Testing of Hypothesis

Basic Terms Population  all possible values Sample  a portion of the population Statistical inference  generalizing from a sample to a population with calculated degree of certainty Two forms of statistical inference Hypothesis testing Estimation Parameter  a characteristic of population, e.g., population mean µ Statistic  calculated from data in the sample, e.g., sample mean

Distinctions Between Parameters and Statistics Source Population Sample Notation Greek (e.g., μ) Roman (e.g., xbar) Vary No Yes Calculated

Hypothesis Testing Steps Null and alternative hypotheses Test statistic P-value and interpretation Significance level (optional) / critical region

Null and Alternative Hypotheses Convert the research question to null and alternative hypotheses The null hypothesis (H0) is a claim of “no difference in the population” The alternative hypothesis (Ha) claims “H0 is false” Collect data and seek evidence against H0

Illustrative Example: “Body Weight” The problem: In the 1970s, 20–29 year old men in the U.S. had a mean μ body weight of 170 pounds. Standard deviation σ was 40 pounds. We test whether mean body weight in the population now differs. Null hypothesis H0: μ = 170 (“no difference”) The alternative hypothesis can be either Ha: μ > 170 (one-sided test) or Ha: μ ≠ 170 (two-sided test)

Test Statistic This is an example of a one-sample test of a mean when σ is known. Use this statistic to test the problem:

For the illustrative example, μ0 = 170 We know σ = 40 Take a sample of n = 64. Therefore If we found a sample mean of 173, then

Decision Making : If the calculated. statistic value falls with Decision Making : If the calculated statistic value falls with in critical region we reject Null Hypothesis. Otherwise we will accept Null Hypothesis.

A sample of 200 people has a mean age 21 with a normal population standard deviation 5. Test the hypothesis that the population mean is 18.9 at 5% level of significance.

A certain stimulus administered to each of 12 patients resulted in the following increases of blood pressure: 5,2,8,-1,3,0,6,-2,1,5,0,4. Can it be concluded that the stimulus will be, in general, accompanied by an increase in blood pressure?

A cosmetics company fills its best-selling 8ounce jars of facial cream by an automatic dispensing machine. The machine is set to dispense a mean of 8.1 ounces per jar. Uncontrollable factors in the process can shift the mean away from 8.1and cause either under fill or overfill, both of which are undesirable. In such a case the dispensing machine is stopped and recalibrated. Regardless of the mean amount dispensed, the standard deviation of the amount dispensed always has value 0.22 ounce. A quality control engineer routinely selects 30jars from the assembly line to check the amounts filled. On one occasion, the sample mean is x¯=8.2ounces and the sample standard deviation is s=0.25 ounce. Determine if there is sufficient evidence in the sample to indicate, at the 1% level of significance, that the machine should be recalibrated.

A machine which produce mica insulating washers for use in electric devices is said to turn out washers having a thickness of 10 mil (1mil=0.001 inch). A sample of 10 washers has an average thickness of 9.52 mils with a standard deviation of 0.60 mil. Find out the significance of such a deviation.