IS 310 Business Statistics CSU Long Beach

IS 310 Business Statistics CSU Long Beach

Hypothesis Testing What is a Hypothesis? A statement about population that may or may not be true. Examples: O The average nicotine content of a new brand of cigarette is 0.05 milligrams. O The average life of a new battery is over 72 months. O The proportion of all registered voters in the US who favor a Presidential candidate is 0.55. What is Hypothesis Testing? A procedure to test a hypothesis

Hypothesis Testing Hypothesis testing is used under several situations: O To test if a new drug is more effective O To test if a new process improves a product O To test if employee training improves employee job rating O To test if a new bonus plan increases sales performance O To test if new advertising increases sales volume And a host of similar situations

Hypothesis Testing • Developing Null and Alternative Hypotheses • Type I and Type II Errors • Population Mean: s Known • Population Mean: s Unknown

Developing Null and Alternative Hypotheses • The Null hypothesis is denoted by H • o • The Alternative hypothesis is denoted by H • a • The Null hypothesis is written in one of the following three ways: • H : µ = a value • o or • H : µ ≤ a value • o or • H : µ ≥ a value • o

Developing Null and Alternative Hypotheses • The Alternative hypothesis is written in such a way that it is different from the Null hypothesis. • H : µ = 25 • o • H : µ ≠ 25 or H : µ < 25 or H : µ > 25 • a aa • H : µ ≥ 25 • o • H : µ < 25 • a

Developing Null and Alternative Hypotheses • Rules for the Null and Alternative Hypotheses: • 1. The Null hypothesis must have an equal sign. • 2. The Alternative hypothesis must be different from • the Null hypothesis.

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). One-tailed (lower-tail) One-tailed (upper-tail) Two-tailed

Examples of Null and Alternative Hypotheses • Example (10-Page 340; 11-Page 350): • A particular automobile model currently gets an average fuel efficiency of 24 MPG. A new fuel injection system has been developed that increases the MPG. We want to test this claim. • H : µ = 24 or µ ≤ 24 • o • H : µ > 24 • a

Examples of Null and Alternative Hypotheses • Example (10-Page 340/341; 11-Page 351): • Soft drink containers are filled with an average of at least 67.6 fluid ounces. We want to test if these containers do indeed hold 67.6 ounces. • H : µ ≥ 67.6 • o • H : µ < 67.6 • a

Examples of Null and Alternative Hypotheses • Example (10-Page 341) • A particular part must have an average (or mean) length of two inches. The part is not accepted if it is less than or more than two inches. You want to write the two hypotheses for this situation. • H : µ = 2 • o • H : µ ≠ 2 • a

Examples of Null and Alternative Hypothesis • Example (11-Page 352) • Back to the soft drink example. • The company does not want to over-fill or under-fill bottles. In this case, the bottles must have exactly 67.6 fluid ounces. • H µ = 67.6 • 0 • H µ ≠ 67.6

Sample Problems on Null and Alternate Hypotheses • Problem # 1 (10-Page 342; 11-Page 353) • H : µ ≤ 600 H : µ > 600 • 0 a • Problem # 2 (10-Page 342; 11-Page 353) • H : µ ≤ 14 H : µ > 14 • 0 a

Sample Problems on Null and Alternative Hypotheses Problem # 3 (10-Page 342; 11-Page 353) H : µ = 32 H : µ ≠ 32 0 a Problem # 4 (10-Page 342; 11-Page 353) H : µ = 220 H :µ < 220 0 a

Type I Error • Because hypothesis tests are based on sample data, • we must allow for the possibility of errors. • A Type I error is rejecting H0 when it is true. • The probability of making a Type I error when the null hypothesis is true as an equality is called the level of significance. • Applications of hypothesis testing that only control the Type I error are often called significance tests.

Hypothesis Testing Steps for One Population Mean (known σ) • Steps: • 1. Develop the Null and the Alternative hypothesis. • 2. Determine the level of significance (  ). • 3. Get a sample of data and calculate test statistic as follows: • _ • x - µ • z = ----------- (µ is the value in H ) • σ /√ n o • 4. Obtain the critical value of z from Table 1 for the given value of . • 5. Draw area of rejection. • 5. Compare the critical value of z with the test statistic. If the • test statistic falls in the area of rejection, reject the Null • hypothesis.

