- \n
- What is Wald-Wolfowitz Test?<\/a><\/li>\n
- Formula for the Wald-Wolfowitz Test?<\/a><\/li>\n
- Exercise 1<\/a><\/li>\n
- Solution Steps<\/a><\/li>\n<\/ol>\n
- \n
- Step 1:\u00a0 State the null and alternate hypothesis<\/a><\/li>\n
- Step 2:\u00a0 Merge and sort the values in order<\/a><\/li>\n
- Step 3: Calculate the Mean<\/a><\/li>\n
- Step 4: Calculate the Variance<\/a><\/li>\n
- Step 5: Calculate Z statistic<\/a><\/li>\n
- Step 6: Draw your conclusion<\/a><\/li>\n<\/ul>\n
<\/div>\n<\/p>\n
1. What is Wald-Wolfowitz Test?<\/h4>\n
Wald-Wolfowitz Test (also called Wald-Wolfowitz run test) is a non-parametric hypothesis test used to test the randomness of a two-valued data sequence. It tests to see if the sequence are mutually independent.<\/p>\n
<\/p>\n
2. Formula for Wald-Wolfowitz Test<\/h4>\n
The three formulas for the Wald-Wolfowitz run test\u00a0 are given below:<\/p>\n
![](\"https:\/\/4.bp.blogspot.com\/-qAoN4VZh0qg\/Wx4nhctZlzI\/AAAAAAAAB60\/bYfLBpE-cJYlQiUf88s9PyC5ynFWNLd4wCLcBGAs\/s200\/Wald-Wolfowitz.jpg\")
<\/a><\/div>\nwhere the mean is given by the formula<\/p>\n
![](\"https:\/\/2.bp.blogspot.com\/-SKCKTaGF0RU\/Wx4oZsdDb3I\/AAAAAAAAB7A\/xLxAOAKAlnEPqljHHk6S7kqUZ_1OIV-ZgCLcBGAs\/s320\/Mean.jpg\")
<\/a><\/div>\nand the variance is given by the formula<\/p>\n
<\/div>\n![](\"https:\/\/3.bp.blogspot.com\/-C32KEWw7Jhk\/Wx4qdIesEpI\/AAAAAAAAB7M\/-kHm967PjiUU4cHXlP6r3wZUAZyWupNrwCLcBGAs\/s400\/Variance.jpg\")
<\/a><\/div>\n<\/div>\n<\/div>\nNote<\/b>: Variance is the square of the standard deviation. So we calculated variance. To get the standard deviation, we must take the square root of the variance.<\/p>\n
Let’s now solve an example!<\/p>\n
<\/p>\n
3. Example 1<\/h4>\n
There are two IVM(Innoson Vehicle Manufacturers) buses, one with 48 passengers,\u00a0 and another with 38 passengers.<\/em>
\nLet X and Y denote the number of miles travelled per day for the 48-passenger and 38-passenger buses respectively. Innoson would like to test the equality of the two distributions.<\/em>
\nThat is, if:<\/em><\/p>\nH0<\/sub>: F(z) = G(z)<\/em><\/p>\n
The company observed the following data on a random sample of n1 = 10 buses carrying 48 passengers and n2 = 11 buses carying 38 passengers.<\/em><\/p>\n
X: 104 253 300 308 315 323 331 396 414 452 <\/em>
\nY:\u00a0 184 196 197 248 260 279 355 386 393 432 450<\/em><\/p>\nUsing normal approximation to R, conduct a Wald-Wolfowitz test at 0.05 level of significance<\/em><\/p>\n
<\/p>\n
4. Solution<\/h4>\n
We would solve this problem step by step.<\/p>\n
<\/p>\n
Step 1: State the null and the alternate hypothesis and rejection criteria<\/h5>\n
H0<\/sub>: F(z) = G(z)
\nH1<\/sub>: F(z) = G(z)<\/p>\nRejection criteria: Reject the null hypothesis if<\/p>\n
![](\"https:\/\/4.bp.blogspot.com\/-wCzqfD7xXHQ\/Wx42-o35KdI\/AAAAAAAAB7Y\/1Y2PgnnAsFEQfaoh5IPxotG_3iNsrbDaACLcBGAs\/s200\/Rejection%2Bcriteria.jpg\")
<\/a><\/div>\n<\/p>\n
Step 2: Merge the two lists and sort in ascending order<\/h5>\n
104\u00a0 184\u00a0 196\u00a0 197\u00a0 248<\/span>\u00a0<\/span> 253\u00a0 260\u00a0 279<\/span>\u00a0 300\u00a0 308\u00a0 315\u00a0 331\u00a0 355\u00a0 386\u00a0 393\u00a0<\/span> 394\u00a0 414\u00a0 432\u00a0 450\u00a0<\/span> 452<\/p>\n
<\/p>\n
Step 3: Count the number of runs: R, n1<\/sub> and n2<\/sub><\/h5>\n
Number of runs R = 9
\nn1 = 10
\nn2 =\u00a0 11<\/p>\n<\/p>\n
Step 4: Calculate the mean<\/h5>\n
We calculate the mean using the formula and we have the results below<\/p>\n
- Step 2:\u00a0 Merge and sort the values in order<\/a><\/li>\n
- Formula for the Wald-Wolfowitz Test?<\/a><\/li>\n
![](\"https:\/\/3.bp.blogspot.com\/-boz92sdE2YE\/Wx458V8nwOI\/AAAAAAAAB7k\/B5s_4NL3T1cJAH4gOIaFCE5Thez0K1RIgCLcBGAs\/s320\/Mean%2BCalculation.jpg\")
<\/a><\/div>\n![](\"https:\/\/2.bp.blogspot.com\/-W2m54VPUWH0\/Wx47IypzmEI\/AAAAAAAAB7w\/yJvoBBu5Ci89NN4_bsPT3RdCpyDPALt_QCLcBGAs\/s320\/Variance%2BCalculation.jpg\")
<\/a><\/div>\n![](\"https:\/\/1.bp.blogspot.com\/-Sthv8i8c2bY\/Wx49VjwJUYI\/AAAAAAAAB78\/gp0s5LN7AmQA8yKdvAnj0Vf7KW_wVYy0ACLcBGAs\/s200\/Z%2Bstatistic%2Bcalculation.jpg\")
<\/a><\/div>\n