Chapter 8 Hypothesis Tests v What are Hypothesis

Chapter 8 Hypothesis Tests v What are Hypothesis Tests? A set of methods and procedure to study the reliability of claims about population parameters. Examples of Hypotheses: The air quality of San Diego meets federal standards. All the transactions of the audited firm follow the GAAP. Your supplier provides a product with less than 1% defective rate. Students at CSUSM travels longer than 30 minutes on average to school for education. Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Step 1: Formulating Hypotheses v At the first step, two hypotheses shall be formulated for testing. Null Hypothesis ( H 0 ) The statement about the population value that will be tested. The null hypothesis will be rejected only if the sample data provide substantial contradictory evidence. Alternative Hypothesis ( HA ) The hypothesis the includes all population values not covered by the null hypothesis. The alternative hypothesis is deemed to be true if the null hypothesis is rejected. Based on the sample data, we either reject H 0, or we do not reject H 0. Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Determining the null hypothesis v A null hypothesis is the basis for testing. v Represents the situation that is assumed to be true unless the evidence is strong enough to convince the decision maker it is not true. § Legal system: what do you think of a person if there is not sufficient evidence that (s)he is guilty? § In the case of examining parts, if you are the buyer, what would be your assumption when there is no strong evidence? § In the case of fire inspection, what would be your assumption when examining a house’ condition? v If we accept null hypothesis by mistake, it is not so big a problem as mistakenly accept the alternative hypothesis. Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Example 8 -1 (p. 305) Student Work Hours: In today’s economy, many university students work many hours, often full time, to help pay for the high costs of a college education. Suppose a university in the Midwest was considering changing its class schedule to accommodate students working long hours. The registrar has stated a change was needed because the mean number of hours worked by undergraduate students at the university is more than 20 per week. § Step 1: determine the population value of interest: mean hours worked, . § Step 2: Define the situation that is assumed to be true unless substantial information exists to suggest otherwise: Would change only when strongly suggested § Step 3: Formulate the hypotheses pair. H 0: 20, HA: >20 Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Research Hypothesis v The hypothesis the decision maker attempts to demonstrate to be true. Because this is the hypothesis deemed to be the most important to the decision maker, it will not be declared true unless the sample data strongly indicate that it is true. HA v Research projects: § Students’ some habits may affect their GPA? You cannot prove it unless you have strong evidence § A company’s supplier has supplied more defective products than specified in the contract. § Women are discriminated and paid less for the same job description. § Professors in CSU systems are underpaid by the state agencies, which has caused difficulties in recruiting quality professors. Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Types of errors v As a result of hypothesis testing, you will need to decide whether § to reject null hypothesis; or § to accept the null hypothesis (normally stated as “failed to reject null hypothesis) In either case, you may or may not make the right decision. v Type I error § Rejecting the null hypothesis when it is, in fact, true. v Type II error § Failing to reject the null hypothesis when it is, in fact, false. Which error is more serious? see figure 8 -1 (page 307) for the relationship between decisions and states of nature. Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Exercise v Problem 8. 7 (Page 323) Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Constructing the hypotheses Pair v Constructing the hypotheses pair is the basis for testing v There are totally 3 types of hypothesis. v Example: 1. The mean price of a beach house in Carlsbad is at least $1 million dollars H 0: μ ≥ $1 million HA: μ < $1 million 2. The mean gas price in CA is no higher than $3 per gallon H 0: μ ≤ $3 per gallon HA: μ > $3 per gallon 3. The mean weight of a football quarterback is $200 lbs. H 0: μ = 200 lbs HA: μ 200 lbs Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Exercise v Problem 8. 1 (Page 323) Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

How to decide the cutoff? : Level of Significance One-tailed vs. Two tailed. = Maximum allowed probability of type I error = Total blue area. v One-tailed test: § Upper tail test (e. g. ≤ $1000) Reject when the sample mean is too high § Lower tail test (e. g. ≥$800) Reject when the sample mean is too low v Two-tailed test: § =$1000 Nov 3, 2005 Reject when the sample mean is either too high or too low BUS 304 – Chapter 8 Hypothesis for Mean

Information needed in hypothesis tests v When is known § The claimed range of mean (i. e. H 0 and HA) § When to reject: level of significance • i. e. if the probability is too small (even smaller than ), I reject the hypothesis. § Sample size n § Sample mean v When is unknown § The claimed range of mean (i. e. H 0 and HA) § When to reject: level of significance • i. e. if the probability is too small (even smaller than ), I reject the hypothesis. § Sample size n § Sample mean § Sample variance (or standard deviation): s 2 or s Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Upper tail test H 0 : μ ≤ 3 HA: μ > 3 Reject when the sample mean is too high z v Level of Significance: § Generally given in the task § The maximum allowed probability of type I error § In other words, the size of the blue area v The cutoff z-score. z § The corresponding z-score which makes P(z> z )= § In other words, P(0<z< z ) = 0. 5 - v Decision rule § If zx > z , reject H 0 § If zx ≤ z , do not reject H 0 Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Example v Problem 8. 3 (P 323) Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Lower tail test H 0 : μ ≥ 3 HA: μ < 3 Reject when the sample mean is too low v The cutoff z score is negative § z <0 v Decision rule: § If zx < z , reject H 0 § If zx ≥ z , do not reject H 0 v The hypothesis is rejected only when you get a sample mean too low to support it. v Exercise: Problem 8. 5 (Page 323) assuming that =210 Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Two-tailed tests H 0 : μ = 3 HA: μ 3 /2 v The null hypothesis is rejected when the sample mean is too high or too low v Given a required level of significance § There are two cutoffs. (symmetric) § The sum of the two blue areas is . § So each blue area has the size /2. § The z-scores: Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Decision Rule for twotailed tests H 0 : μ = 3 HA: μ 3 /2 v Decision rule for two-tailed tests § If zx > z /2, reject H 0 § Or, if zx < -z /2, reject H 0 § Otherwise, do not reject H 0 Exercise 8. 8 Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Hypothesis testing Steps When is known v Step 1: Construct the hypotheses pair H 0 / HA. v Step 2: Write down the decision rule § One-tailed? Upper or lower? § Two-tailed? v Step 3: Find out the cutoff z-score (normal table) Drawing always help! v Step 4: Find out the z-score for sample mean v Step 5: compare the z-scores and use decision rule to make your decision. Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

When is unknown v Now we use the sample standard deviation (i. e. s) to estimate the population standard deviation v The distribution is a t-distribution, Not Normal ! You should check the t-table P 597 Pay attention to the degree of freedom: n-1 v The rest of the calculations are the same. Exercise 8. 5 – lower tail test Exercise 8. 14 – upper tail test Exercise 8. 16 – two-tailed test Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Hypothesis testing Steps When is unknown v Step 1: Construct the hypotheses pair H 0 / HA. v Step 2: Write down the decision rule § One-tailed? Upper or lower? § Two-tailed? v Step 3: Find out the cutoff t –score (t-table, page 597) v Step 4: Find out the t -score for sample mean v Step 5: compare the t -scores and use decision rule to make your decision. Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean

Use of PHStat Nov 3, 2005 BUS 304 – Chapter 8 Hypothesis for Mean