One-Sample Tests of Hypothesis Chapter 10 .

Learning Objectives LO10-1 Define a hypothesis. LO10-2 Explain the process of testing a hypothesis. LO10-3 Apply the six-step procedure for testing a hypothesis. LO10-4 Distinguish between a one-tailed and a two-tailed test of hypothesis. LO10-5 Conduct a test of a hypothesis about a population mean. LO10-6 Compute and interpret a p-value. LO10-7 Use a t statistic to test a hypothesis. LO10-8 Compute the probability of a Type II error.

LO10-1 Define a hypothesis. Examples: Pay is related to performance: People who are paid more perform better. Consumers prefer Coke over all other cola drinks. Billboard advertising is more effective than advertising in paper-based media. Consumer confidence in the economy is increasing.

Hypothesis Testing LO10-2 Explain the process of testing a hypothesis. HYPOTHESIS TESTING A procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement.

Step 1: State the Null and the Alternate Hypothesis LO10-2 Step 1: State the Null and the Alternate Hypothesis NULL HYPOTHESIS A statement about the value of a population parameter developed for the purpose of testing numerical evidence. It is represented by H0. ALTERNATE HYPOTHESIS A statement that is accepted if the sample data provide sufficient evidence that the null hypothesis is false. It is represented by H1.

Step 2: State a Level of Significance: Errors in Hypothesis Testing The significance level of a test: Defined as the probability of rejecting the null hypothesis when it is actually true. This is denoted by the Greek letter “”. Also known as Type I Error. We select this probability prior to collecting data and testing the hypothesis. A typical value of “” is 0.05.

Step 2: State a Level of Significance: Errors in Hypothesis Testing Another possible error: The probability of not rejecting the null hypothesis when it is actually false. This is denoted by the Greek letter “β”. Also known as Type II Error. We cannot select this probability. It is related to the choice of , the sample size, and the data collected.

Step 2: State a Level of Significance: Errors in Hypothesis Testing

Step 3: Identify the Test Statistic LO10-2 Step 3: Identify the Test Statistic TEST STATISTIC A value, determined from sample information, used to determine whether to fail to reject or reject the null hypothesis. To test hypotheses about population means we use the z or t-statistic.

Step 4: Formulate a Decision Rule: One-Tail vs. Two-Tail Tests LO10-2 Step 4: Formulate a Decision Rule: One-Tail vs. Two-Tail Tests CRITICAL VALUE Based on the selected level of significance, the critical value is the dividing point between the region where the null hypothesis is rejected and the region where it is not rejected. If the test statistic is greater than or less than the critical value (in the region of rejection), then reject the null hypothesis.

Step 5: Take a Sample, Arrive at a Decision LO10-2 Step 5: Take a Sample, Arrive at a Decision Identify an unbiased sample. Collect the data on the relevant variables. Calculate test statistics. Compare the test statistic to the critical value. Make a decision, i.e., reject or fail to reject the null hypothesis.

Step 6: Interpret the Result LO10-2 Step 6: Interpret the Result What does the decision to reject or fail to reject the null hypothesis mean in the context of the study? Examples: “Based on the data, there is no evidence to support the hypothesis that pay is related to performance.” “Based on the data, there is evidence that billboard advertising if more effective than paper-based media advertising”.

LO10-5 Conduct a test of a hypothesis about a population mean. Hypothesis Test of a Population Mean, Known Population Standard Deviation – Example Jamestown Steel Company manufactures and assembles desks and other office equipment. The weekly production of the Model A325 desk at the Fredonia Plant follows the normal probability distribution with a mean of 200 and a standard deviation of 16. Recently, new production methods have been introduced and new employees hired. The VP of manufacturing would like to investigate whether there has been a change in the weekly production of the Model A325 desk.

