about heating costs with the new insulation are:
H0: µ ≥ 78
HA: µ < 78
This test is also a one-tailed test since the landlord is interested only in lowering heating
costs. Notice that the direction of the inequality in the alternate hypothesis points to the
left, same as the area of the curve that forms the rejection region.
Study Design:
After stating the hypotheses, the researcher’s next step is to design the study. In designing
the study, the researcher selects an appropriate statistical test, chooses a level of
significance, and formulates a plan for conducting the study..
Hypothesis TestingThe Right HypothesisIn business, or an.docxadampcarr67227
Hypothesis Testing
The Right Hypothesis
In business, or any other discipline, once the question has been asked there must be a statement as to what will or will not occur through testing, measurement, and investigation. This process is known as formulating the right hypothesis. Broadly defined a hypothesis is a statement that the conditions under which something is being measured or evaluated holds true or does not hold true. Further, a business hypothesis is an assumption that is to be tested through market research, data mining, experimental designs, quantitative, and qualitative research endeavors. A hypothesis gives the businessperson a path to follow and specific things to look for along the road.
If the research and statistical data analysis supports and proves the hypothesis that becomes a project well done. If, however, the research data proved a modified version of the hypothesis then re-evaluation for continuation must take place. However, if the research data disproves the hypothesis then the project is usually abandoned.
Hypotheses come in two forms: the null hypothesis and the alternate hypothesis. As a student of applied business statistics you can pick up any number of business statistics textbooks and find a number of opinions on which type of hypothesis should be used in the business world. For the most part, however, and the safest, the better hypothesis to formulate on the basis of the research question asked is what is called the null hypothesis. A null hypothesis states that the research measurement data gathered will not support a difference, relationship, or effect between or amongst those variables being investigated. To the seasoned research investigator having to accept a statement that no differences, relationships, and/or effects will occur based on a statistical data analysis is because when nothing takes place or no differences, effects, or relationship are found there is no possible reason that can be given as to why. This is where most business managers get into trouble when attempting to offer an explanation as to why something has not happened. Attempting to provide an answer to why something has not taken place is akin to discussing how many angels can be placed on the head of a pin—everyone’s answer is plausible and possible. As such business managers need to accept that which has happened and not that which has not happened.
Many business people will skirt the null hypothesis issue by attempting to set analternative hypothesis that states differences, effects and relationships will occur between and amongst that which is being investigated if certain conditions apply.Unfortunately, however, this reverse position is as bad. The research investigator might well be safe if the data analysis detects differences, effect or relationships, but what if it does not? In that case the business manager is back to square one in attempting to explain what has not happened. Although the hypothesis situation may seem c.
PAGE
O&M Statistics – Inferential Statistics: Hypothesis Testing
Inferential Statistics
Hypothesis testing
Introduction
In this week, we transition from confidence intervals and interval estimates to hypothesis testing, the basis for inferential statistics. Inferential statistics means using a sample to draw a conclusion about an entire population. A test of hypothesis is a procedure to determine whether sample data provide sufficient evidence to support a position about a population. This position or claim is called the alternative or research hypothesis.
“It is a procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement” (Mason & Lind, pg. 336).
This Week in Relation to the Course
Hypothesis testing is at the heart of research. In this week, we examine and practice a procedure to perform tests of hypotheses comparing a sample mean to a population mean and a test of hypotheses comparing two sample means.
The Five-Step Procedure for Hypothesis Testing (you need to show all 5 steps – these contain the same information you would find in a research paper – allows others to see how you arrived at your conclusion and provides a basis for subsequent research).
Step 1
State the null hypothesis – equating the population parameter to a specification. The null hypothesis is always one of status quo or no difference. We call the null hypothesis H0 (H sub zero). It is the hypothesis that contains an equality.
State the alternate hypothesis – The alternate is represented as H1 or HA (H sub one or H sub A). The alternate hypothesis is the exact opposite of the null hypothesis and represents the conclusion supported if the null is rejected. The alternate will not contain an equal sign of the population parameter.
Most of the time, researchers construct tests of hypothesis with the anticipation that the null hypothesis will be rejected.
Step 2
Select a level of significance (α) which will be used when finding critical value(s).
The level you choose (alpha) indicates how confident we wish to be when making the decision.
For example, a .05 alpha level means that we are 95% sure of the reliability of our findings, but there is still a 5% chance of being wrong (what is called the likelihood of committing a Type 1 error).
The level of significance is set by the individual performing the test. Common significance levels are .01, .05, and .10. It is important to always state what the chosen level of significance is.
Step 3
Identify the test statistic – this is the formula you use given the data in the scenario. Simply put, the test statistic may be a Z statistic, a t statistic, or some other distribution. Selection of the correct test statistic will depend on the nature of the data being tested (sample size, whether the population standard deviation is known, whether the data is known to be normally distributed).
