4. Reconsider the overall distance data for golf balls in Is there evidence to support the claim that the standard deviation of overall distance is the same for both brands of balls (use a = 0.05)? Explain how this question can be answered with a 95% condence interval on 01/02. The "spring-like effect" in a golf club could be determined by measuring the coefficient of restitution (the ratio of the outbound velocity to the inbound velocity of a golf ball fi red at the clubhead). Twelve randomly selected drivers produced by two clubmakers are tested and the coefficient of restitution measured. The data follow: Club 1: 0.8406, 0.8104, 0.8234, 0.8198, 0.8235, 0.8562, 0.8123, 0.7976, 0.8184, 0.8265, 0.7773, 0.7871 Club 2: 0.8305, 0.7905, 0.8352, 0.8380, 0.8145, 0.8465, 0.8244, 0.8014, 0.8309, 0.8405, 0.8256, 0.8476 Assume that two golf clubs' coefficient restitution are independent and normally distributed and the unknown variances are equal. (a) Is there evidence to support the claim that there is a difference in the mean coefficients of restitution for clubl and club2 at a = 0.05 using P-value approach. (b) Construct a 95% two-sided CI on the mean difference in coefficient of restitution for the two brands of golf clubs, and explain how this interval could be used to answer the question in (a). (c) What is the power of the statistical test in part (a) to detect a true difference in mean coefficient of restitution of 0.2? (d) What sample size would be required to detect a true difference in mean coefficient of restitution of 0.1 with power of approximately 0.8? (a) Since the parameter of interest is the difference in mean coefficient of restitution, we want to test Ho: H1 - H2 = 0 VS. H1: 141 - H2 # 0 The test statistic is (X -X2) - 40 - 0.8161-0.8271 t = =-1.367 Spy + 0.01971, + V12 12(n1 - 1)s, + (n2 - 1)$2 where Sp 1 1(0.0217)~ +11(0.0175) n1 + n2 - 2 = 0.01971 22 Because to.1,22 = 1.321 a = 0.05, thus we fail to reject the null hypothesis. In other words, we do not have sufficient evidence to support the claim that there is a difference in the mean coefficients of restitution for Club 1 and Club 2 at significance level of 0.05. (b) two-sided 95% confidence interval is computed as below: X1 - X2 - to.025,22 5p In1 -+-