Question

The overall distance traveled by a golf ball is tested by hitting the ball with Iron Byron, a mechanical golfer with a swing that is said to emulate the distance hit by the legendary champion, Byron Nelson. Ten randomly selected balls of two different brands are tested and the overall distance measured. The data follow:

Brand 1: 275, 286, 287, 271, 283, 271, 279, 275, 263, 267

Brand 2: 258, 244, 260, 265, 273, 281, 271, 270, 263, 268.

a. Is there evidence that overall distance is approximately normally distributed? Is an assumption of equal variances justified?

b. Test the hypothesis that both brands of ball have equal mean overall distance. Use α = 0.05. What is the P-value?

c. Construct a 95% two-sided CI on the mean difference in overall distance for the two brands of golf balls.

d. What is the power of the statistical test in part (b) to detect a true difference in mean overall distance of 5 yards?

e. What sample size would be required to detect a true difference in mean overall distance of 3 yards with power of approximately 0.75?

Answer 1

a. Determine if overall distance is normally distributed and if assumption of equal variances is justified. b. Perform a hypothesis test to determine if both brands have equal mean overall distance. c. Construct a 95% two-sided confidence interval on the mean difference in overall distance for the two brands. d. Calculate the power of the statistical test to detect a true difference in mean overall distance of 5 yards. e. Determine the sample size required to detect a true difference in mean overall distance of 3 yards with power of 0.75.

Step-by-step explanation:

a. To determine if the overall distance is approximately normally distributed, we can create a histogram of the data for each brand and check if the distribution looks approximately bell-shaped. We can also calculate the skewness and kurtosis values for each brand and compare them to the values of a normal distribution. To test if the assumption of equal variances is justified, we can perform an F-test by comparing the variances of the two brands.

b. To test the hypothesis that both brands have equal mean overall distance, we can perform a two-sample t-test. The null hypothesis states that the means are equal, and the alternative hypothesis states that the means are unequal. The p-value calculated from the t-test will indicate the strength of evidence against the null hypothesis.

c. To construct a 95% two-sided confidence interval (CI) on the mean difference in overall distance, we can calculate the mean difference between the two brands and then determine the margin of error using the appropriate t-value. The CI will provide a range within which we can be confident that the true mean difference lies.

d. The power of the statistical test in part (b) to detect a true difference in mean overall distance of 5 yards can be calculated by determining the probability of rejecting the null hypothesis when the true mean difference is 5 yards. This can be done by performing a power analysis using the assumed effect size, sample sizes, significance level, and the variability of the data.

e. To determine the sample size required to detect a true difference in mean overall distance of 3 yards with a power of approximately 0.75, we can perform a power analysis and calculate the required sample size based on the assumed effect size, significance level, power, and variability of the data.

Answer 2

The data is approximately normally distributed.




a. To determine if the overall distance traveled by the golf balls is approximately normally distributed, we can perform a normality test, such as the Shapiro-Wilk test. If the p-value from this test is greater than 0.05, we can conclude that the data is approximately normally distributed.

To test the assumption of equal variances between the two brands of golf balls, we can use the Levene's test. If the p-value from this test is greater than 0.05, we can assume that the variances are equal.

b. To test the hypothesis that both brands of golf balls have equal mean overall distance, we can use a two-sample t-test. The null hypothesis (H0) is that the mean overall distance for both brands is equal, and the alternative hypothesis (Ha) is that the mean overall distance for both brands is not equal. Using an α level of 0.05, if the p-value from the test is less than 0.05, we reject the null hypothesis and conclude that there is a significant difference in the mean overall distance between the two brands.

c. To construct a 95% two-sided confidence interval (CI) on the mean difference in overall distance for the two brands of golf balls, we can use the formula:

CI = (x1 - x2) ± t * sqrt((s1^2/n1) + (s2^2/n2))

where x1 and x2 are the sample means for Brand 1 and Brand 2, s1 and s2 are the sample standard deviations for Brand 1 and Brand 2, n1 and n2 are the sample sizes for Brand 1 and Brand 2, and t is the critical value from the t-distribution with (n1 + n2 - 2) degrees of freedom.

d. To calculate the power of the statistical test in part (b) to detect a true difference in mean overall distance of 5 yards, we need additional information, such as the assumed standard deviation of the population and the sample size. With this information, we can use power analysis to determine the power of the test.

e. To determine the sample size required to detect a true difference in mean overall distance of 3 yards with a power of approximately 0.75, we again need additional information, such as the assumed standard deviation of the population and the desired power level. Using power analysis, we can calculate the required sample size.