- And more tennis Suppose the tennis player in Exercise 25...

- And more tennis Suppose the tennis player in Exercise 25 serves 80 times in a match. a) What are the mean and standard deviation of the number of good first serves expected? b) Verify that you can use a Normal model to approximate the distribution of the number of good first serves. c) Use the $68-95-99.7$ Rule to describe this distribution. d) What's the probability she makes at least 65 first serves?

- And more tennis Suppose the tennis player in Exercise
25 serves 80 times in a match.
a) What are the mean and standard deviation of the number of good first serves expected?
b) Verify that you can use a Normal model to approximate the distribution of the number of good first serves.
c) Use the $68-95-99.7$ Rule to describe this distribution.
d) What's the probability she makes at least 65 first serves?

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. This concept is used to analyze events where each trial has two outcomes (success or failure) and is foundational in probability and statistics.

Expected Value

The expected value, or mean, of a distribution quantifies the average outcome one would anticipate over many repetitions of the experiment. In the context of the binomial distribution, it is calculated as the product of the number of trials and the probability of success, representing the center of the distribution.

Standard Deviation

The standard deviation measures the spread or variability of a distribution around its mean. For a binomial distribution, it is found by taking the square root of the product of the number of trials, the probability of success, and the probability of failure. This metric helps in understanding the dispersion of the outcomes.

Normal Approximation to the Binomial

The normal approximation to the binomial distribution involves using a normal distribution to approximate the behavior of a binomial process when the number of trials is sufficiently large. The approximation is justified by the central limit theorem and is typically used when both np and n(1-p) are greater than or equal to a certain threshold, ensuring the binomial distribution is not too skewed.

Empirical Rule (68-95-99.7 Rule)

The 68-95-99.7 rule is an empirical guideline that describes how data are distributed in a normal distribution. It states that approximately 68% of the data fall within one standard deviation of the mean, 95% within two, and 99.7% within three. This rule provides a quick way to estimate probabilities and understand the dispersion in normally distributed data.

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