Basic Business Statistics, 10/e

Chapter 5 Fundamentals of Hypothesis Testing: One-Sample Tests.

Learning Outcomes. In this chapter, you learn: The basic principles of hypothesis testing How to use hypothesis testing to test a mean or proportion The assumptions of each hypothesis-testing procedure, how to evaluate them, and the consequences if they are seriously violated.

What is a Hypothesis?. A hypothesis is a claim about a population parameter: population mean population proportion.

The Null Hypothesis, H 0. States the claim or assertion to be tested Example: The average number of TV sets in U.S. Homes is equal to three ( ) Is always about a population parameter, not about a sample statistic..

The Null Hypothesis, H 0. Begin with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty Refers to the status quo or historical value Always contains “=” , “≤” or “≥ ” sign May or may not be rejected.

The Alternative Hypothesis, H 1. Is the opposite of the null hypothesis e.g., The average number of TV sets in U.S. homes is not equal to 3 ( H 1 : μ ≠ 3 ) Challenges the status quo May or may not be proven Is generally the hypothesis that the researcher is trying to prove.

The Hypothesis Testing Process. Claim: The population mean age is 50. H 0 : μ = 50, H 1 : μ ≠ 50 Sample the population and find sample mean..

The Hypothesis Testing Process. Suppose the sample mean age was X = 20. This is significantly lower than the claimed mean population age of 50. If the null hypothesis were true, the probability of getting such a different sample mean would be very small, so you cannot reject the null hypothesis . In other words, getting a sample mean of 20 is so unlikely if the population mean was 50, you conclude that the population mean must not be 50..

The Hypothesis Testing Process. Sampling Distribution of X.

The Test Statistic and Critical Values. If the sample mean is close to the assumed population mean, the null hypothesis is not rejected. If the sample mean is far from the assumed population mean, the null hypothesis is rejected. How far is “far enough” to reject H 0 ? The critical value of a test statistic creates a “line in the sand” for decision making -- it answers the question of how far is far enough..

The Test Statistic and Critical Values. Critical Values “Too Far Away” From Mean of Sampling Distribution.

Possible Errors in Hypothesis Test Decision Making.

Possible Errors in Hypothesis Test Decision Making.

Possible Results in Hypothesis Test Decision Making.

Type I & II Error Relationship. Type I and Type II errors cannot happen at the same time A Type I error can only occur if H 0 is true A Type II error can only occur if H 0 is false If Type I error probability (  ) , then Type II error probability ( β ).

Factors Affecting Type II Error. All else equal, β when the difference between hypothesized parameter and its true value β when  β when σ β when n.

Level of Significance and the Rejection Region. Level of significance = a.

-1.96 from Ho -2.58 -3.30 u = .05 u = .01 = .001 I .96 2.58 3.30.

Hypothesis Tests for the Mean.  Known.  Unknown.

Z Test of Hypothesis for the Mean ( σ Known). Convert sample statistic ( ) to a Z STAT test statistic.

Critical Value Approach to Testing. For a two-tail test for the mean, σ known: Convert sample statistic ( ) to test statistic (Z STAT ) Determine the critical Z values for a specified level of significance  from a table or computer Decision Rule: If the test statistic falls in the rejection region, reject H 0 ; otherwise do not reject H 0.

Do not reject H 0. Reject H 0. Reject H 0. There are two cutoff values (critical values) , defining the regions of rejection.

6 Steps in Hypothesis Testing. State the null hypothesis, H 0 and the alternative hypothesis, H 1 Choose the level of significance,  , and the sample size, n Determine the appropriate test statistic and sampling distribution Determine the critical values that divide the rejection and nonrejection regions Collect data and compute the value of the test statistic Make the statistical decision and state the managerial conclusion. If the test statistic falls into the non rejection region, do not reject the null hypothesis H 0 . If the test statistic falls into the rejection region, reject the null hypothesis . Express the managerial conclusion in the context of the problem.

Hypothesis Testing Example. Test the claim that the true mean # of TV sets in US homes is equal to 3..