The sampling distribution of the test statistic is divided into t.
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Sustainability has become an increasingly critical topic as the world recognizes the need to protect our planet and its resources for future generations. Sustainability means meeting our current needs without compromising the ability of future generations to meet theirs. It involves long-term planning and consideration of the consequences of our actions. The goal is to create strategies that ensure the long-term viability of People, Planet, and Profit.
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1. Statistics is all about data but data alone is not interesting. It is the interpretation of the data that
we are interested in…
Data Science field is evolving like never before. Many companies are now looking for professionals
who can sift their goldmine data and help them drive swift business decisions efficiently. It also
gives the edge to many working professionals to switch their careers to the Data Science field.
Having this AI, Data Science buzz around many college students also wants to pursue their careers
in the Data Science field. And this hype around Data Science is correctly proclaimed by Thomas
H. Davenport and D.J. Patil in one of the Harvard Business Review articles that,
“Data Scientist: The Sexiest Job of the 21st Century”
In today’s analytics world building machine learning models has become relatively easy (thanks to
more robust and flexible tools and algorithms), but still the fundamental concepts are very
confusing. One of such concepts is Hypothesis Testing.
In this post, I’m attempting to clarify the basic concepts of Hypothesis Testing with illustrations.
What is Hypothesis Testing? What are we trying to achieve? Why do we need to perform
Hypothesis Testing? We must know the answers to all these questions before we proceed.
2. Statistics is all about data. Data alone is not interesting. It is the interpretation of the data that we
are interested in. Using Hypothesis Testing, we try to interpret or draw conclusions about the
population using sample data.
A Hypothesis Test evaluates two mutually exclusive statements about a population to determine
which statement is best supported by the sample data. Whenever we want to make claims about the
distribution of data or whether one set of results are different from another set of results in applied
machine learning, we must rely on statistical hypothesis tests.
There are two possible outcomes: if the result confirms the hypothesis, then you’ve made a
measurement. If the result is contrary to the hypothesis, then you’ve made a discovery — Enrico
Fermi
Let’s look at the terminology that we should be aware of in Hypothesis Testing
1. Parameter and Statistic:
A Parameter is a summary description of a fixed characteristic or measure of the target population.
A Parameter denotes the true value that would be obtained if a census rather than a sample were
undertaken
Ex: Mean (μ), Variance (σ²), Standard Deviation (σ), Proportion (π)
Population: Population is a collection of objectsthat we want to study/test.Thecollection of objects
could be Cities, Students, Factories, etc. It depends on the study at hand.
In the real world, it’s tough ask to get complete information about the population. Hence, we draw
a sample out of that population and derive the same statistical measures mentioned above. And
these measures are called Sample Statistics. In other words,
A Statistic is a summary description of a characteristic or measure of the sample. The Sample
Statistic is used as an estimate of the population parameter.
Ex: Sample Mean (x̄ ), Sample Variance (S²), Sample Standard Deviation (S), Sample Proportion
(p)
3. 2. Sampling Distribution:
A Sampling Distribution is a probability distribution of a statistic obtained through a large number
of samples drawn from a specific population.
Ex: Suppose a simple random sample of five hospitals is to be drawn from a population of 20
hospitals. The possibilities could be, (20, 19, 18, 17, 16) or (1,2,4,7,8) or any of the 15,504 (using
20C₅ combinations) different samples of size 5 can be drawn.
In general, the mean of the sampling distribution will be approximately equivalent to the population
mean i.e. E(x̄ ) = μ
To know more about Sampling Distribution please do check this below video:
Video from Khan Academy
3. Standard Error (SE):
The standard error (SE) is very similar to the standard deviation. Both are measures of spread. The
higher the number, the more spread out your data is. To put it simply, the two terms are essentially
equal — but there is one important difference. While the standard error uses statistics (sample data)
standard deviation use parameters (population data)
The standard error tells you how far your sample statistic (like the sample mean) deviates from the
actual population mean. The larger your sample size, the smaller the SE. In other words, the larger
your sample size, the closer your sample mean is to the actual population mean.
4. To know more about Standard Error (SE) please do watch below video
Now let’s consider the following example to better understand the remaining concepts.
4. (a). Null Hypothesis (H₀ ):
A statement in which no difference or effect is expected. If the null hypothesis is not rejected, no
changes will be made.
The word “null” in this context meansthat it’sa commonly acceptedfact that researchers tonullify.
It doesn’t mean that the statement is null itself! (Perhaps the term should be called the “nullifiable
hypotheiss” as that might cause less confusion)
4. (b). Alternate Hypothesis (H₁ ):
A statement that some difference or effect is expected. Accepting the alternative hypothesis will
lead to changes in opinions or actions. It is the opposite of the null hypothesis.
