Introduction to Hypothesis Testing Chapter 10

What is a Hypothesis? A tentative explanation for an observation, phenomenon, or scientific problem that can be tested by further investigation. Hypothesis testing is a statistical decision making tool that has applications in various fields: education, economics, medicine, psychology, marketing, sociology, management, etc. There are countless situations where a decision needs to be made, and a statistical hypothesis test will help with the decision.

Examples of where we could use hypothesis testing A new flu vaccine is being tested to determine its effectiveness and safety. These results will be compared to the effectiveness of the vaccine currently in use. Based on a comparison of the test results, a decision will be made either to market the vaccine or to continue marketing the present vaccine.

Examples of where we could use hypothesis testing The developer of a special reflective coating for windows claims that this coating applied to the windows of a house will reduce air conditioning costs by at least 40%. Before a manufacturer agrees to purchase the manufacturing and marketing rights of this product, a test will be conducted to determine if the product is as effective as the developer claims. Based on the test results, the manufacturer will decide if it should purchase the special reflective coating.

Hypothesis Testing When we conduct a hypothesis test, we use sample information to draw a conclusion about the population from which the sample was selected. Therefore, hypothesis testing is an example of inferential statistics.

Hypothesis Testing The essence of hypothesis testing is to evaluate the evidence provided by a sample regarding a claim made about a population parameter. Example: A doctor claims she has a method that is 75% accurate in predicting, several months before birth, the sex of an unborn child. To test her claim, she predicts the sex of a sample of 36 unborn children and is correct in 17 of her predictions. Can the doctor’s claim be supported by the sample data? Is the sample data significantly less than 75%, or did it happen by chance alone?

Hypothesis Testing Our intuition says, no, the doctor’s claim cannot be supported by the sample data. But intuition isn’t good enough when:  Changing public policy  Convincing a jury of someone’s guilt  Marketing a new drug  Deciding to implement a city’s multimillion dollar infrastructure. We need statistical proof to show that a claim can or cannot be supported by sample data.

Null and Alternative Hypotheses We begin a hypothesis test by stating the hypotheses, or claims, of the statistical test. We state 2 hypotheses: a null hypothesis and an alternative hypothesis. The null hypothesis, symbolized by H o, hypothesizes a value for a population parameter. It is initially assumed to be true. It is formulated for the sole purpose of trying to reject it by seeking evidence against it.

Null and Alternative Hypotheses An alternative hypothesis, H a, is a statement about the value of a population parameter which is different from the value stated in the null hypothesis. H a is phrased to indicate that the null hypothesis is not correct.

The U.S. judicial process To better understand the procedure used in statistical hypothesis testing, we can compare it to the judicial process. Once an individual is accused of committing a crime, a trial is conducted to determine if there is evidence to convict the individual. Throughout the trial, the individual is presumed innocent. This can be paralleled with the statistical idea of a null hypothesis, H o. The prosecutor’s task is to present evidence to cause the rejection of the assumption of innocence. In hypothesis testing, the researcher attempts to present sample evidence to cause the rejection of H o.

The U.S. judicial process The essence of hypothesis testing is to evaluate the evidence provided by a sample regarding a claim made about a population parameter. Just as the purpose of a trial is to evaluate the evidence provided by the prosecutor regarding the assumption of the accused person’s innocence.

Hypothesis Testing Procedure After hypotheses are formulated, the researcher designs a procedure to test the null hypothesis. The following 5 steps make up the hypothesis testing procedure:  Step 1. Formulate the H o and the H a.  Step 2. Determine the model to test the null hypothesis. (z-test or t-test)  Step 3. Use a calculator to find a p-value.  Step 4. Compare p-value with α (alpha) level of significance and state the conclusion.  Step 5. Decision rule: either reject H o or fail to reject H o.

Hypothesis Testing Procedure Even though the result of a hypothesis test is fail to reject H o, this is not the same as saying “accept H o ” It is not correct to accept H o because there may be other samples, when tested, that enable us to reject H o. Similarly, if there is not enough evidence to prove an an individual guilty, we say “not guilty.” We don’t say the individual is “innocent.” Just as there is a difference between “not guilty” and “innocent,” there is a difference between “fail to reject H o ” and “accept H o.”

