Ch 11 實習 (1)

Jia-Ying Chen2 Introduction 所謂假設檢定，係指在尚未蒐集樣本資料、進行 推論之前，就事先對母體的某種特徵性質作一合 理的假設敘述，再利用隨機抽出的樣本及抽樣分 配，配合機率原理，以判斷此項假設是否為真 若抽出的樣本資料與所陳述的假設很不一致，檢 定的結果必然認為這個假設不對，則否定或拒絕 （ reject ）這個假設

Jia-Ying Chen3 Introduction 若抽出的樣本資料與所陳述的假設不會很不一致， 檢定的結果就沒有充分理由斷定這個假設不對， 但也不認為這個假設是對的 假設檢定的主要精神在於尋找證據來拒絕 H 0 ，所 以假設檢定只能檢定 H 0 是否顯著錯誤，而不能判 斷其絕對正確

Jia-Ying Chen4 Developing Null and Alternative Hypotheses Hypothesis testing can be used to determine whether a statement about the value of a population parameter should or should not be rejected. The null hypothesis, denoted by H 0, is a tentative assumption about a population parameter. The alternative hypothesis, denoted by H 1, is the opposite of what is stated in the null hypothesis. Hypothesis testing is similar to a criminal trial. The hypotheses are: H 0 : The defendant is innocent H 1 : The defendant is guilty

Jia-Ying Chen5 A Summary of Forms for Null and Alternative Hypotheses about a Population Mean The equality part of the hypotheses always appears in the null hypothesis. In general, a hypothesis test about the value of a population mean  must take one of the following three forms (where  0 is the hypothesized value of the population mean). H 0 :  =  0 H 0 :  =  0 H 0 :  =  0 H 1 :   0 H 1 :   0

Jia-Ying Chen6 Concepts of Hypothesis Testing The critical concepts of hypothesis testing. Example: An operation manager needs to determine if the mean demand during lead time is greater than 350. If so, changes in the ordering policy are needed. There are two hypotheses about a population mean: H 0 : The null hypothesis μ = 350 H 1 : The alternative hypothesis μ > 350 This is what you want to prove

Jia-Ying Chen7 Example 1 A spouse suspects that the average amount of money spent on Christmas gifts for immediate family members is above $1,200. The correct set of hypotheses is: a. H0: μ = 1200 vs. H1: μ < 1200 b. H0: μ > 1200 vs. H1: μ = 1200 c. H0: μ = 1200 vs. H1: μ > 1200 d. H0: μ < 1200 vs. H1: μ = 1200

Jia-Ying Chen8 Concepts of Hypothesis Testing  = 350 Assume the null hypothesis is true (  = 350). Sample from the demand population, and build a statistic related to the parameter hypothesized (the sample mean). Pose the question: How probable is it to obtain a sample mean at least as extreme as the one observed from the sample, if H 0 is correct?

Jia-Ying Chen9 Since the is much larger than 350, the mean  is likely to be greater than 350. Reject the null hypothesis. Concepts of Hypothesis Testing  = 350 Assume the null hypothesis is true (  = 350). In this case the mean  is not likely to be greater than 350. Do not reject the null hypothesis.

Jia-Ying Chen10 Types of Errors Two types of errors may occur when deciding whether to reject H 0 based on the statistic value. Type I error: Reject H 0 when it is true. Type II error: Do not reject H 0 when it is false. Example continued Type I error: Reject H 0 (  = 350) in favor of H 1 (  > 350) when the real value of  is 350. Type II error: Believe that H 0 is correct (  = 350) when the real value of  is greater than 350.

Jia-Ying Chen11 Type I and Type II Errors Since hypothesis tests are based on sample data, we must allow for the possibility of errors. The person conducting the hypothesis test specifies the maximum allowable probability of making a Type I error, denoted by  and called the level of significance. Generally, we cannot control for the probability of making a Type II error, denoted by . Statistician avoids the risk of making a Type II error by using “ do not reject H 0 ” and not “ accept H 0 ”.

Jia-Ying Chen12 ( () Null TrueNull False Fail to reject null Correct Decision Type II error  ) Reject nullType I error  Correct Decision (Power) Decision Table for Hypothesis Testing

Jia-Ying Chen13 Controlling the probability of conducting a type I error Recall: H 0 :  = 350 and H 1 :  > 350. H 0 is rejected if is sufficiently large Thus, a type I error is made if when  = 350. By properly selecting the critical value we can limit the probability of conducting a type I error to an acceptable level. Critical value  = 350

Jia-Ying Chen14 1. Establish hypotheses: state the null and alternative hypotheses. 2. Determine the appropriate statistical test and sampling distribution. 3. Specify the Type I error rate (  4. State the decision rule. 5. Gather sample data. 6. Calculate the value of the test statistic. 7. State the statistical conclusion. 8. Make a managerial decision. Steps in Hypothesis Testing

Jia-Ying Chen15 Testing the Population Mean When the Population Standard Deviation is Known Example A new billing system for a department store will be cost- effective only if the mean monthly account is more than $170. A sample of 400 accounts has a mean of $178. If accounts are approximately normally distributed with  = $65, can we conclude that the new system will be cost effective?

Jia-Ying Chen16 Example 11.1 – Solution Example 11.1 The population of interest is the credit accounts at the store. We want to know whether the mean account for all customers is greater than $170. H 1 :  > 170 The null hypothesis must specify a single value of the parameter  H 0 :  = 170 Testing the Population Mean (  is Known)

Jia-Ying Chen17 Approaches to Testing There are two approaches to test whether the sample mean supports the alternative hypothesis (H 1 ) The rejection region method is mandatory for manual testing (but can be used when testing is supported by a statistical software) The p-value method which is mostly used when a statistical software is available.