To know more about Null and Alternate hypotheses please watch this below video
5. (a). One-Tailed Test:
A one-tailed test is a statistical hypothesis test in which the critical area of a distribution is one-
sided so that it is either greater than or less than a certain value, but not both. If the sample being
tested falls into the one-sided critical area, the alternative hypothesis will be accepted instead of the
null hypothesis.
A one-tailed test is also known as a directional hypothesis or directional test.
Critical Region: The critical region is the region of values that corresponds to the rejection of the
null hypothesis at some chosen probability level.
5. (b). Two-Tailed Test:
A two-tailed test is a method in which the critical area of a distribution is two-sided and tests
whether a sample is greater than or less than a certain range of values. If the sample being tested
5. falls into either of the critical areas, the alternative hypothesis is accepted instead of the null
hypothesis.
By convention, two-tailed tests are used to determine significance at the 5% level, meaning each
side of the distribution is cut at 2.5%
6. Test Statistic:
The teststatistic measures how close the sample has come to the null hypothesis. Its observed value
changes randomly from one random sample to a different sample. A test statistic contains
information about the data that is relevant for deciding whether to reject the null hypothesis or not.
Different hypothesis tests use different test statistics based on the probability model assumed in the
null hypothesis. Common tests and their test statistics include:
6. Image from https://support.minitab.com
In general, the sample data must provide sufficient evidence to reject the null hypothesis and
conclude that the effect exists in the population. Ideally, a hypothesis test fails to reject the null
hypothesis when the effect is not present in the population, and it rejects the null hypothesis when
the effect exists.
By now we understand that the entire hypothesis testing works on based on the sample that is at
hand. We may come to a different conclusion if the sample is changed. There are two types of errors
that relate to incorrect conclusions about the null hypothesis.
7. (a). Type-I Error:
Type-I error occurs when the sample results, lead to the rejection of the null hypothesis when it is
in fact true. Type-I errors are equivalent to false positives.
Type-I errors can be controlled. The value of alpha, which is related to the levelof Significance that
we selected has a direct bearing on Type-I errors.
7. (b). Type-II Error:
Type-II error occurs when based on the sample results, the null hypothesis is not rejected when it
is in fact false. Type-II errors are equivalent to false negatives.
Level of Significance (α):
7. The probability of making a Type-I error and it is denoted by alpha (α). Alpha is the maximum
probability that we have a Type-I error. For a 95% confidence level, the value of alpha is 0.05. This
means that there is a 5% probability that we will reject a true null hypothesis.
P-Value:
The p-valueis used all over statistics, from t-tests to simple regression analysis to tree-based models
almost in all the machine learning models. We all use P-values to determine statistical significance
in a hypothesis test. Despite being so important, the P-value is a slippery concept that people often
interpret incorrectly.
P-values evaluate how well the sample data support the devil’s advocate argument that the null
hypothesis is true. It measures how compatible your data are with the null hypothesis. How likely
the effect observed in your sample data if the null hypothesis is true?
In other words, given the null hypothesis is true, a P-Value is a probability of getting a result as or
more extreme than the sample result by random chance alone.
High P-Values: Your data are likely with a true null
Low P-Values: Your data are unlikely with a true null
Ex: Suppose you are testing the following hypothesis at a significance level (α) of 5% and you got
the p-value as 3%, and your sample statistic is x̄ = 25
H₀ : μ = 20
H₁ : μ > 20
The interpretation of the p-value as follows:
We saw above that α is also known as committing Type-I error. When we say α=5%, we are fine to
reject our null hypothesis 5 out of 100 times even though it is true. Now that our P-value is 3%
which is less than α (we are definitely below the threshold of committing Type-I error), means
obtaining a sample statistic as extreme as possible (x̄ >= 25) given that H₀ is true is very less. In
other words, we can’t obtain our sample statisticas long as we assume H₀ is true. Hence, we reject
H₀ and accept H₁ . Suppose you get P-Value as 6% i.e. the probability of obtaining the sample
8. statistic as extreme as possible is higher given that the null hypothesis is true. So we fail to reject
H₀ , comparing with α we can’t take risk of committing Type-I error more than the agreed level of
significance. Hence, we fail to reject the null hypothesis and reject the alternative hypothesis.
Now that we understood the basic terminology in the Hypothesis Testing, now let’s look at the
steps involved in the Hypothesis Testing and an illustration with an example.
For example, a major department store is considering the introduction of an Internet shopping
service. The new service will be introduced if more than 40 percent of the Internet users shop via
the Internet.
