Questions 1 And 2: Mimi Was 5th Seed In 2014 UMUC Tennis

For Questions 1 2mimi Was The 5th Seed In 2014 Umuc Tennis Open T

(For Questions 1 & 2) Mimi was the 5th seed in the 2014 UMUC Tennis Open held in August. She won 80 of her 100 serving games during the tournament. Based on UMUC Sports Network data, her career record shows she wins 75% of her serving games over 5 years. The assignment involves calculating a confidence interval and conducting hypothesis tests related to her performance.)

Paper For Above instruction

The case of Mimi’s tennis performance offers a compelling illustration of applying statistical inference methods to real-world sports data. Analyzing her performance through confidence intervals and hypothesis testing provides insight into whether her recent tournament results are consistent with her career performance or represent a statistically significant improvement.

Introduction

Statistical inference plays a vital role in sports analytics, enabling practitioners to evaluate athletes’ performances and determine whether observed differences are statistically significant or merely due to chance. In this context, we analyze Mimi’s performance in the 2014 UMUC Tennis Open, applying confidence interval estimation to her winning proportion and hypothesis testing to assess whether her recent success exceeds her recorded career performance. These techniques allow us to interpret her results within a probabilistic framework and make informed decisions based on statistical evidence.

Estimate of the Proportion of Serving Games Won

First, we estimate the true proportion of winning her serving games during the tournament using a confidence interval. With a sample size of 100 games and 80 wins, the sample proportion (p̂) is 0.80. The formula for a confidence interval for a population proportion is:

CI = p̂ ± Zα/2 * √[p̂(1 - p̂) / n]

where Zα/2 is the critical value for a 90% confidence level, which is approximately 1.645. Substituting the values:

Standard error (SE) = √[0.80 * 0.20 / 100] = √[0.16 / 100] = √0.0016 ≈ 0.04

Margin of error (ME) = 1.645 * 0.04 ≈ 0.066

The confidence interval is therefore:

0.80 ± 0.066, which yields (0.734, 0.866)

Rounded to three decimal places, the 90% confidence interval estimate of the true proportion of serving games Mimi won during the tournament is (0.734, 0.866).

Hypothesis Test to Compare Tournament Result with Career Performance

Next, to assess whether Mimi’s tournament performance significantly exceeds her career win rate of 75%, we conduct a hypothesis test for the proportion.

a) Calculating the Test Statistic

The null hypothesis is:

H0: p = 0.75

and the alternative hypothesis is:

Ha: p > 0.75

Using the sample proportion p̂ = 0.80, sample size n=100, and null hypothesis value p0 = 0.75, the z-test statistic is calculated as:

z = (p̂ - p0) / √[p0(1 - p0) / n]

Plugging in: z = (0.80 - 0.75) / √[0.75 * 0.25 / 100] = 0.05 / √[0.1875 / 100] = 0.05 / √0.001875 ≈ 0.05 / 0.0433 ≈ 1.15

Thus, the test statistic is approximately 1.15, rounded to two decimal places.

b) Determining the P-value

Since the alternative hypothesis is a right-tailed test, the p-value corresponds to the probability that Z exceeds 1.15:

P-value = P(Z > 1.15) = 1 - Φ(1.15)

From standard normal distribution tables, Φ(1.15) ≈ 0.8749. Therefore,

P-value ≈ 1 - 0.8749 = 0.1251

Rounded to three decimal places, the p-value is 0.125.

c) Conclusion at Significance Level

At a significance level α = 0.05, since the p-value (0.125) exceeds 0.05, we fail to reject the null hypothesis. This indicates that there is not sufficient statistical evidence to conclude that her recent tournament performance is significantly better than her career average of 75%.

Broader Sports Analytics Context

This analysis exemplifies how confidence intervals offer a range estimate for an athlete's true performance metric, accounting for variability in data collection. Hypothesis testing provides a formal framework to assess whether observed differences are likely due to actual performance changes or random variation. In sports analytics, such methods assist coaches, analysts, and athletes in making data-driven decisions to optimize training strategies and expectations.

Conclusion

In conclusion, while Mimi’s recent tournament results appear promising, statistical analysis indicates that her performance is consistent with her longstanding career record. The use of confidence intervals shows a plausible range, and hypothesis testing confirms that the observed difference is not statistically significant at conventional levels. This underscores the importance of rigorous statistical methods in sports performance evaluation, ensuring that interpretations are grounded in probabilistic evidence rather than anecdotal impressions.

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