Math 101: Elementary Statistics Tests of Hypothesis

Transcription

1 Tests of Hypothesis Department of Mathematics and Computer Science University of the Philippines Baguio November 15, 2018

2 Basic Concepts of Statistical Hypothesis Testing A statistical hypothesis is an assertion or conjecture concerning one or more populations. The null hypothesis (Ho) is a statistical hypothesis stating that there is no difference between a parameter and a given value. The alternative hypothesis (Ha) is a statistical hypothesis stating the existence of a difference between a parameter and a specific value.

3 Null Hypothesis vs. Alternative Hypothesis State the null and alternative hypotheses for each of the following situations. Illustration 1. A researcher thinks that if expectant mothers use vitamin pills, the birth weight of the babies will increase. The average birth weight of the population is 8.6 pounds. Illustration 2. An engineer hypothesizes that the mean number of defects can be decreased in a manufacturing process of compact disks by using robots instead of humans for certain tasks. The mean number of defective disks per 1000 is 18. Illustration 3. A psychologist feels that playing soft music during a test will change the results of the test. The psychologist is not sure whether the grades will be higher or lower. In the past, the mean of the scores was 73.

4 Null Hypothesis vs. Alternative Hypothesis A one-tailed test of hypothesis is a test where the alternative hypothesis specifies a one-directional difference for the parameter of interest. It can be either right-tailed or left-tailed. A two-tailed test of hypothesis is a test where the alternative hypothesis does not specify a directional difference for the parameter of interest.

5 Basic Concepts of Statistical Hypothesis Testing A statistical test uses the data obtained from a sample to make a decision about whether the null hypothesis should be rejected. A test statistic is the numerical value obtained from a statistical test.

6 Basic Concepts of Statistical Hypothesis Testing The Type I error is the error made by rejecting the null hypothesis when it is true. The probability of a Type I error is denoted by α. The Type II error is the error made by accepting (not rejecting) the null hypothesis when it is false. The probability of a Type II error is denoted by β. Null Hypothesis Decision True False Reject Ho Type I error Correct decision Accept Ho Correct decision Type II error

7 Basic Concepts of Statistical Hypothesis Testing Illustration. Consider a jury trial.the defendant is either guilty or innocent, and he or she will be convicted or acquitted. Identify the four possible outcomes. The level of significance, α, is the maximum probability of Type I error the researcher is willing to commit.

8 Basic Concepts of Statistical Hypothesis Testing The critical value separates the critical region from the noncritical region. The critical or rejection region is the range of values of the test value that indicates that there is a significant difference and that the null hypothesis should be rejected. The noncritical or nonrejection region is the range of values of the test value that indicates that the difference was probably due to chance and that the null hypothesis should not be rejected.

9 Basic Concepts of Statistical Hypothesis Testing Steps in Hypothesis Testing 1. State the null hypothesis (Ho) and the alternative hypothesis (Ha). 2. Find the critical value(s) using the given α. 3. Select the appropriate test statistic and establish the critical region using the critical value. 4. Collect the data and compute the value of the test statistic from the sample data. 5. Make the decision. Reject Ho if the value of the test statistic belongs in the critical region. Otherwise, do not reject Ho.

10 Testing a Hypothesis on the Population Mean Testing a Hypothesis on the Population Mean Ho Test Statistic Ha Critical Region a. σ known µ < µ o z < z α µ = µ o Z = X µ o σ/ n b. σ unknown µ > µ o z > z α µ µ o z > z α/2 µ < µ o t < t α µ = µ o t = X µ o S/ n µ > µ o t > t α v = n 1 µ µ o t > t α/2

11 Testing a Hypothesis on the Population Mean Remarks: 1. The above tests are exact α-level tests for samples from a normal distribution. However, they provide good approximate α-level test when the distribution is not normal provided that the sample size is large, i.e. n > If σ is unknown and n > 30, use the test in (a) replacing the test statistic by Z = X µ o S/ n

12 Illustration. A researcher reports that the average salary of assistant professors is more than $42,000.A sample of 30 assistant professors has a mean salary of $43,260.At α = 0.05, test the claim that assistant professors earn more than $42,000 per year. The standard deviation of the population is $5230. Solution: Step 1. Ho: µ = $42,000 and Ha: µ >$42,000. Step 2 and 3. Since the given level of significance is α = 0.05, and it is a right-tailed test, the critical value is z = Step 4. The test value is given by Z = X µ σ/ n = / 30 = Since the test value, z=1.32, is less than the critical value, the decision is to not reject the null hypothesis.thus, there is not enough evidence to support the claim that assistant professors earn more on average than $ 42,000 per year.

13 Testing a Hypothesis on the Population Mean EXAMPLES: 1. It is claimed that an automobile is driven on the average less than 25,000 kilometers per year. To test this claim, a random sample of 100 automobile owners are asked to keep a record of the kilometers they travel. Would you agree with this claim if the random sample showed an average of 23,500 kilometers and a standard deviation of 3,900 kilometers. Use a 0.01 level of significance.