Hypothesis Testing Example. 3. Determine the appropriate technique σ is assumed known so this is a Z test . 4. Determine the critical values For  = 0.05 the critical Z values are ±1.96 5. Collect the data and compute the test statistic Suppose the sample results are n = 100, X = 2.84 ( σ = 0.8 is assumed known) So the test statistic is:.

Reject H 0. Do not reject H 0. 6. Is the test statistic in the rejection region?.

6 (continued). Reach a decision and interpret the result.

p-Value Approach to Testing. p-value: Probability of obtaining a test statistic equal to or more extreme than the observed sample value given H 0 is true The p-value is also called the observed level of significance It is the smallest value of  for which H 0 can be rejected.

p-Value Approach to Testing: Interpreting the p-value.

The 5 Step p-value approach to Hypothesis Testing.

p-value Hypothesis Testing Example. Test the claim that the true mean # of TV sets in US homes is equal to 3..

p-value Hypothesis Testing Example. 3. Determine the appropriate technique σ is assumed known so this is a Z test . 4. Collect the data, compute the test statistic and the p-value Suppose the sample results are n = 100, X = 2.84 ( σ = 0.8 is assumed known) So the test statistic is:.

p-Value Hypothesis Testing Example: Calculating the p-value.

5. Is the p-value < α ? Since p-value = 0.0456 < α = 0.05 Reject H 0 6. (continued) State the managerial conclusion in the context of the situation. There is sufficient evidence to conclude the average number of TVs in US homes is not equal to 3..

Questions:. . The owner of a bread factory claims that the mean weight of bread is 280g. The measurement for 10 breads produced the following data: 276 285 288 279 286 281 277 274 285 282 With variance 22.23g 2 , test at significance level of 1% whether the claim is true or false. Show working on critical value approach and p-value approach..

Questions:. . 2. Generally, a car is driven 20,000 km/year in Malaysia. To test the claim, a sample at 150 car owners are picked randomly and asked to state the distance that their car had traveled in a year. Do you agree to the claim of the following sample data: x = 22,500 km, σ = 3800 km. Use the significance level of 5%. Show working on critical value approach and p-value approach..

Do You Ever Truly Know σ ?. Probably not! In virtually all real world business situations, σ is not known. If there is a situation where σ is known then µ is also known (since to calculate σ you need to know µ.) If you truly know µ there would be no need to gather a sample to estimate it..

Hypothesis Testing: σ Unknown. If the population standard deviation is unknown, you instead use the sample standard deviation s. Because of this change, you use the t distribution instead of the Z distribution to test the null hypothesis about the mean. When using the t distribution you must assume the population you are sampling from a normal distribution. All other steps, concepts, and conclusions are the same..

t Test of Hypothesis for the Mean ( σ Unknown). The test statistic is:.

Example: Two-Tail Test (  Unknown). The average cost of a hotel room in Penang is said to be RM168 per night. To determine if this is true, a random sample of 25 hotels is taken and resulted in an X of RM172.50 and a S of RM15.40. Test the appropriate hypotheses at  = 0.05. (Assume the population distribution is normal).

a = 0.05 n = 25, df = 25-1=24  is unknown, so use a t statistic Critical Value: ± t 24,0.025 = ± 2.0639.

One-Tail Tests. In many cases, the alternative hypothesis focuses on a particular direction.

Reject H 0. Do not reject H 0. There is only one critical value, since the rejection area is in only one tail.

Reject H 0. Do not reject H 0. Upper-Tail Tests. a.

Example: Upper-Tail t Test for Mean (  unknown).

Reject H 0. Do not reject H 0. Suppose that  = 0.10 is chosen for this test and n = 25. Find the rejection region:.

Obtain sample and compute the test statistic Suppose a sample is taken with the following results: n = 25, X = 53.1, and S = 10 Then the test statistic is:.

Reject H 0. Do not reject H 0. Example: Decision.

Example: Utilizing The p-value for The Test. Calculate the p-value and compare to  (p-value below calculated using excel spreadsheet on next page).

Hypothesis Tests for Proportions. Involves categorical variables Two possible outcomes Possesses characteristic of interest Does not possess characteristic of interest Fraction or proportion of the population in the category of interest is denoted by π.