How to Set Up a Hypothesis Test LO10-3 LO10-5 How to Set Up a Hypothesis Test In actual practice, the status quo is set up as H0. If the claim is “boastful” the claim is set up as H1 (we apply the Missouri rule – “show me”). Remember, H1 has the burden of proof. In problem solving, look for key words and convert them into symbols. Some key words include: “improved, better than, as effective as, different from, has changed, etc.” Keywords Inequality Symbol Part of: Larger (or more) than > H1 Smaller (or less) than < No more than  H0 At least ≥ Has increased Is there difference? ≠ Has not changed = Has “improved”, “is better than”, “is more effective” See keywords

Hypothesis Setups for Testing a Mean () LO10-5 LO10-3 Hypothesis Setups for Testing a Mean ()

Important Things to Remember about H0 and H1 LO10-5 Important Things to Remember about H0 and H1 H0 is the null hypothesis; H1 is the alternate hypothesis. H0 and H1 are mutually exclusive and collectively exhaustive. H0 is always presumed to be true. H1 has the burden of proof. A random sample (n) is used to “reject H0.” If we conclude “do not reject H0,” this does not necessarily mean that the null hypothesis is true, it only suggests that there is not sufficient evidence to reject H0; rejecting the null hypothesis, suggests that the alternative hypothesis may be true given the probability of Type I error. Equality is always part of H0 (e.g. “=”, “≥”, “≤”). Inequality is always part of H1 (e.g. “≠”, “<”, “>”).

LO10-3 LO10-3 Hypothesis Test of a Population Mean, Known Population Standard Deviation – Example Step 1: State the null and alternate hypotheses. H0:  = 200 H1:  ≠ 200 (Note: The keyword in the problem “has changed.”) Step 2: Select the level of significance. α = 0.01 as stated in the problem.

LO10-3 LO10-3 Hypothesis Test of a Population Mean, Known Population Standard Deviation – Example Step 3: Select the test statistic. Use z-distribution since σ is known.

LO10-3 LO10-3 Hypothesis Test of a Population Mean, Known Population Standard Deviation – Example Step 4: Formulate the decision rule. Reject H0 if |z| >z/2 Step 5: Make a decision and interpret the result. H0 is not rejected because 1.55 does not fall in the rejection region. Step 6: Interpret the result. We conclude that the population mean is not different from 200. So we would report to the vice president of manufacturing that the sample evidence does not show that the production rate at the plant has changed from 200 per week.

LO10-4 Distinguish between a one-tailed and a two-tailed test of hypothesis. Hypothesis Test of a Population Mean, Known Population Standard Deviation – One-tail Example Suppose in the previous problem the vice president wants to know whether there has been an increase in the number of units assembled. To put it another way, can we conclude, because of the improved production methods, that the mean number of desks assembled in the last 50 weeks was more than 200? Recall: σ=16, n=200, α=.01

One-Tailed Test versus Two-Tailed Test LO10-4 One-Tailed Test versus Two-Tailed Test

LO10-4 Testing for a Population Mean, Known Population Standard Deviation – One-Tail Example Step 1: State the null hypothesis and the alternate hypothesis. H0:  ≤ 200 H1:  > 200 (Note: The keyword in the problem “an increase.”) Step 2: Select the level of significance. α = 0.01 as stated in the problem. Step 3: Select the test statistic. Use z-distribution since σ is known.

LO10-4 Testing for a Population Mean, Known Population Standard Deviation – One-Tail Example Step 4: Formulate the decision rule. Reject H0 if z > z. Step 5: Make a decision. Because 1.55 does not fall in the rejection region, H0 is not rejected. Step 6. Interpret the result. Based on the evidence, we cannot conclude that the average number of desks assembled increased in the last 50 weeks.

p-Value in Hypothesis Testing LO10-6 Compute and interpret a p-value. p-Value in Hypothesis Testing A p-value is the probability of observing a sample value as extreme as, or more extreme than, the value observed (the test statistic), given that the null hypothesis is true. In testing a hypothesis, we can also compare the p-value to the significance level (). Decision rule using the p-value: Reject H0 if p-value < significance level.

p-Value in Hypothesis Testing – Example LO10-6 p-Value in Hypothesis Testing – Example Recall the last problem where the hypothesis and decision rules were set up as: H0:  ≤ 200 H1:  > 200 Reject H0 if z >z, where z = 1.55 and z = 2.33. Reject H0 if p-value < : 0.0606 is not < 0.01. Conclude: Fail to reject H0.

What does it mean when the p-value < ? LO10-6 What does it mean when the p-value < ? If p-value =.10, we have some evidence that H0 is not true. If p-value =.05, we have strong evidence that H0 is not true. If p-value =.01, we have very strong evidence that H0 is not true. If p-value =.001, we have extremely strong evidence that H0 is not true.

