Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean1 Chapter 8 Hypothesis Tests  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, 2005BUS304 – Chapter 8 Hypothesis for Mean2 Step 1: Formulating Hypotheses  At the first step, two hypotheses shall be formulated for testing. Alternative Hypothesis ( H A ) 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. 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. Based on the sample data, we either reject H 0, or we do not reject H 0.

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean3 Determining the null hypothesis  A null hypothesis is the basis for testing.  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?  If we accept null hypothesis by mistake, it is not so big a problem as mistakenly accept the alternative hypothesis.

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean4 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, H A :  >20

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean5 Research Hypothesis  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.  H A  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, 2005BUS304 – Chapter 8 Hypothesis for Mean6 Types of errors  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.  Type I error  Rejecting the null hypothesis when it is, in fact, true.  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, 2005BUS304 – Chapter 8 Hypothesis for Mean7 Exercise  Problem 8.7 (Page 323)

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean8 Constructing the hypotheses Pair  Constructing the hypotheses pair is the basis for testing  There are totally 3 types of hypothesis.  Example: 1.The mean price of a beach house in Carlsbad is at least $1million dollars 2.The mean gas price in CA is no higher than $3 per gallon 3.The mean weight of a football quarterback is $200lbs. H 0 : μ ≥ $1million H A : μ < $1million H 0 : μ ≤ $3 per gallon H A : μ > $3 per gallon H 0 : μ = 200lbs H A : μ  200lbs

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean9 Exercise  Problem 8.1 (Page323)

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean10 One-tailed vs. Two tailed.  One-tailed test:  Upper tail test (e.g.  ≤ $1000)  Lower tail test (e.g.  ≥$800)  Two-tailed test:   =$1000 Reject when the sample mean is too high Reject when the sample mean is too low Reject when the sample mean is either too high or too low How to decide the cutoff?  : Level of Significance = Maximum allowed probability of type I error = Total blue area.

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean11 Information needed in hypothesis tests  When  is known  The claimed range of mean  (i.e. H 0 and H A )  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  When  is unknown  The claimed range of mean  (i.e. H 0 and H A )  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, 2005BUS304 – Chapter 8 Hypothesis for Mean12 Upper tail test  The cutoff z-score. z   The corresponding z-score which makes P(z> z  )=   In other words, P(0<z< z  ) =  Reject when the sample mean is too high H 0 : μ ≤ 3 H A : μ > 3  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 zz  Decision rule  If z x > z , reject H 0  If z x ≤ z , do not reject H 0

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean13 Example  Problem 8.3 (P323)

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean14 Lower tail test  The cutoff z score is negative  z  <0  Decision rule:  If z x < z , reject H 0  If z x ≥ z , do not reject H 0  The hypothesis is rejected only when you get a sample mean too low to support it.  Exercise: Problem 8.5 (Page 323) assuming that  =210 Reject when the sample mean is too low H 0 : μ ≥ 3 H A : μ < 3

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean15 Two-tailed tests  The null hypothesis is rejected when the sample mean is too high or too low  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: H 0 : μ = 3 H A : μ  3  /2

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean16 Decision Rule for two- tailed tests  Decision rule for two-tailed tests  If z x > z  /2, reject H 0  Or, if z x < -z  /2, reject H 0  Otherwise, do not reject H 0 Exercise 8.8  /2 H 0 : μ = 3 H A : μ  3

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean17 Hypothesis testing Steps When  is known  Step 1: Construct the hypotheses pair H 0 / H A.  Step 2: Write down the decision rule  One-tailed? Upper or lower?  Two-tailed?  Step 3: Find out the cutoff z-score (normal table) Drawing always help!  Step 4: Find out the z-score for sample mean  Step 5: compare the z-scores and use decision rule to make your decision.

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean18 When  is unknown  Now we use the sample standard deviation (i.e. s) to estimate the population standard deviation  The distribution is a t-distribution, Not Normal ! You should check the t-table P597 Pay attention to the degree of freedom: n-1  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, 2005BUS304 – Chapter 8 Hypothesis for Mean19 Hypothesis testing Steps When  is unknown  Step 1: Construct the hypotheses pair H 0 / H A.  Step 2: Write down the decision rule  One-tailed? Upper or lower?  Two-tailed?  Step 3: Find out the cutoff t –score (t-table, page 597)  Step 4: Find out the t -score for sample mean  Step 5: compare the t -scores and use decision rule to make your decision.

Nov 3, 2005BUS304 – Chapter 8 Hypothesis for Mean20 Use of PHStat