Chapter 8 Inferences Based on a Single Sample: Tests of Hypothesis - PowerPoint PPT Presentation

Chapter 8 Inferences Based on a Single Sample: Tests of Hypothesis

The Elements of a Test of Hypothesis 7 elements 1. The Null hypothesis 2. The alternate, or research hypothesis 3. The test statistic 4. The rejection region 5. The assumptions 6. The Experiment and test statistic calculation 7. The Conclusion

The Elements of a Test of Hypothesis Does a manufacturer’s pipe meet building code? Null hypothesis – Pipe does not meet code (H 0 ):  < 2400 Alternate hypothesis – Pipe meets specifications (H a ):  > 2400

The Elements of a Test of Hypothesis   Test statistic to be used x 2400 x 2400   z   n x Rejection region Determined by Type I error, which is the probability of rejecting the null hypothesis when it is true, which is  . Here, we set  =.05 Region is z>1.645, from z value table

The Elements of a Test of Hypothesis Assume that s is a good approximation of  Sample of 60 taken, , s=200  x 2460   Test statistic is 2400 2460 2400 60 x     2 . 12 z 28 . 28 200 50 s n Test statistic lies in rejection region, therefore we reject H 0 and accept H a that the pipe meets building code

The Elements of a Test of Hypothesis Type I vs Type II Error Conclusions and Consequences for a Test of Hypothesis True State of Nature Conclusion H 0 True H a True Accept H 0 Correct decision Type II error (probability  ) (Assume H 0 True) Reject H 0 Type I error Correct decision (probability  ) (Assume H a True)

The Elements of a Test of Hypothesis 1. The Null hypothesis – the status quo. What we will accept unless proven otherwise. Stated as H 0 : parameter = value 2. The Alternative (research) hypothesis (H a ) – theory that contradicts H 0 . Will be accepted if there is evidence to establish its truth 3. Test Statistic – sample statistic used to determine whether or not to reject Ho and accept H a

The Elements of a Test of Hypothesis The rejection region – the region that will lead to 4. H 0 being rejected and H a accepted. Set to minimize the likelihood of a Type I error The assumptions – clear statements about the 5. population being sampled The Experiment and test statistic calculation – 6. performance of sampling and calculation of value of test statistic The Conclusion – decision to (not) reject H 0 , 7. based on a comparison of test statistic to rejection region

Large-Sample Test of Hypothesis about a Population Mean Null hypothesis is the status quo, expressed in one of three forms H 0 :  = 2400 H 0 :  ≤ 2400 H 0 :  ≥ 2400 It represents what must be accepted if the alternative hypothesis is not accepted as a result of the hypothesis test

Large-Sample Test of Hypothesis about a Population Mean Alternative hypothesis can take one of 3 forms: H a :  <2400 One-tailed, upper tail H a :  >2400 One-tailed, upper tail H a :  2400 Two-tailed

Large-Sample Test of Hypothesis about a Population Mean Rejection Regions for Common Values of  Alternative Hypotheses Lower-Tailed Upper-Tailed Two-Tailed  = .10 z < -1.28 z > 1.28 z < -1.645 or z > 1.645  = .05 z < -1.645 z > 1.645 Z < -1.96 or z > 1.96  = .01 z < -2.33 z > 2.33 Z < -2.575 or z > 2.575

Large-Sample Test of Hypothesis about a Population Mean If we have: n=100, = 11.85, s = .5, and we x want to test if  12 with a 99% confidence  level, our setup would be as follows:  H 0 : = 12 H a :  12   x 12 Test statistic  z  x Rejection region z < -2.575 or z > 2.575 (two-tailed)

Large-Sample Test of Hypothesis about a Population Mean CLT applies, therefore no assumptions about population are needed Solve      x 12 x 12 11 . 85 12 11 . 85 12 . 15        z . 3    10 . 5 10 s n 100 x Since z falls in the rejection region, we conclude that at .01 level of significance the observed mean differs significantly from 12

Observed Significance Levels: p- Values The p-value, or observed significance level, is the smallest  that can be set that will result in the research hypothesis being accepted.