Testing for the Population Mean: Population Standard Deviation Unknown LO10-7 Use a t statistic to test a hypothesis. Testing for the Population Mean: Population Standard Deviation Unknown When the population standard deviation (σ) is unknown, the sample standard deviation (s) is used in its place The t-distribution is used as the test statistic, which is computed using the formula:

LO10-7 Testing for the Population Mean: Population Standard Deviation Unknown – Example The McFarland Insurance Company Claims Department reports the mean cost to process a claim is $60. An industry comparison showed this amount to be larger than most other insurance companies, so the company instituted cost-cutting measures. To evaluate the effect of the cost-cutting measures, the Supervisor of the Claims Department selected a random sample of 26 claims processed last month. The sample information is reported below. At the .01 significance level, is it reasonable to conclude that a claim is now less than $60?

LO10-7 Testing for a Population Mean: Population Standard Deviation Unknown – Example Step 1: State the null hypothesis and the alternate hypothesis. H0:  ≥ $60 H1:  < $60 (Note: The keyword in the problem is “now less than.”) Step 2: Select the level of significance. α = 0.01 as stated in the problem. Step 3: Select the test statistic. Since σ is unknown, use a t-distribution with n-1 (26 – 1 = 25) degrees of freedom.

t-Distribution Table (Portion) LO10-7 t-Distribution Table (Portion)

LO10-7 Testing for a Population Mean: Population Standard Deviation Unknown – Example Step 4: Formulate the decision rule. Reject H0 if t < -t,n-1. Step 5: Make a decision. Because -1.818 does not fall in the rejection region, H0 is not rejected at the .01 significance level. Step 6: Interpret the result. We have not demonstrated that the cost- cutting measures reduced the mean cost per claim to less than $60. The difference of $3.58 ($56.42 - $60) between the sample mean and the population mean could be due to sampling error.

LO10-7 Testing for a Population Mean: Population Standard Deviation Unknown – Example The current rate for producing 5 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. A sample of 10 randomly selected hours from last month revealed the mean hourly production on the new machine was 256 units, with a sample standard deviation of 6 per hour. At the .05 significance level, can Neary conclude that the new machine is faster?

LO10-7 Testing for a Population Mean: Population Standard Deviation Unknown – Example Step 1: State the null and the alternate hypothesis. H0: µ ≤ 250 H1: µ > 250 Step 2: Select the level of significance. It is .05. Step 3: Select the test statistic. The population standard deviation is not known. Use the t-distribution with n - 1 (10 – 1 = 9) degrees of freedom.

LO10-7 Testing for a Population Mean: Population Standard Deviation Unknown - Example Step 4: State the decision rule. There are 10 – 1 = 9 degrees of freedom. The null hypothesis is rejected if t > 1.833. Step 5: Make a decision. The null hypothesis is rejected. Step 6: Interpret the results. The mean number produced is more than 250 per hour.

LO10-8 Compute the probability of a Type II error. Recall that Type I Error, or the level of significance, is defined as the probability of rejecting the null hypothesis when it is actually true. It is denoted by the Greek letter alpha, “”. Type II Error is defined as the probability of “accepting” the null hypothesis when it is actually false. It is denoted by the Greek letter beta, “β”.

LO10-8 Type II Error – Example A manufacturer purchases steel bars to make cotter pins. Past experience indicates that the mean tensile strength of all incoming shipments is 10,000 psi and that the standard deviation, σ, is 400 psi. In order to make a decision about incoming shipments of steel bars, the manufacturer set up this rule for the quality-control inspector to follow: “Take a sample of 100 steel bars. At the .05 significance level if the sample mean strength falls between 9,922 psi and 10,078 psi, accept the lot. Otherwise the lot is to be rejected.”

Type I and Type II Errors Illustrated LO10-8 Type I and Type II Errors Illustrated If we take a sample of 100 bars and the sample mean is 9,900 psi, this value is our best estimate of the true population mean. And, based on the Region A graph, we should reject the null hypothesis with a 0.05 probability of a Type I error. However, there is always sampling error. For a distribution with a population mean of 9,900, it is possible that a sample would have a sample mean greater than 9,922. See Region B. So we could commit a Type II error: Fail to reject a false null hypothesis. Region B and the computed z-statistic show that the probability of a Type II error is 0.2912 when our estimate of the population mean is 9,900.