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Golf Equipment, Inc. has developed a new coating designed to resist cuts and provide a more durable golf ball. There is concern about how this new coating might affect driving distances, which are crucial for maintaining competitive performance standards and customer satisfaction. Some board members hypothesize that the mean driving distance of the new golf ball could either be longer or shorter than that of the current golf ball, raising questions about its suitability for production under USGA regulations and golfers' expectations.

To evaluate this, a controlled experiment was conducted where 40 golf balls for each model (new coating and current model) were tested using a mechanical hitting machine to eliminate variability due to human error. The goal is to compare the driving distances statistically, ensuring that any observed differences are due to the properties of the balls themselves rather than external factors.

This analysis encompasses descriptive statistics, confidence intervals for the population means, hypothesis testing for the difference in means, and an expression of the final recommendation regarding the production of the new balls based on statistical evidence.

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Introduction

In competitive sports, particularly golf, performance metrics such as driving distance are critical indicators of equipment efficacy and player satisfaction. Recently, Golf Equipment Inc. developed a new coating intended to improve durability and cut resistance. However, before adopting this new golf ball for mass production, it is essential to verify that its performance, specifically driving distance, is comparable to the existing model. This investigation employs statistical methods to compare the mean driving distances of the two golf ball models based on sample data obtained from controlled tests.

Descriptive Statistics and Initial Observations

The data collected from testing 40 golf balls of each model was summarized using descriptive statistics, including measures such as the mean, median, standard deviation, minimum, maximum, and quartiles. These summaries reveal initial insights into the data distribution and variability.

For the current golf ball, the average driving distance was approximately [Insert Mean], with a standard deviation of [Insert SD]. The new coating golf ball showed a slightly different average of [Insert Mean], with a standard deviation of [Insert SD]. These differences suggest some variation in performance, but further analysis is needed to determine if this difference is statistically significant.

Confidence Intervals for Population Means

Using Excel’s Descriptive Statistics feature, the 95% confidence intervals for each model's population mean driving distance were calculated. The confidence interval for the current model was approximately [Lower Bound, Upper Bound], indicating a range within which the true mean is likely to fall with 95% confidence. The new coating model had a confidence interval of [Lower Bound, Upper Bound].

These overlapping confidence intervals suggest that there could be no significant difference in driving distances, but formal hypothesis testing is necessary for confirmation.

Hypothesis Testing for Mean Difference

The null hypothesis (H0) posits that there is no difference in the mean driving distances between the two golf ball models: H0: μ1 = μ2. The alternative hypothesis (Ha) considers that the means are different: Ha: μ1 ≠ μ2. This is a two-tailed test because the direction of the difference is not specified.

Applying t-test for two independent samples assuming unequal variances (Welch's t-test) at α=0.05, the calculated t-statistic was [Insert T-value], with degrees of freedom [Insert df]. The resulting p-value was [Insert p-value], which exceeds/falls below the significance level.

Analysis of Test Results and Confidence Interval for the Difference

The hypothesis test yields a p-value of [Insert p-value], indicating whether to reject or fail to reject H0. If the p-value is greater than 0.05, we conclude that there is not enough evidence to suggest a significant difference between the two models' driving distances. Conversely, if the p-value is less than 0.05, the difference is statistically significant.

To complement this, the 95% confidence interval for the difference in means was computed manually using the formula:

(X̄1 - X̄2) ± t* √(s²₁/n₁ + s²₂/n₂)

Where t* is the critical value from the t-distribution based on degrees of freedom, and the other symbols denote sample means, standard deviations, and sample sizes.

The resulting interval was [Lower Bound, Upper Bound]. If this interval contains zero, it indicates no statistically significant difference in the mean driving distances between the models.

Conclusions and Final Recommendations

Based on the analysis, if the confidence interval for the difference contains zero and the p-value exceeds 0.05, we conclude that the new coating does not significantly impact driving distance performance. This suggests that the new golf ball offers comparable distance performance to the current model and could be considered suitable for production, provided other performance criteria are also met.

However, if the analysis shows a significant difference, the decision to introduce the new ball must weigh the practical importance of this difference against durability benefits. Given that the primary concern is maintaining performance while enhancing durability, a non-significant difference supports endorsing the new coating.

Therefore, the final recommendation is to proceed with production of the new golf ball with the new coating, contingent upon the confirmatory statistical evidence indicating no significant reduction in driving performance and ensuring compliance with USGA standards.

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