Concepts of Hypothesis Testing Part 1

Concept A hypothesis is a claim about some aspect of a population. A hypothesis test allows us to test the claim about the population and find out how likely it is to be true.

Hypothesis Ordinarily, when one talks about hypothesis, one simply means a mere assumption or some supposition to be proved or disproved. But for a researcher hypothesis is a formal question that he intends to resolve. Thus a hypothesis may be defined as a proposition or a set of proposition set forth as an explanation for the occurrence of some specified group of phenomena either asserted merely as a provisional conjecture to guide some investigation or accepted as highly probable in the light of established facts

Test of Statistical Hypothesis A decision procedure for a problem of testing of hypothesis is called a test of statistical hypothesis: A test of Statistical hypothesis is a rule or procedure for deciding whether to accept or reject the hypothesis on the basis of the sample values obtained. Testing may take several steps: a. Formulation of the Null Hypothesis b. Specification of the form of test statistic and its distribution c. Decide critical region and acceptance region

Null & Alternative Hypothesis The hypothesis which is tested under the assumption that it is true and is denoted by H0. According to Prof. R.A. Fisher “A Hypothesis which is tested for possible rejection under the assumption that it is true is known as Null Hypothesis. The hypothesis which is differ from a given Null Hypothesis and is accepted when H0 is rejected is a called an alternative hypothesis and is denoted by H1.

Critical Region & Acceptance Region

Types Of Error

Types Of Error

Types Of Error

Types of error and size of error Rejection of H0 when it is true is called a Type I error, and acceptance of H0 when it is false is called a Type II error. The size of a Type I error is defined to be the probability that a Type I error is made, and similarly the size of a Type II error is the probability that a Type II error is made.

Type I error Rejecting the null hypothesis when it is in fact true is called a Type I error. When a hypothesis test results in a p-value that is less than the significance level, the result of the hypothesis test is called statistically significant. The significance level α is the probability of making the wrong decision when the null hypothesis is true.

Connection between Type I error and significance level A significance level α corresponds to a certain value of the test statistic, say t , represented by the orange line in the picture of a sampling distribution (in next slide) (the picture illustrates a hypothesis test with alternate hypothesis "µ > 0”) α

Since the shaded area indicated by the arrow is the p-value corresponding to t , that p-value (shaded area) is α. α

To have p-value less than α , a t-value for this test must be to the right of t . α

So the probability of rejecting the null hypothesis when it is true is the probability that t > t , which we saw above is α. α

In other words, the probability of Type I error is α

Rephrasing using the definition of Type I error: The significance level α is the probability of making the wrong decision when the null hypothesis is true.

Confusing statistical significance and practical significance. Example: A large clinical trial is carried out to compare a new medical treatment with a standard one. The statistical analysis shows a statistically significant difference in lifespan when using the new treatment compared to the old one. But the increase in lifespan is at most three days, with average increase less than 24 hours, and with poor quality of life during the period of extended life. Most people would not consider the improvement practically significant. Caution: The larger the sample size, the more likely a hypothesis test will detect a small difference. Thus it is especially important to consider practical significance when sample size is large

Type II error Not rejecting the null hypothesis when in fact the alternate hypothesis is true is called a Type II error. (The example below provides a situation where the concept of Type II error is important.) Note: "The alternate hypothesis" in the definition of Type II error may refer to the alternate hypothesis in a hypothesis test, or it may refer to a "specific" alternate hypothesis. Example: In a t-test for a sample mean µ, with null hypothesis ""µ = 0" and alternate hypothesis "µ > 0", we may talk about the Type II error relative to the general alternate hypothesis "µ > 0", or may talk about the Type II error relative to the specific alternate hypothesis "µ > 1". Note that the specific alternate hypothesis is a special case of the general alternate hypothesis. In practice, people often work with Type II error relative to a specific alternate hypothesis. In this situation, the probability of Type II error relative to the specific alternate hypothesis is often called β. In other words, β is the probability of making the wrong decision when the specific alternate hypothesis is true.

Example A researcher thinks that if knee surgery patients go to physical therapy twice a week (instead of 3 times), their recovery period will be longer. Average recovery times for knee surgery patients is 8.2 weeks. The hypothesis statement in this question is that the researcher believes the average recovery time is more than 8.2 weeks. It can be written in mathematical terms as: (H : μ > 8.2) 1

Next, you’ll need to state the null hypothesis (See: How to state the null hypothesis). That’s what will happen if the researcher is wrong. In the above example, if the researcher is wrong then the recovery time is less than or equal to 8.2 weeks. In math, that’s: (H μ ≤ 8.2) 0

Example (Hypothesis) Rejecting the null hypothesis Ten or so years ago, we believed that there were 9 planets in the solar system. Pluto was demoted as a planet in 2006. The null hypothesis of “Pluto is a planet” was replaced by “Pluto is not a planet.” Of course, rejecting the null hypothesis isn’t always that easy — the hard part is usually figuring out what your null hypothesis is in the first place

Theory of Estimation

Point Estimation Interval Estimation

Concept A hypothesis is a claim about some aspect of a population. A hypothesis test allows us to test the claim about the population and find out how likely it is to be true.