Example 10.3 State the null and alternative hypotheses for the following: A college newspaper claims that full-time college students work an average of 20 hours a week. A marketing professor who believes this claim is too high decides to conduct a study to test the newspaper’s claim. H o : The population mean number of hours that full-time college students work is 20 hours per week. Symbolically: μ=20 H a : The population mean number of hours that full-time college students work is less than 20 hours per week. Symbolically: μ<20

Directional alternative hypothesis A directional alternative hypothesis is an alternative hypothesis that considers only one specified direction of difference away from the value stated in the null hypothesis. A directional alternative hypothesis is stated using words equivalent to: less than or greater than. Example 10.3: The population mean is less than 20.

Example 10.4 State the null and alternative hypothesis for the following research: A new environmentally safe organic fertilizer developed by a botanist is claimed to have no effect on the average yield of 50 vegetables per plant. The US Agriculture Dept. wants to test the botanist’s claim again the suspicion the new fertilizer has an effect on the average yield of 50 vegetables per plant. H o : The population mean yield per plant is 50 vegetables. Symbolically: μ=50

Example 10.4 The Dept. of Agriculture would like to test whether the average yield of a plant has been affected (either increased or decreased). Both directions (greater and less than) must be considered. H a : The population mean yield per plant is not 50 vegetables. Symbolically: μ≠50

Non-Directional H a H a : Using the new fertilizer the population average yield per plant is not 50 vegetables. If we need to consider both directions for an alternative hypothesis, it is called a non-directional alternative hypothesis. The words such as “too high,” “too low,” “more than,” “less than” are not explicitly stated. In a non-directional alternative hypothesis, H a will be stated using phrases like “is not” or “ is not equal to.”

Development of a Decision Rule The objective of a hypothesis test is to use sample data to decide if one should reject or fail to reject the null hypothesis. How do we make this decision? A decision rule is formulated comparing p-value with α (alpha) level of significance. Note: p α (p less than or equal to α) Reject Hull Hypothesis p > α (p greater than α) Fail to Reject Null Hypothesis

Development of a Decision Rule All hypothesis tests in this course will involve a either a normal distribution or t-distribution. This picture will change based on hypothesis test details. Hypothesis test results are the result of an experiment involving a sample. It will be a sample mean and will be positioned on the horizontal axis.

Development of a Decision Rule In the test pictured here, possible significant results occur to the right of the critical value, and non-significant results are to the left. If a statistical result falls to the right of the critical value, the conclusion will be to reject H o. If a statistical result falls to the left of the critical value, the conclusion will be to fail to reject H o.

Development of a Decision Rule The area of the shaded region to the right of the critical value is referred to as the level of significance. The level of significance is represented by α (alpha). α is given to you in a hypothesis test. The level of significance, α, is given as a percent, either 5% or 1%. The critical values we will use is a p-value.

Example A consumer testing agency is conducting a hypothesis to determine the validity off an advertised claim made by the Eye Saver Light Bulb Co. that the mean life of its new 60 watt bulb is 1800 hours. To test this claim, the agency randomly selects 400 bulbs. Use a standard error, of 5 hrs. The consumer agency believes the claim is too high and selects a significance level of 5%. The consumer agency wants to test the claim using a non-directional test and selects a significance level of 5%.

Example Mean = 1800 and standard error = 5 Step 1: Formulate hypotheses H o : The population mean life of the new 60 watt bulb is 1800 hrs; µ=1800 H a : The population mean life of the new 60 watt bulb is less than 1800; µ<1800

Example Mean = 1800 and standard error = 5 Step 2: Determine the model to test the null hypothesis The mean of the sampling distribution is The standard error of the distribution is given to be 5 but you need to find the population standard deviation. If standard deviation of the population is given we will not be required to calculate the standard error.

Example The consumer agency wants to test the claim using a non-directional test and selects a significance level of 5%. Formulate hypotheses H o : The population mean life of the new 60 watt bulb is 1800 hrs; µ=1800 H a : The population mean life of the new 60 watt bulb is not 1800; µ≠1800

We have a non-directional alternative hypothesis test because of the words not 1800 hours. Since there are two shaded areas, the level of significance, α=5%, must be split in half and compared with p-value. Example ½ α = 2.5%

Compare p-value with α–level of significance: p α Reject Hull Hypothesis p>α Fail to Reject Null Hypothesis Example 10.10

Two-Tailed Test, 2TT There are two regions for rejecting the null hypothesis the left tail and the right tail. We call this a two-tailed test. The level of significance, α, is divided equally between the two tails. Each tail has half the significance level, α/2.

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