10- Chapter Ten McGraw- © 2005 The McGraw-Hill Companies, Inc., All 10- Chapter Ten One-Sample Tests of Hypothesis GOALS When you have completed this chapter, you will be able to: ONE Define a hypothesis and hypothesis testing TWO Describe the five step hypothesis testing procedure THREE Distinguish between a one-tailed and a two-tailed test of hypothesis FOUR Conduct a test of hypothesis about a population mean 10- Chapter Ten continued One-Sample Tests of Hypothesis GOALS When you have completed this chapter, you will be able to: FIVE Conduct a test of hypothesis about a population proportion SIX Define Type I and Type II errors SEVEN Compute the probability of a Type II error 10- A statement about the value of a population parameter developed for the purpose of testing The mean monthly income for systems analysts is $6,325 What is a Hypothesis? Twenty percent of all customers at Bovine’s Chop House return for another meal within a month What is a Hypothesis? 10- Hypothesis testing Based on sample evidence and probability theory Used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected What is Hypothesis Testing? 10- S t e p : S t a t e n u ll a n d a lt e r n a t e h y p o t h e s e s S t e p : S e le c t a le v e l o f s ig n ific a n c e S t e p : I d e n t ify t h e t e s t s t a t is t ic S t e p : F o r m u la t e a d e c is io n r u le S t e p : T a k e a s a m p le , a r r iv e a t a d e c is io n D o n o t r e je c t n u ll R e je c t n u ll a n d a c c e p t a lt e r n a t e Hypothesis Testing 10- Step One: State the null and alternate hypotheses Null Hypothesis H0 A statement about the value of a population parameter Alternative Hypothesis H1: A statement that is accepted if the sample data provide evidence that the null hypothesis is false 10- Step One: State the null and alternate hypotheses Three possibilities regarding means H0: = H1: = H0: < H1: > H0: > H1: < The null hypothesis always contains equality hypotheses about means 10- Type I Error Level of Significance The probability of rejecting the null hypothesis when it is actually true; the level of risk in so doing Rejecting the null hypothesis when it is actually true Type II Error Accepting the null hypothesis when it is actually false Step Two: Select a Level of Significance 10- 10 Step Two: Select a Level of Significance Null Hypothesis Researcher Accepts Rejects Ho Ho Ho is true Correct decision Type I error Type II Error Correct decision Ho is false Risk table 10- 22 The processors of Fries’ Catsup indicate on the label that the bottle contains 16 ounces of catsup The standard deviation of the process is 0.5 ounces A sample of 36 bottles from last hour’s production revealed a mean weight of 16.12 ounces per bottle At the 05 significance level is the process out of control? That is, can we conclude that the mean amount per bottle is different from 16 ounces? Example 10- 23 Step Make a decision and interpret the results Step State the decision rule Reject H0 if z > 1.96 or z < -1.96 or if p < 05 Step Identify the test statistic Because we know the population standard deviation, the test statistic is z Step State the null and the alternative hypotheses H0: = 16 H1: 16 Step Select the significance level The significance level is 05 EXAMPLE 10- 24 Step 5: Make a decision and interpret the results The p(z > 1.44) X 16.12 16.00 z 1.44 is 1499 for a n 0.5 36 two-tailed test oComputed z of 1.44 < Critical z of 1.96, op of 1499 > of 05, Do not reject the null hypothesis We cannot conclude the mean is different from 16 ounces Example 10- 25 Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown Here is unknown, so we estimate it with the sample standard deviation s As long as the sample size n > 30, z can be approximated using X z s/ n Testing for the Population Mean: Large Sample, Population Standard Deviation Unknown 10- 26 Roder’s Discount Store chain issues its own credit card Lisa, the credit manager, wants to find out if the mean monthly unpaid balance is more than $400 The level of significance is set at 05 A random check of 172 unpaid balances revealed the sample mean to be $407 and the sample standard deviation to be $38 Should Lisa conclude that the population mean is greater than $400, or is it reasonable to assume that the difference of $7 ($407$400) is due to chance? Example 10- 27 Step Make a decision and interpret the results Step H0 is rejected if z > 1.65 or if p < 05 Step Because the sample is large we can use the z distribution as the test statistic Step H0: µ < $400 H1: µ > $400 Step The significance level is 05 Example Step Make a decision and interpret the results 10- 28 z oComputed z of 2.42 > Critical z of 1.65, op of 0078 < of 05 Reject H0 Lisa can conclude that the mean unpaid balance is greater than $400 X s n $407 $400 $38 172 2.42 The p(z > 2.42) is 0078 for a onetailed test 10- 29 Testing for a Population Mean: Small Sample, Population Standard Deviation Unknown The test statistic is the t distribution t X s/ n The critical value of t is determined by its degrees of freedom equal to n-1 Testing for a Population Mean: Small Sample, Population Standard Deviation Unknown The current rate for producing amp fuses at Neary Electric Co is 250 per hour A new machine has been purchased and installed that, according to the supplier, will increase the production rate The production hours are normally distributed A sample of 10 randomly selected hours from last month revealed that the mean hourly production on the new machine was 256 units, with a sample standard deviation of per hour 10- 30 At the 05 significance level can Neary conclude that the new machine is faster? Example 10- 31 The null hypothesis is rejected if t > 1.833 or, using the p-value, the null hypothesis is rejected if p < 05 Step State the decision rule There are 10 – = degrees of freedom Step State the null and alternate hypotheses H0: µ < 250 H1: µ > 250 Step Find a test statistic Use the t distribution since is not known and n < 30 Step Select the level of significance It is 05 10- 32 Step Make a decision and interpret the results t of 3.162 >Critical t of 1.833 op of 0058 < a of 05 Reject Ho t X s n 256 250 10 3.162 The p(t >3.162) is 0058 for a onetailed test oComputed The mean number of amps produced is more than 250 per hour Example 10- 33 Proportion The sample proportion is p and is the population proportion The fraction or percentage that indicates the part of the population or sample having a particular trait of interest Number of successes in the sample p Number sampled Test Statistic for Testing a Single Population Proportion z p (1 ) n 10- 34 In the past, 15% of the mail order solicitations for a certain charity resulted in a financial contribution A new solicitation letter that has been drafted is sent to a sample of 200 people and 45 responded with a contribution At the 05 significance level can it be concluded that the new letter is more effective? Example 10- 35 Step Find a test statistic The z distribution is the test statistic Step State the decision rule The null hypothesis is rejected if z is greater than 1.65 or if p < 05 Step Make a decision and interpret the results Step State the null and the alternate hypothesis H0: p < 15 H1: p > 15 Step Select the level of significance It is 05 Example 10- 36 Step 5: Make a decision and interpret the results 45 15 p z 200 2.97 (1 ) 15(1 15) n 200 p( z > 2.97) = 0015 Because the computed z of 2.97 > critical z of 1.65, the p of 0015 < of 05, the null hypothesis is rejected More than 15 percent responding with a pledge The new letter is more effective Example ... Calculated from the probability distribution function or by computer Using the p-Value in Hypothesis Testing 10- 15 Interpreting p-values > p > 0 p SOME evidence Ho is not true > p STRONG evidence... purpose of testing The mean monthly income for systems analysts is $6,325 What is a Hypothesis? Twenty percent of all customers at Bovine’s Chop House return for another meal within a month What... hypothesis when it is actually true; the level of risk in so doing Rejecting the null hypothesis when it is actually true Type II Error Accepting the null hypothesis when it is actually false