Jia-Ying Chen18 The rejection region is a range of values such that if the test statistic falls into that range, the null hypothesis is rejected in favor of the alternative hypothesis. The Rejection Region Method

Jia-Ying Chen19 Example 11.1 – solution continued Define a critical value for that is just large enough to reject the null hypothesis. Reject the null hypothesis if The Rejection Region Method for a Right - Tail Test

Jia-Ying Chen20 Allow the probability of committing a Type I error be  (also called the significance level). Find the value of the sample mean that is just large enough so that the actual probability of committing a Type I error does not exceed  Watch … Determining the Critical Value for the Rejection Region

Jia-Ying Chen21 P(commit a Type I error) = P(reject H 0 given that H 0 is true)  = P( given that H 0 is true) Example – solution continued … is allowed to be  we have: Determining the Critical Value – for a Right – Tail Test

Jia-Ying Chen22 Determining the Critical Value – for a Right – Tail Test  = 0.05 Example – solution continued

Jia-Ying Chen23 Determining the Critical value for a Right - Tail Test Conclusion Since the sample mean (178) is greater than the critical value of , there is sufficient evidence to infer that the mean monthly balance is greater than $170 at the 5% significance level. Conclusion Since the sample mean (178) is greater than the critical value of , there is sufficient evidence to infer that the mean monthly balance is greater than $170 at the 5% significance level.

Jia-Ying Chen24 Instead of using the statistic, we can use the standardized value z. Then, the rejection region becomes One tail test The standardized test statistic

Jia-Ying Chen25 Example - continued We redo this example using the standardized test statistic. Recall:H 0 :  = 170 H 1 :  > 170 Test statistic: Rejection region: z > z.05  The standardized test statistic

Jia-Ying Chen26 Example - continued The standardized test statistic Conclusion Since Z = 2.46 > 1.645, reject the null hypothesis in favor of the alternative hypothesis. Conclusion Since Z = 2.46 > 1.645, reject the null hypothesis in favor of the alternative hypothesis.

Jia-Ying Chen27 The p-value provides information about the amount of statistical evidence that supports the alternative hypothesis. – The p-value of a test is the probability of observing a test statistic at least as extreme as the one computed, given that the null hypothesis is true. – 虛無假設成立，獲得檢定統計量及更極端數值之機率。檢定統計量 –Let us demonstrate the concept on Example P-value Method

Jia-Ying Chen28 The probability of observing a test statistic at least as extreme as 178, given that  = 170 is… The p-value P-value Method

Jia-Ying Chen29 Because the probability that the sample mean will assume a value of more than 178 when  = 170 is so small (.0069), there are reasons to believe that  > 170. …it becomes more probable under H 1, when Note how the event is rare under H 0 when but... Interpreting the p-value

Jia-Ying Chen30 We can conclude that the smaller the p-value the more statistical evidence exists to support the alternative hypothesis. We can conclude that the smaller the p-value the more statistical evidence exists to support the alternative hypothesis. Interpreting the p-value

Jia-Ying Chen31 The p-value can be used when making decisions based on rejection region methods as follows: Define the hypotheses to test, and the required significance level α. Perform the sampling procedure, calculate the test statistic and the p-value associated with it. Compare the p-value to α. Reject the null hypothesis only if p-value < α ; otherwise, do not reject the null hypothesis. The p-value  = 0.05 The p-value and the Rejection Region Methods

Jia-Ying Chen32 If we reject the null hypothesis, we conclude that there is enough evidence to infer that the alternative hypothesis is true. If we do not reject the null hypothesis, we conclude that there is not enough statistical evidence to infer that the alternative hypothesis is true. If we reject the null hypothesis, we conclude that there is enough evidence to infer that the alternative hypothesis is true. If we do not reject the null hypothesis, we conclude that there is not enough statistical evidence to infer that the alternative hypothesis is true. The alternative hypothesis is the more important one. It represents what we are investigating. Conclusions of a Test of Hypothesis

Jia-Ying Chen33 Example 2 If a hypothesis is not rejected at the 0.10 level of significance, it: a.must be rejected at the 0.05 level. b.may be rejected at the 0.05 level. c.will not be rejected at the 0.05 level. d.must be rejected at the level.

Jia-Ying Chen34 Example 3 Calculate the value of the test statistic, set up the rejection region, determine the p-value, interpret the result, and draw the sampling distribution.

Jia-Ying Chen35 Solution Rejection region z<z 0.1 =-1.28

Jia-Ying Chen36 Example 4 A business student claims that on average an MBA students is required to prepare more than five cases per week. To examine the claim, a statistics professor asks a random sample of 10 MBA students to report the number of cases they prepare weekly. The results are exhibited here. Can the professor conclude at the 5% significance level that the claim is true, assuming that the number of cases is normally distributed with a standard deviation of 1.5?

Jia-Ying Chen37 Solution

Jia-Ying Chen38 Example 5 A random sample of 12 second-year university students enrolled in a business statistics course was drawn. At the course’s completion, each student was asked how many hours he or she spent doing homework in statistics. The data are listed here. It is known that the population standard deviation is σ=8. The instructor has recommended that students devote 3 hours per week for the duration of the 12-week semester, for a total of 36 hours. Test to determine whether there is evidence that the average student spent less than the recommend amount of time. Compute the p-value of the test

Jia-Ying Chen39 Solution