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 mean monthly cell phone bill of this city is $42. The mean dividend return of Oracle stock is higher than $3 per share. The mean price of a Cannon Powershot G6 camera on Internet is less than $430. Why do we do hypothesis tests? BUS304 – Chapter 8 Hypothesis for Mean

Constructing a null hypothesis H0 A null hypothesis is the basis for testing. Null Hypothesis H0 Mathematical statement of the assumption to be tested Example: The average number of TV sets in U.S. Homes is at least three ( H0:  ≥ 3 ) The null hypothesis is always about the population parameter, not about a sample statistic Conventionally, it always contains an equal sign. e.g.  ≥ 4,  ≤ 6, or  = 10 BUS304 – Chapter 8 Hypothesis for Mean

Alternative Hypothesis The opposite of null hypothesis Written as HA. Example: The mean price of a beach house in Carlsbad is at least $1million dollars The mean gas price in CA is no higher than $3 per gallon The mean weight of a football quarterback is $200lbs. H0: μ ≥ $1million HA: μ < $1million H0: μ ≤ $3 per gallon HA: μ > $3 per gallon H0: μ = 200lbs HA: μ  200lbs BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean Exercise Problem 8.1 (Page323) BUS304 – Chapter 8 Hypothesis for Mean

Hypothesis Testing Process We want to test whether the null hypothesis is true. In statistics, we can never say a hypothesis is wrong for sure. We can only evaluate the probability that the hypothesis is true If the probability is too small, we say we reject the null hypothesis Otherwise, we say we fail to reject the null hypothesis. sample The mean height of male students at Cal State San Marcos is 6 feet Not likely. Reject the hypothesis BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean Types of errors Type I error Rejecting the null hypothesis when it is, in fact, true. It may happen when you decide to reject the hypothesis. -- you decide to reject the hypothesis when your result suggests that the hypothesis is not likely to be true. However, there is a chance that it is true but you get a bad sample. Type II error Failing to reject the null hypothesis when it is, in fact, false. It may happen when you decide not to reject. Whatever your decision is, there is always a possibility that you make at least one mistake. The issue is which type error is more serious and should not be made. BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean Exercise Problem 8.7 (Page 323) BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean Two kinds of tests One-tailed test: Upper tail test (e.g.  ≤ $1000) Lower tail test (e.g.  ≥$800) Reject when the sample mean is too high Reject when the sample mean is too low Two-tailed test:  =$1000 Reject when the sample mean is either too high or too low BUS304 – Chapter 8 Hypothesis for Mean

Information needed in hypothesis tests When  is known The claimed range of mean  (i.e. H0 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 When  is unknown The claimed range of mean  (i.e. H0 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): s2 or s BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean Upper tail test H0: μ ≤ 3 HA: μ > 3 Reject when the sample mean is too high z 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 The cutoff z-score. z The corresponding z-score which makes P(z> z)=  In other words, P(0<z< z) = 0.5 -  Decision rule If zx > z, reject H0 If zx ≤ z, do not reject H0 BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean Example Problem 8.3 (P323) BUS304 – Chapter 8 Hypothesis for Mean

An alternative way to test: use p-value The probability of getting the sample mean or higher. Reject if the p-value is too small i.e. even smaller than  It is too insignificant. Exercise: Use the p-value method to test the hypotheses in Problem 8.3 Think: what is the probability of making type 1 and type 2 errors if you reject the hypothesis If you fail to reject the hypothesis H0: μ ≤ 3 HA: μ > 3 The p-value of the sample mean BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean More Exercise Problem 8.4 BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean Lower tail test Reject when the sample mean is too low H0: μ ≥ 3 HA: μ < 3 The cutoff z score is negative z <0 Decision rule: If zx < z, reject H0 If zx ≥ z, do not reject H0 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 BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean Two-tailed tests H0: μ = 3 HA: μ  3 /2 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: BUS304 – Chapter 8 Hypothesis for Mean

Decision Rule for two-tailed tests /2 H0: μ = 3 HA: μ  3 Decision rule for two-tailed tests If zx > z/2, reject H0 Or, if zx < -z/2, reject H0 Otherwise, do not reject H0 Exercise 8.8 BUS304 – Chapter 8 Hypothesis for Mean

BUS304 – Chapter 8 Hypothesis for Mean 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 BUS304 – Chapter 8 Hypothesis for Mean

Summary of Hypothesis testing Steps Step 1: Construct the hypotheses pair H0 and HA. Step 2: Whether  is given? Given: use z-score (page 595) Unknown: use t-score (page 597) Need to have s (sample standard deviation) Degree of freedom: n-1 Step 3: Determine the decision rule One-tailed? Upper or lower? Two-tailed? Write down the decision rule based on the type of tests. Step 5: Find out the cutoff z-score or t-score ( ) Drawing always help! Step 6: Find out the z-score or t-score for sample mean ( ) Step 7: compare and make the right decision. BUS304 – Chapter 8 Hypothesis for Mean