Last Qtr Stats & Prob.

WEEK 1

What is a statistical hypothesis? Statistical hypothesis is an assertion or conjecture concerning one or more populations.

-What is the hypothesis denoted by Ho/H1? Null hypothesis is referred to any hypothesis we wish to test and is denoted by Ho. The rejection of null hypothesis leads to the acceptance of an alternative hypothesis. Alternative hypothesis is represented by H1. An understanding of the different roles played by the null hypothesis (H0) and the alternative hypothesis (H1) is crucial to one’s understanding of the rudiments of hypothesis testing.  The alternative hypothesis H1 usually represents the question to be answered or the theory to be tested, and thus its specification is crucial.  The null hypothesis H0 nullifies or opposes H1 and is often the logical complement to H1. What is test statistics? The test statistic on which we base our decision is X, the number of individuals in our test group who receive protection from the new vaccine for a period of at least 2 years. The possible values of X, from 0 to 20, are divided into two groups: those numbers less than or equal to 8 and those greater than 8. All possible scores greater than 8 constitute the critical region. The last number that we observe in passing into the critical region is called the critical value. In our example, the critical value is the number 8. What is type I & II error? Rejection of the null hypothesis when it is true is called a type I error. Non-rejection of the null hypothesis when it is false is called a type II error.

The probability of committing a type I error, also called the level of significance, is denoted by the Greek letter α. The probability of committing a type II error, denoted by β. The probability of committing a type I error, also called the level of significance (sometimes called the size of the test), is denoted by the Greek letter α. The probability of committing a type II error, denoted by β. Rejection of the null hypothesis when it is true is called a type I error.

Non-rejection of the null hypothesis when it is false is called a type II error. The null hypothesis H0 will often be stated using the equality sign. With this approach, it is clear how the probability of type I error is controlled. Example  A manufacturer of a certain brand of rice cereal claims that the average saturated fat content does not exceed 1.5 grams per serving. Identify the parameter to be tested. Saturated fat content* Serving Amount of rice cereal Number of cereal brands Explanation The manufacturer’s claim should be rejected only if μ is greater than 1.5 milligrams and should not be rejected if μ is less than or equal to 1.5 milligrams. Null hypothesis is referred to any hypothesis we wish to test and is denoted by Ho. The rejection of null hypothesis leads to the acceptance of an alternative hypothesis. Alternative hypothesis is represented by H1. An understanding of the different roles played by the null hypothesis (H0) and the alternative hypothesis (H1) is crucial to one’s understanding of the rudiments of hypothesis testing. The alternative hypothesis H1 usually represents the question to be answered or the theory to be tested, and thus its specification is crucial. The null hypothesis H0 nullifies or opposes H1 and is often the logical complement to H1. Examples are given in order for the students to identify and rewrite the null and alternative hypotheses into symbols. Example # 1 : Bottled Fruit Juice Content The owner of a factory that sells a particular bottled fruit juice claims that the average capacity of a bottle of their product is 250mL. In the example, the owner’s statement (called claim) is a general statement. The claim is that the capacity of all their bottled products is 250mL per bottle. A consumer group may generalize that the bottled product is short of the claim. If this can be proven, then the factory owner is lying. The evidence has to be established. So the consumer group gets interested to know if, in reality, each bottle contains 250 mL. Thus, the two hypotheses would be: H0: The bottled drinks contain 250 mL per bottle. (This is the claim.)

H1: The bottled drinks do not contain 250 mL per bottle. (This is the opposite of the claim.) But these statements should be written in symbols. For now, let us drop the unit measure and simply write: H0:   1= 250    and  H1: μ≠ 250  Example # 2 : Working Students A university claims that working students earn an average of Php 20 per hour. H0: The working students earn an average of Php 20 per hour. (This is the claim.) H1: The working students do not earn an average of Php 20 per hour. (This is the opposite of the claim.) Ask the students to write the null and alternative hypotheses in symbols. Answer:  H0: μ= 250 and  H1: μ≠ 250  Example # 3 : Songs on an MP3 player Suppose that  is the average number of songs on an MP3 player owned by a student. Write down the description of the null hypothesis H0: μ= 228. Answer:   H0: The average number of songs on an MP3 player is 228. Example # 4 : Songs on an MP3 player In example number 3, write down the description of the alternative hypothesis H1: μ≠ 228. Answer:   H0: The average number of songs on an MP3 player is not 228. Understanding Errors Task: Study the following examples carefully and the notes that follow. Discuss for better understanding of hypothesis testing. Example 1: Maria’s Age Maria insists that she is 3o years old when, in fact, she is 32 years old. What error is Mary committing? Solution: Mary is rejecting the truth. She is committing a Type I error. Example 2: Stephen’s Hairline Stephen says that he is not bald. His hairline is just receding. Is he committing an error? If so, what type of error? Solution: Yes. A receding hairline indicates balding. This is a Type I error. Stephen’s action may be to find remedial measures to stop falling hair.

Example 3: Monkey-Eating Eagle Hunt A man plans to go hunting the Philippine monkey-eating eagle believing that it is a proof of his mettle. What type of error is this? Solution: Hunting the Philippine eagle is prohibited by law. Thus, it is not a good sport. It is a Type II error. Since hunting the Philippine monkey-eating eagle is against the law, the man may find himself in jail if he goes out of his way hunting endangered species. In decisions that we make, we form conclusions and these conclusions are the bases of our actions. But this is not always the case in Statistics because we make decisions based on sample information. The best that we can do is to control the probability with which an error occurs. The probability of committing a Type I error is denoted by the Greek letter   (alpha) while the probability of committing a Type II error is denoted by  (beta).

WEEK 3

WEEK 4

WEEK 5