An Example Problem on Hypothesis Testing (10-Page 345; 11-Page 357) • Hilltop Coffee claims that a large can contains 3 pounds of coffee, on the average. Federal Trade Commission (FTC) wants to test the manufacturer’s claim at  = 0.01. • A sample of 36 cans of coffee is selected and the value of the sample mean is calculated as 2.92. The standard deviation is known as σ = 0.18. • Step 1: H : µ = 3 or µ ≥ 3 • o • H : µ < 3 • a

Example Problem • Step 2: • Determine the level of significance,  = 0.01 • Step 3: Calculate the test statistic as • 2.92 – 3 - 0.08 • z = ------------- = -------- = - 2.67 • 0.18/√36 0.03 • Step 4: The critical value of z = - 2.33 (from Table 1) • Step 5: Draw the area of rejection (Figure 9.3 on page • 350. • Step 6: Compare critical z with test statistic. Reject H • o • No statistical evidence to support Hilltop Coffee’s claim.

Example Problem • Problem # 18 (10-Page 358; 11-Page 369) • Do this problem in class!

Hypothesis Testingp-values The concept of p-values calculates the strength of rejection of null hypothesis, H 0 When a null hypothesis is rejected, one can calculate the strength of rejection by computing the p-value. p-value represents the area to the right or left of the test statistic, depending on how the alternative hypothesis is written. The lower the p-value, greater the strength of rejection.

p-Value Look at the following examples: Example 1 Example 2 Test-statistic, z= 3.0 Test-statistic, z = 1.97 Critical z-value = 1.96 Critical z-value = 1.96 Reject H Reject H 0 0 p-value = 0.0014 p-value = 0.0244 In both examples, the null hypothesis is rejected. However, the strength of rejection is much stronger in Example 1

Hypothesis Testing Steps for One Population Mean (Unknown σ) • Steps: • 1. Develop the Null and the Alternative Hypothesis. • 2. Determine the level of significance (). • 3. Get a sample of data and calculate the test statistic • - • x - µ • t = ------------- ( µ is the value in H ) • s/√n o • 4. Obtain the critical value of t from Table 2 for the given value • of . • 5. Draw area of rejection. • 6. Compare the critical value of t with the test statistic. If test • statistic falls in the area of rejection, reject the null • hypothesis.

An Example Problem • Problem # 27 (10-Page 364; 11-Page 375) • a . H : µ = 238 H : µ < 238 • o a • b. Sample size, n = 100 degree of freedom = 100 – 1 = 99 • _ • Sample mean, x = 231 s = 80 • _ • x - µ 231- 238 • t = ---------- = -------------- = - 0.88 • s/√n 80/10 • p-value is between 0.2 and 0.1 • c. Critical value of t from Table 2 is -1.660 • Since test statistic does not fall in the area of rejection, do not reject the null hypothesis

Example Problem (continued) • What does the test result mean? • The average (or mean) weekly unemployment insurance benefit in Virginia is NOT below the national average at 5% level of significance. • or • There is statistical evidence at 5% level of significance to support that the mean weekly unemployment insurance benefit in Virginia is NOT below the national average.

p -Values and the t Distribution • The format of the t distribution table provided in most • statistics textbooks does not have sufficient detail • to determine the exact p-value for a hypothesis test. • However, we can still use the t distribution table to identify a range for the p-value. • An advantage of computer software packages is that the computer output will provide the p-value for the t distribution.

p-Value for Problem # 28 • Test statistic was 2.48 • If we look at the row with 91 degree of freedom and try to locate 2.48, we find it is between 0.01 and 0.005. Therefore, the p-value is between 0.01 and 0.005. • p-value = Between 0.01 and 0.005

Example Problem • Problem # 30 (10-Page 364; 11-Page 375) • Do this problem in class!

Hypothesis Testing for Population Proportion • Thus far, we performed hypothesis testing on population means. Hypothesis testing can be conducted on population proportions. • Examples: • 1. The manufacturer of a new product claims that 75 percent of potential customers like the product. • 2. A worker union claims that at least 60 percent of workers support the union. • 3. A city council claims that a majority of residents will approve a half-percent increase in sales tax.

Null and Alternative Hypotheses for Population Proportion • H : p = a value or p ≥ a value or p ≤ a value • o • H : p ≠ a value or p < a value or p > a value • a

Test Statistic • The test statistic is: • _ • p - p • z = ------------------ • √[p (1 – p)/n] • Other steps in the test are similar to those we have covered earlier.

Sample Problem • Problem #37 (10-Page 369; 11-Page 380) • H : p = 0.125 H : p > 0.125 • o a • _ • Given: n = 400 p = 52/400 = 0.13  = 0.05 • _ • Test statistic, z = ( p – p)/√[p (1 – p)/n] • = (0.13 – 0.125)/√[0.125 (1 – 0.125)/400] = 0.302 • Critical z at 0.05 = 1.645 • Since test statistic does not fall in the area of rejection, do not reject the null hypothesis. • There is no evidence that union efforts have increased union membership.

End of Chapter 9, Part A

IS 310 Business Statistics CSU Long Beach