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Standard deviation is a key concept in statistics that measures how spread out numbers are in a data set. When data points are close to the mean, the standard deviation is low, and when they are spread out over a large range of values, it is high.
In the context of Rafael Nadal's first serve speeds, the standard deviation is 6 mph. This tells us that typically, the speeds of his first serves vary 6 mph away from the average speed of 115 mph. This variation gives us an idea of the consistency (or inconsistency) in his serve speeds.

• A smaller standard deviation would mean that his serves are very consistent, closely hovering around 115 mph.
• A larger standard deviation would indicate that his serve speeds vary significantly from serve to serve, either much slower or much faster than 115 mph.

In practical terms, knowing the standard deviation helps tennis coaches understand his performance variation, which can be worked on to create strategies for improvement.

Probability quantifies the chance of an event occurring. When dealing with continuous data, like serve speeds, probabilities help determine how likely a particular outcome is.
In Nadal's case, we use probabilities to calculate how likely his serve exceeds certain speeds, like 120 mph. This uses the normal distribution properties to find the likelihood of different outcomes:

• To calculate probabilities, we first convert serve speeds from the normal distribution to the standard normal distribution, leading to a calculation of a Z-score.
• A Z-score represents the number of standard deviations a data point is from the mean.

The problem given asks us to find the probability, or likelihood, that a randomly chosen serve will be a certain speed or faster, allowing athletes and coaches to make informed decisions based on likely outcomes.

The standard normal variable, often denoted as Z, is a very important statistical concept used to compare data points from different normal distributions. By standardizing data, we transform it into a standard normal curve centered at 0 with a standard deviation of 1.
Converting a variable into a standard normal variable allows us to use the standard normal distribution table, which lists probabilities of Z-values:

• To convert Nadal's serve speed (Y) to a standard normal variable (Z), use the formula:

\[Z = \frac{Y - \mu}{\sigma}\]
This transformation helps determine how unusual or typical that speed is compared to the average.

• For example, with a Z-score of 0.83, we find that a serve speed of 120 mph is 0.83 standard deviations above the mean.

Using a standard normal variable makes it easier to calculate probabilities and understand how a data point relates to the rest of the data.