Types of Hypotheses Research Hypothesis –a statement of what the researcher believes will be the outcome of an experiment or a study. Statistical Hypotheses –a more formal structure derived from the research hypothesis. Substantive Hypotheses –a statistically significant difference does not imply or mean a material, substantive difference.
Statistical Hypotheses Two Parts –a null hypothesis –an alternative hypothesis Null Hypothesis – nothing new is happening Alternative Hypothesis – something new is happening Notation –null: H 0 –alternative: H a
Null and Alternative Hypotheses: Example A manufacturer is filling 40 oz. packages with flour. The company wants the package contents to average 40 ounces.
One-tailed Tests One-tailed and Two-tailed Tests Two-tailed Test
Steps in Testing Hypotheses 1. Establish hypotheses: state the null and alternative hypotheses. 2. Determine the appropriate statistical test and sampling distribution. 3. Specify the Type I error rate ( 4. State the decision rule. 5. Gather sample data. 6. Calculate the value of the test statistic. 7. State the statistical conclusion. 8. Make a managerial decision.
Rejection and Non Rejection Regions =40 oz Non Rejection Region Rejection Region Critical Value Rejection Region Critical Value
Decision Table for Hypothesis Testing ( () Null TrueNull False Fail to reject null Correct Decision Type II error ) Reject nullType I error Correct Decision
One-tailed Tests =40 oz Rejection Region Non Rejection Region Critical Value =40 oz Rejection Region Non Rejection Region Critical Value
Two-tailed Tests =12 oz Rejection Region Non Rejection Region Critical Values Rejection Region
A survey of CPAs across US found that the average net income for sole proprietor CPAs is $ Because this survey is now more than sever years old an accounting researcher wants to test this figure by taking a random sample of 112 sole proprietor accountants in US to determine whether the net income figure changed. Result of survey derives mean of $ and the standard deviation net incomes for sole proprietor CPAs is $ Test hypothesis using 5 %significance level.
Rejection Region Non Rejection Region =0 Rejection Region
CPA Net Income Example: Critical Value Method Rejection Region Non Rejection Region =0 Rejection Region 72,22377,605
CPA Net Income Example: Critical Value Method Rejection Region Non Rejection Region =0 Rejection Region 72,22377,605
In an attempt to determine why customer service is important to managers in the UK, researcher surveyed managing directors of manufacturing plants in Scotland. One of the reason proposed was that customer service is a mean of retaining customer. On a scale measurement it derives mean of suppose US researcher believe American manufacturing managers would not rate this reason as highly at 0.05 alpha value determine whether US manager rate this reason significantly lower than the US data based on a sample of 32 and sample mean of with standard deviation of
Rejection Region Non Rejection Region 0 =.05
Rejection Region Non Rejection Region 0 =.05
Rejection Region Non Rejection Region 0 =
Rejection Region Non Rejection Region 0 =
Rejection Region Non Rejection Region 0 =.05 p-Value
z Test of Population Proportion
A manufacturer believes exactly 8% of its products contain at least one minor flaw. Suppose a company researcher want to test this belief. He selects a sample of 200 products and determines that 33 items have at least one minor flaw. Test the hypothesis that whether the proportion of products with at least one minor flaw is 0.08 using 0.1 significance level.
Critical Values Non Rejection Region Rejection Regions
Critical Values Non Rejection Region Rejection Regions
A survey of the morning beverage market shows that the primary breakfast beverage for 17% of Americans is milk. A milk producer in Wisconsin, where milk is plentiful, believes the figure is higher for Wisconsin. To test this idea, she contacts a random sample of 550 Wisconsin residents and asks which primary beverage they consumed for breakfast that day. Suppose 115 replied that milk was primary beverage. Using level of significance of 0.05 test the idea that the milk figure is higher for City.
Critical Value Non Rejection Region Rejection Region
Critical Value Non Rejection Region Rejection Region
A large manufacturer investigated the service it received from supplier and discovered that, in the past 32% of all material shipments were received late. However the company recently installed a JIT system in which supplier are linked more closely to the manufacturing process. A random sample of 118 deliveries since the JIT was installed reveals that 22 deliveries were late. Use the sample information to test whether the proportion of late deliveries was reduced significantly. α = 0.05
Estimating the Mean of a Normal Population: Unknown The population has a normal distribution. The value of the population standard deviation is unknown. z distribution is not appropriate for these conditions t distribution is appropriate
The t Distribution Developed by British statistician, William Gosset A family of distributions -- a unique distribution for each value of its parameter, degrees of freedom (d.f.) Symmetric, Unimodal, Mean = 0, Flatter than a z t formula
Comparison of Selected t Distributions to the Standard Normal Standard Normal t (d.f. = 25) t (d.f. = 1) t (d.f. = 5)
Table of Critical Values of t df t t t t t tt With df = 24 and = 0.05, t =
US farmers’ production company builds large harvesters. For a harvester to be properly balanced when operation, a 25 pound plate is installed. The machine that produces these plates is set to yield plates that average 25 pound. The distribution of plates produced from the machine is normal. However, the shop supervisor is worried that the machine is out of adjustment and is producing plates that do not average 25 pnd. To test this concern, he randomly selects 20 plates and produced weights in following table. =.05 Weights in Pounds of a Sample of 20 Plates
Unknown, =.05 Weights in Pounds of a Sample of 20 Plates
Critical Values Non Rejection Region Rejection Regions
Critical Values Non Rejection Region Rejection Regions
Department of Agriculture show that the average size of farms has increased since The mean size of the farm was 174 acres, by 2000 the size was 471 acres. Suppose an researcher believes the average size of farms increased from the 2000 mean figure 471 acres. To test this he randomly sampled 23 farms and find mean of acres and std of acres. =.05
Critical Value Non Rejection Region Rejection Region
Critical Value Non Rejection Region Rejection Region
The average price per square foot for shops has been $. A real estate firm want to determine whether that figure has changed, it sampled 19 shops finds mean price 31.67$ with std of 1.29$. =.05, Test hypothesis.