# Suppose μ1 and μ2 are true mean stopping distances at 50 mph for cars of a certain type equipped with two different types of braking systems. Use the two-sample t test at significance level 0.01 to test H0: μ1 − μ2 = −10 versus Ha: μ1 − μ2 < −10 for the following data: m = 8, x = 115.6, s1 = 5.04, n = 8, y = 129.3, and s2 = 5.32. Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.) t = ________ P-value = _________

This is a test of two independent groups to find the true mean stopping distances for cars of a certain type equipped with two different types of braking systems. The null hypothesis is that the difference between the means is equal to negative ten (H0: μ1 − μ2 = - 10), while the alternative hypothesis is that the difference is less than negative ten (Ha: μ1 − μ2 < - 10). This is a left-tailed t-test. Step-by-step explanation: 1. Determine the test statistic by using the t-test formula: t = (x1 - x2)/√(s1²/n1 + s2²/n2) From the given information, x1 = 115.6 x2 = 129.3 s1 = 5.04 s2 = 5.32 n1 = 8 n2 = 8 t = (115.6 - 129.3)/√(5.04²/8 + 5.32²/8) t = -2.041 Therefore, the test statistic is -2.04. 2. Determine the degree of freedom by using the formula: df = [s1²/n1 + s2²/n2]²/(1/n1 - 1)(s1²/n1)² + (1/n2 - 1)(s2²/n2)² df = [5.04²/8 + 5.32²/8]²/[(1/8 - 1)(5.04²/8)² + (1/8 - 1)(5.32²/8)²] df = 45.064369/3.22827484 df = 14 Therefore, the degree of freedom is 14. 3. Determine the p-value from the t-test calculator: The p-value is 0.030. 4. Compare the p-value with the chosen significance level (alpha): Since alpha (0.01) < p-value (0.03), we fail to reject the null hypothesis. Hence, we can conclude that there is not enough evidence to suggest that the difference in true mean stopping distances at 50 mph between the two types of braking systems is less than negative ten.