Question

Question 25 9 pts A car dealer was interested in comparing two brands of tires to see if they yielded different wear length (in thousands of miles). The dealer selected eight cars at random and used each of the brands of tires on each car. The wear length was recorded as follows: CAR BRANDA BRANDB 1 45 47 2 43 40 3 54 57 4 63 61 5 39 43 6 47 45 7 56 58 8 50 55 Use the sign test to see if the distribution of wear length is the same for both brands of tires. Use = 0.05. The null and alternate hypotheses are: Ho: The distributions of wear length are identical for the two brands of tires, and P(A exceeds B) = p = 0.50. Ha: The distributions of wear length are not identical for the two brands of tires, and p 0.50. The test statistics is equal to [Select] The p-value for the test statistics is equal to [Select] Conclusion [Select]

Options are

1. 4/3/8

2. 0.384 /0.05/ 0.726

3. the distributions are different/ the distributions are identical/ Reject Ho

Answer 1

The null and alternative hypothesis are defined as follows: H.: The distributions of wear length are identical (i.e p=0.5) H,: The distributions of wear length are not identical (i.e p +0.5) The table for calculation is shown as below: Car 1 2 D=A-B -2 3 -3 3 Brand B 47 40 57 61 43 45 Brand A 45 43 54 63 39 47 56 50 4 2 5 6 -4 2 -2 -5 7 8 58 55 a) The steps to calculate the test statistics are given as follows:

1. Calculate the number of positive signs (**) of the difference (D). 2. Calculate the number of negative signs (**) of the difference (D). 3. Calculate the minimum of the number of positive signs and the number of negative Sign. Test statistics is (M)= min(r*,r"). From the above table, The number of positive signs (3+)= 3. the number of negative signs (**) = M = min(3,5) = 3 Hence, the test statistics is 3.

(b). The test statistics M - Binomial(n,p). Here, n is the total number of cars. M - Binomail(8,0.5) The Probability Mass Function of random variable M is defined as follows: P(M = m) = "Cm.p" (1- p)***** m = 0,1,2, .....8 The table of probability is given as below:

m=Number of success P(success) 0 0.0039 1 0.0313 2 0.1094 3 0.2188 4 0.2734 5 0.2188 6 0.1094 7 0.0313 8 0.0039 The p-value for the test statistics (two-sided) is calculated as follows: p-value=2PM Sm). 2P(M 53)=2[P(0)+ P(1)+P(2)+P(3)] = 2 [0.0039+0.0313+0.1094 +0.2188] = 0.7268 Hence, p-value for the test statistics is 0.7268

The level of significance a = 0.05.The p-value is greater than 0.05. Hence, the null hypothesis is not rejected here. Conclusion: The distributions of wear length are identical for the two brands of tires.