Observed Significance Levels: p- Values Steps: Determine value of test statistic z The p-value is the area to the right of z if H a is one-tailed, upper tailed The p-value is the area to the left of z if H a is one-tailed, lower tailed The p-valued is twice the tail area beyond z if H a is two-tailed.

Observed Significance Levels: p- Values When p-values are used, results are reported by setting the maximum  you are willing to tolerate, and comparing p-value to that to reject or not reject H 0

Small-Sample Test of Hypothesis about a Population Mean When sample size is small (<30) we use a different sampling distribution for determining the rejection region and we calculate a different test statistic The t-statistic and t distribution are used in cases of a small sample test of hypothesis about  All steps of the test are the same, and an assumption about the population distribution is now necessary, since CLT does not apply

Small-Sample Test of Hypothesis about a Population Mean  Small-Sample Test of Hypothesis about One-Tailed Test Two-Tailed Test H 0 : H 0 :       0 0       H a : (or H a: ) H a :    0 0 0     x Test Statistic: Test Statistic: x  0  t 0 t s n s n   t  Rejection region: Rejection region: t t t   2 t     (or when H a : t  0 where t  and t  /2 are based on (n-1) degrees of freedom

Large-Sample Test of Hypothesis about a Population Proportion Large-Sample Test of Hypothesis about p One-Tailed Test Two-Tailed Test p  p  H 0 : H 0 : p p 0 0 p  p  p  H a : (or H a: ) H a : p p p 0 0 0 ˆ   ˆ p p p p Test Statistic: Test Statistic:   0 0 z z   ˆ p p  ˆ    where, according to H 0 , and p q n 1 q p 0 0 p 0 0   Rejection region: Rejection region: z z z   z  p  2 (or when z  p z  0

Large-Sample Test of Hypothesis about a Population Proportion Assumptions needed for a Valid Large-Sample Test of Hypothesis for p • A random sample is selected from a binomial population • The sample size n is large (condition satisfied if  3  falls between 0 and 1 p ˆ 0 p

Calculating Type II Error Probabilities: More about  Type II error is associated with  , which is the probability that we will accept H 0 when H a is true Calculating a value for  can only be done if we assume a true value for  There is a different value of  for every value of 

Calculating Type II Error Probabilities: More about  Steps for calculating  for a Large-Sample Test about  1. Calculate the value(s) of corresponding to the x borders of the rejection region using one of the following: s Upper-tailed test:        x z z 0   0 0 x n s Lower-tailed test:        x z z 0   0 0 x n Two-tailed test: s        x z z 0 L   0 0 x n s        x z z 0 U   0 0 x n

Calculating Type II Error Probabilities: More about  2. Specify the value of in H a for  a which  is to be calculated. 3. Convert border values of to x 0  z values using the mean , and a the formula   x 0  a z  x 4. Sketch the alternate distribution, shade the area in the acceptance region and use the z statistics and table to find the shaded area, 

Calculating Type II Error Probabilities: More about  The Power of a test – the probability that the test will correctly lead to the rejection of H 0 for a particular value of  in H a . Power is calculated as 1-  .

Tests of Hypothesis about a Population Variance Hypotheses about the variance use the Chi- Square distribution and statistic   n  2 1 s The quantity has a sampling  2 distribution that follows the chi-square distribution assuming the population the sample is drawn from is normally distributed.

Tests of Hypothesis about a Population Variance 2 Small-Sample Test of Hypothesis about  One-Tailed Test Two-Tailed Test H 0 : H 0 :  2   2  2   2 0 0  2   2  2   2 2 2    H a : (or H a: ) H a : 0 0 0     n  2 n  2 1 s Test Statistic: Test Statistic: 1 s  2   2  2  2  0 0 2 2     2   2 Rejection region: Rejection region:       1 2   1 2  2 2     2   2   2 (or when H a : Or    2  0  2 where is the hypothesized variance and the distribution of is based  2 0 on (n-1) degrees of freedom