Step1: Formulate the Hypotheses:
The appropriate way to formulate the hypotheses is:
H₀ : π ≤ 0.40
H₁ : π > 0.40
If the null hypothesis H₀ is rejected, then the alternative hypothesis H₁ will be accepted and the
new Internet shopping service will be introduced. On the other hand, if we fail to reject H₀ then
the new service should not be introduced unless additional evidence is obtained. This test of the
9. null hypothesis is a one-tailed test, because the alternative hypothesis is expressed directionally:
The proportion of Internet users who use the Internet for shopping is greater than 0.40.
Step2: Select an appropriate Test:
To test the null hypothesis, it is necessary to select an appropriate statistical technique. For this
example, the z statistic, which follows the standard normal distribution would be appropriate.
z = (p-π)/σₚ, where σₚ=sqrt(π(1-π)/n)
Step3: Choose Level of Significance, α:
We understood that Level of Significance refers to Type-I error. In our example, a Type-I error
would occur if we concluded, based on the sample data, that the proportion of customers preferring
the new service plan was greater than 0.40, when in fact it was less than or equal to 0.40.
The Type-II error would occur if we concluded, based on the sample data, that the proportion of
customers preferring the new service plan was less than or equal to 0.40 when, in fact, it was greater
than 0.40.
It is necessary to balance the two types of errors. As a compromise, α is often set at 0.05; sometimes
it is 0.01; other values of α are rare. We will consider 0.05 for our example.
Step4: Collect Data and Calculate Test Statistic:
Sample size is determined after taking into account the desired α and other qualitative
considerations, such as budget constraints to collect the sample data. For our example, let's say, 30
users were surveyed and 17 indicated that they used the Internet for shopping.
Thus, the value of the sample proportion is p=17/30=0.567.
The value of σₚ=sqrt((0.40)(0.60)/30)=0.089.
The test statistic z can be calculated as
z=(p-π)/σₚ=(0.567–0.40)/0.089=1.88
10. Step5: Determine the Probability (or Critical Value):
Using standard normal tables from the above, the probability of obtaining a z value of 1.88 is
0.96995 i.e. P(z≤1.88)=0.96995. But we wanted to calculate the probability to the right of z
(because we are interested in obtaining the probability value that falls in the rejection region or
critical region), i.e. 1–0.96995=0.03005. This Probability is directly comparable to α (since α is
committing a Type-I error and the probability value that we calculated also falls in the critical
region)
11. If you wanted to understand how to look up for the probability values for the given z scores, please
watch below video:
Alternatively, the critical value of z, which will give an area to the right side of the critical value of
0.05, is between 1.64 (at 1.64 the probability is 0.94950) and 1.65 (at 1.65 the probability is
0.95053) and equals 1.645 (the probability is 0.95, i.e. from the left of the normaldistribution, which
means to the right it is 0.05).
Note that in determining the critical value of the test statistic, the area in the tail beyond the critical
value is either α or α/2. It is α for a one-tailed test and α/2 for a two-tailed test. Our example is a
one-tailed test.
If you wanted to understand how to look up for the critical value of α, Please watch below video:
Step 6 and 7: Compare the probability (or Critical value) and make the decision:
The probability associated with the calculated or observed value of the test statistic is 0.03005. This
is the probability of getting a P-Value of 0.567 (sample proportion = p) when π=0.40. This is less
than the level of significance of 0.05. Hence, the null hypothesis is rejected.
Alternatively, the calculated value of the test statistic z=1.88 lies in the rejection region, beyond the
value of 1.645. Again, the same conclusion to reject the null hypothesis is reached.
Note that two ways of testing the null hypothesis are equivalent but mathematically opposite in the
direction of comparison. If the probability associated with the calculated or observed value of the
test statistic (TSCAL) is less than the level of significance (α), the null hypothesis is rejected.
However, if the absolute value of the calculated value of the test statistic is greater thanthe absolute
value of the critical value of the test statistic (TSCR), the null hypothesis is rejected. The reason for
this sign shift is that the larger the absolute value of TSCAL, the smaller the probability of obtaining
a more extreme value of the test statistic under the null hypothesis.
if the probability of TSCAL < significance level (α), then reject H₀ .
But, if |TSCAL| > |TSCR|, then reject H₀
Step8: Conclusion:
12. In our example, we conclude that there is evidence that the proportion of Internet users who shop
via the Internet is significantly greater than 0.40. Hence, the recommendation to the department
store would be to introduce the new Internet shopping service.
This example refers to one sample test of proportions. However, there are several types of tests exist
depends on the knowledge about the population and the problem at hand.
For Example, We have a t-test, Z-test. Chi-Square Test, Mann-Whitney Test, Wilcoxon Test, etc.
With this, I would like to conclude the Part-I of “Everything You Need To Know about Hypothesis
Testing”. I will discuss the Parametric and Non-Parametric tests and which test to use in what
scenario in Part-II. Until then Happy Learning…