Hypothesis Ordinarily, when one talks about hypothesis, one simply means a mere assumption or some supposition to be proved or disproved. But for a researcher hypothesis is a formal question that he intends to resolve. Thus a hypothesis may be defined as a proposition or a set of proposition set forth as an explanation for the occurrence of some specified group of phenomena either asserted merely as a provisional conjecture to guide some investigation or accepted as highly probable in the light of established facts

Test of Statistical Hypothesis A decision procedure for a problem of testing of hypothesis is called a test of statistical hypothesis: A test of Statistical hypothesis is a rule or procedure for deciding whether to accept or reject the hypothesis on the basis of the sample values obtained. Testing may take several steps: a. Formulation of the Null Hypothesis b. Specification of the form of test statistic and its distribution c. Decide critical region and acceptance region

Null & Alternative Hypothesis The hypothesis which is tested under the assumption that it is true and is denoted by H0. According to Prof. R.A. Fisher “A Hypothesis which is tested for possible rejection under the assumption that it is true is known as Null Hypothesis. The hypothesis which is differ from a given Null Hypothesis and is accepted when H0 is rejected is a called an alternative hypothesis and is denoted by H1.

Critical Region & Acceptance Region

Types Of Error

Types Of Error

Types Of Error

Types of error and size of error Rejection of H0 when it is true is called a Type I error, and acceptance of H0 when it is false is called a Type II error. The size of a Type I error is defined to be the probability that a Type I error is made, and similarly the size of a Type II error is the probability that a Type II error is made.

Type I error Rejecting the null hypothesis when it is in fact true is called a Type I error. When a hypothesis test results in a p-value that is less than the significance level, the result of the hypothesis test is called statistically significant. The significance level α is the probability of making the wrong decision when the null hypothesis is true.

Connection between Type I error and significance level A significance level α corresponds to a certain value of the test statistic, say t , represented by the orange line in the picture of a sampling distribution (in next slide) (the picture illustrates a hypothesis test with alternate hypothesis "µ > 0”) α

Since the shaded area indicated by the arrow is the p-value corresponding to t , that p-value (shaded area) is α. α

To have p-value less than α , a t-value for this test must be to the right of t . α

So the probability of rejecting the null hypothesis when it is true is the probability that t > t , which we saw above is α. α

In other words, the probability of Type I error is α

Rephrasing using the definition of Type I error: The significance level α is the probability of making the wrong decision when the null hypothesis is true.

Confusing statistical significance and practical significance. Example: A large clinical trial is carried out to compare a new medical treatment with a standard one. The statistical analysis shows a statistically significant difference in lifespan when using the new treatment compared to the old one. But the increase in lifespan is at most three days, with average increase less than 24 hours, and with poor quality of life during the period of extended life. Most people would not consider the improvement practically significant. Caution: The larger the sample size, the more likely a hypothesis test will detect a small difference. Thus it is especially important to consider practical significance when sample size is large

Type II error Not rejecting the null hypothesis when in fact the alternate hypothesis is true is called a Type II error. (The example below provides a situation where the concept of Type II error is important.) Note: "The alternate hypothesis" in the definition of Type II error may refer to the alternate hypothesis in a hypothesis test, or it may refer to a "specific" alternate hypothesis. Example: In a t-test for a sample mean µ, with null hypothesis ""µ = 0" and alternate hypothesis "µ > 0", we may talk about the Type II error relative to the general alternate hypothesis "µ > 0", or may talk about the Type II error relative to the specific alternate hypothesis "µ > 1". Note that the specific alternate hypothesis is a special case of the general alternate hypothesis. In practice, people often work with Type II error relative to a specific alternate hypothesis. In this situation, the probability of Type II error relative to the specific alternate hypothesis is often called β. In other words, β is the probability of making the wrong decision when the specific alternate hypothesis is true.

Example A researcher thinks that if knee surgery patients go to physical therapy twice a week (instead of 3 times), their recovery period will be longer. Average recovery times for knee surgery patients is 8.2 weeks. The hypothesis statement in this question is that the researcher believes the average recovery time is more than 8.2 weeks. It can be written in mathematical terms as: (H : μ > 8.2) 1

Next, you’ll need to state the null hypothesis (See: How to state the null hypothesis). That’s what will happen if the researcher is wrong. In the above example, if the researcher is wrong then the recovery time is less than or equal to 8.2 weeks. In math, that’s: (H μ ≤ 8.2) 0

Example (Hypothesis) Rejecting the null hypothesis Ten or so years ago, we believed that there were 9 planets in the solar system. Pluto was demoted as a planet in 2006. The null hypothesis of “Pluto is a planet” was replaced by “Pluto is not a planet.” Of course, rejecting the null hypothesis isn’t always that easy — the hard part is usually figuring out what your null hypothesis is in the first place

Theory of Estimation

Point Estimation Interval Estimation