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Statistics 312 22 Hypothesis Tests
p-Value Approach to Hypothesis Testing
The p-value is the probability of obtaining a test statistic equal to or more extreme than the result obtained from the sample data, given that the null hypothesis H0 is really true.
Steps: (Remember to define the parameter as the initial step for any inference.)
1. State the null hypothesis, H0. This statement concerning the mean will include an equal sign.
2. State the alternative hypothesis, H1. This statement concerning the mean will involve ≠ or > or <.
3. Choose the level of significance, .
4. Choose the sample size, n.
5. Determine the appropriate statistical technique and corresponding test statistic to use. With means, we require the same conditions as when calculating a confidence interval:
a) At least 30 observations are selected
or
b) At least 15 observations are selected and the population distribution is fairly symmetrical
or
c) The population distribution is normally distributed.
If one of these conditions is satisfied and we have a simple random sample from the population of interest, then we will use either of the following as our test statistic:
Z Test Statistic for ( Known)
t Test Statistic for ( Unknown)
6. Collect the data and compute the sample value of the appropriate test statistic.
7. Calculate the p-value based on the test statistic. This involves
(a)Sketching the distribution under the null hypothesis H0.
(b)Placing the test statistic on the horizontal axis.
(c)Shading in the appropriate area under the curve, based on the alternative hypothesis H1. These would be the possible values of the test statistic even further into the tail(s) of the distribution than the value obtained from the sample data; i.e., the possible values of the test statistic that are more unusual than the value obtained.
8. Compare the p-value to .
9. Make the statistical decision.
•If the p-value is greater than or equal to , the null hypothesis is not rejected.
•If the p-value is smaller than , the null hypothesis is rejected.
10. Express the statistical decision in terms of the problem.
Two-Tailed Tests: We have a two-tailed test if the null and alternative hypotheses are in the form:
H0: Parameter = hypothesized value (e.g., = 27)
H1: Parameter hypothesized value (e.g., 27)
A two-tailed test is one that rejects the null hypothesis if the estimate of the parameter is significantly higher or lower than the hypothesized value of the population parameter; i.e., if the test statistic based on the estimate of the parameter is larger or smaller than is reasonably expected.
Ex Stress test scores have a known standard deviation of = 25. Additionally, the mean on the stress test is 55 for the general population. A SRS of 140 college students was taken and the stress test given to each person in the sample. Run a test to determine if there is evidence that the mean amount of stress for college students differs from the general population.
Parameter:
1. H0:
2. H1:
3. =
4. n= 140.
5. Z or t test? Why? What do we need to assume?
6-7. Data, test statistic, p-value
MINITAB: Stat>Basic Statistics>1-Sample Z
8. Compare the p-value to .
9. Make the statistical decision.
10. Express the statistical decision in terms of the problem.
Ex Lara C. Blair worked at Upper Crust Trattoria. She took a sample of 17 tables with more than 2 customers and figured out the tip as a percent of the bill. The mean and standard deviation of that sample were 17.07 and 6.73. She collected this data to see if it is reasonable to think that the mean tip percent is 20%.
Parameter:
1. H0
2. H1
3.
4. n= 17.
5. Z or t test? Why? What do we need to assume?
6-7. Data, test statistic, p-value
MINITAB: Stat>Basic Statistics>1-Sample t
8. Compare the p-value to .
9. Make the statistical decision.
10. Express the statistical decision in terms of the problem.
One-Tailed Tests
H0: Parameter = hypothesized value (e.g., = 27)
versus
H1: Parameter > hypothesized value (e.g., > 27) Right-tailed test
or
H1: Parameter < hypothesized value (e.g., < 27) Left-tailed test
In a one-tailed test, the p-value is depends on the form of the alternative hypothesis. To compute the appropriate p-value for a one-tailed test (t may replace z)
H1: parameter > hypothesized value
p-value (right-tailed test) = P(z > Z)
= the area under the normal curve to the right of the test statistic, Z.
H1: parameter < hypothesized value
p-value (left-tailed test) = P(z < Z)
= the area under the normal curve to the left of the test statistic, Z.
Note: In a one-tailed test, it is not necessary for the p-value to be a "tail area."
Ex Following are combinations of the value of a t-test statistic based on 15 df and its associated p-value.
H1: > 27; t = 2.49
H1: > 27; t = –2.49
H1: < 27; t = 2.49
H1: < 27; t = –2.49
Ex Coastal Paint Supply has been in the business of manufacturing and testing indoor latex paints for many years. Their new formula is designed to be sold as “quick-dry” paint. That is, if they got the chemical formulation of the paint right, it should dry, on average, in less than 2 hours (120 minutes). To evaluate their paint, they painted n = 13 test surfaces and measured the length of time it took for each surface to dry, yielding a mean of 114 minutes. Based on many previous paint studies, Coastal Paint Supply is willing to assume that drying times follow a normal distribution with = 10 minutes. Is there strong evidence (use = 0.05) that their paint is “quick-dry”?
Ex The use of polymers in medicine, especially in the area of drug delivery, is one of the fastest growing areas of polymer chemistry. In a project, (Comparison of Drug Levels from Drug Delivery Devices), B. McClure described an experiment in which different formulations were used in a new delivery device. As part of the evaluation, a sample was taken from one formulation and readings taken on the amount of drug delivered (mg). The desired result was that the mean drug reading exceeds 500 mg. Using a level of significance of .05, test to see if the formulation appears to be delivering sufficient amounts of the drug.
Test of Normality
H0: The population of amount of drug delivered is normally distributed.
H1: The population of amount of drug delivered is NOT normally distributed.
Ex: Santa Maria school officials have stated that the mean weight of kindergarten students in their district is over 45 lbs. A sample of 17 students is taken from Ontiveras and (1) a test is performed to see if the data supports the claim, (2) a test is performed to see if the mean is actually less than 45 lbs., and (3) a two-tailed test is done. Notice what happens to the p-values in each case.
MTB > TTest 45 'Weight';
SUBC> Alternative 1.
T-Test of the Mean
Test of mu = 45.00 vs mu > 45.00
Variable N Mean StDev SE Mean T P
Weight 17 39.76 6.15 1.49 -3.51 1.00
MTB > TTest 45 'Weight';
SUBC> Alternative -1.
T-Test of the Mean
Test of mu = 45.00 vs mu < 45.00
Variable N Mean StDev SE Mean T P
Weight 17 39.76 6.15 1.49 -3.51 0.0015
MTB > TTest 45 'Weight';
SUBC> Alternative 0.
T-Test of the Mean
Test of mu = 45.00 vs mu not = 45.00
Variable N Mean StDev SE Mean T P
Weight 17 39.76 6.15 1.49 -3.51 0.0029
Read pp. 398-411 (Skip Critical Value), Prob 9.5ab,9.6ab,9.7a,9.8ab,9.9; provide a clear interpretation of each test result; check normality for all problems.