14 Examples 2. According to Dietary Goals for the United States (1977), high sodium intake may be related to ulcers, stomach cancer, and migraine headaches. The human requirement for salt is only 230 milligrams per day, which is surpassed in the most single servings of ready-to-eat cereals. A random sample of 20 similar servings of Special K had mean sodium content of 244 milligrams of sodium and standard deviation of 24.5 milligrams. Is there sufficient evidence to believe that the average sodium content for single servings of Special K exceeds the human requirement for salt at α = 0.025? at α = 0.05? at α = 0.10? Assume normality. 3. Test Ho: µ = 50 vs. Ha: µ 50 if a random sample of 16 subjects had mean 48 and standard deviation of 5.8 at 0.05 level of significance. Assume that the sample was taken from a normal population with standard deviation of 6.

15 Testing the Difference Between Two Population Means Based on Two Independent Samples Ho Test Statistic Ha Critical Region a. σ1 2 and σ2 2 known µ 1 µ 2 < d o z < z α µ 1 µ 2 = d o Z = (X 1 X 2 ) d o σ1 2 + σ2 2 n 1 n 2 b. σ1 2 = σ2 2 but unknown t = (X 1 X 2 ) d o µ 1 µ 2 > d o z > z α µ 1 µ 2 d o z > z α/2 S p 1 n n 2 µ 1 µ 2 < d o t < t α µ 1 µ 2 = d o v = n 1 + n 2 2 µ 1 µ 2 > d o t > t α S 2 p = (n 1 1)S (n 2 1)S 2 2 n 1 + n 2 2 µ 1 µ 2 d o t > t α/2

16 Testing the Difference Between Two Population Means Based on Two Independent Samples c. σ 2 σ2 and Ho Test Statistic Ha Critical Region unknown 1 2 µ 1 µ 2 < d o t < t α µ 1 µ 2 = d o t = (X 1 X 2 ) d o S1 2 + S2 2 n 1 n ( 2 S 2 1 /n 1 + S2 2/n 2 2) v = (S1 2/n 1) 2 n (S2 2 /n 2) 2 n 2 1 µ 1 µ 2 > d o t > t α µ 1 µ 2 d o t > t α/2 Remark: If σ 2 1 and σ2 2 are unknown and n 1 and n 2 are greater than 30, we will approximate the t value above (c) by z.

17 Testing the Difference Between Two Population Means EXAMPLES: 1. A statistics test was given to 50 girls and 75 boys. The girls made an average of 80 with a standard deviation of 4 and the boys had an average of 86 with a standard deviation of 6. Is there sufficient evidence of 0.05 level of significance that the average grades of girls and boys differ? 2. A study was made to determine if the subject matter in a physics course is better understood when a lab constitutes part of the course. Students were allowed to choose between a 3-unit course without lab and a 4-unit course with lab. In the section with lab, a sample of 12 students had an average grade of 84 with a standard deviation of 4, and in the section without lab, a sample of 18 students had an average grade of 77 with a standard deviation of 6. Would you say that the laboratory course increases the average grade by more than 5 points? Use a 0.01 level of significance and assume the populations to be approximately normally distributed with equal variances.

18 Testing the Difference Between Two Population Means 3. The following data represent the running time (in minutes) of a random sample of films produced by two motion picture companies: Company Company Test the hypothesis that the average running time of films produced by Company 2 exceeds the average running time of films produced by Company 1 by 10 minutes against the one-sided alternative that the difference is more than 10 minutes. Use a 0.1 level of significance and assume the distributions of times to be approximately normal with unequal variances.

19 Testing the Difference Between Two Population Means Based on Two Related Samples Ho Test Statistic Ha Critical Region t = d do S d / n µ D < d o t < t α µ D = d o v = n 1 µ D > d o t > t α µ D d o t > t α/2 where n d i i=1 d i = x i y i d = v = n 1 n n n ( n ) 2 di 2 d i i=1 i=1 S d = n = number of pairs n(n 1)

20 Example The weights of a random sample of 7 people who followed a new diet were recorded before and after a two-week period: Person Weight Before Weight After Assume the distribution of weights to be approximately normal. At α = 0.05, test the claim that this diet will reduce a person s weight by 4.5 kilograms on the average in a period of 2 weeks.

21 Example Twenty college freshmen were divided into 10 pairs, each member of the pair having approximately the same IQ. One of each pair was selected at random and assigned to a mathematics section using programmed materials only. The other member of each pair was assigned to a section in which the professor lectured. At the end of the semester each group was given the same examination and the following results were recorded. Pair Programmed Materials Lectures Assume the distribution of scores to be approximately normal. At α = 0.05, test the claim that the mean scores will be higher in the class where a professor lectures.

22 Homework 1. The average length of time for students to register at a certain college has been 50 minutes with a standard deviation of 10 minutes. A new registration procedure using modern computing machines is being tried. If a random sample of 12 students had an average registration time of 42 minutes with a standard deviation of 11.9 minutes under the new system, test the hypothesis that the population mean is now less than 50 minutes, using a level of significance of.10,.05 and.01. Assume the population of times to be normal.

23 Homework 2. A taxi company is trying to decide whether the use of radial tires instead of regular belted tires improves fuel economy. Twelve cars were driven twice over a prescribed test course, each time using a different type of tire (radial and belted) in random order. The mileage, in kilometers per liter, were recorded as follows: Kilometers per Liter Cars Radial Tires Belted Tires At the level of significance, can we conclude that cars equipped with radial tires give better fuel economy than those equipped with belted tires? Assume the populations to be normally distributed.