Math  /  Data & Statistics

Question9. Rafa serves again! Tennis superstar Rafael Nadal's first-serve speeds in a recent season can be modeled by a normal distribution with mean 115 mph and standard deviation 6 mph . Use the empirical rule to approximate the following: (a) The proportion of Rafa's first serves that were faster than 121 mph (b) The percent of Rafa's first serves with speeds between 109 and 133 mph
10. Cholesterol modeled Cholesterol levels for teenage boys can be modeled by a normal distribution with mean 150mg/dl150 \mathrm{mg} / \mathrm{dl} and standard deviation

Studdy Solution

STEP 1

What is this asking? We want to figure out how often Rafa serves faster than 121 mph and how often his serves fall between 109 and 133 mph, knowing his average serve speed and how much they typically vary. Watch out! Remember the empirical rule applies *only* to normal distributions!
Don't forget what percentages go with each standard deviation from the mean.

STEP 2

1. Set up the problem
2. Calculate the proportion of serves faster than 121 mph
3. Calculate the proportion of serves between 109 and 133 mph

STEP 3

Alright, so we know Rafa's serves follow a **normal distribution**.
This is key because it lets us use the **empirical rule**!
His average serve speed, the **mean**, is μ=115\mu = 115 mph, and the **standard deviation**, which tells us how spread out his serve speeds are, is σ=6\sigma = 6 mph.

STEP 4

The **empirical rule**, also known as the 68-95-99.7 rule, tells us approximately what percentage of data falls within certain ranges of a normal distribution: - About **68%** of the data falls within **one** standard deviation of the mean. - About **95%** of the data falls within **two** standard deviations of the mean. - About **99.7%** of the data falls within **three** standard deviations of the mean.

STEP 5

We want to know how often Rafa serves faster than 121 mph.
First, let's see how far 121 mph is from the mean: 121115=6121 - 115 = 6 mph.
This is exactly **one standard deviation** (σ\sigma) above the mean!

STEP 6

The empirical rule tells us that about 68% of Rafa's serves fall within one standard deviation of the mean (between 109 mph and 121 mph).
This means the remaining 32% of serves are *outside* this range.
Since the normal distribution is symmetric, half of that 32%, or **16%**, will be *above* 121 mph.

STEP 7

Now, we're looking for serves between 109 mph and 133 mph.
Let's find how far these values are from the mean: 115109=6115 - 109 = 6 mph, which is one standard deviation *below* the mean. 133115=18133 - 115 = 18 mph, which is 18÷6=318 \div 6 = 3 standard deviations *above* the mean.

STEP 8

We know that 68% of serves are within one standard deviation of the mean.
Since 109 mph is one standard deviation below the mean, the percentage of serves between 109 mph and 115 mph is half of 68%, which is 34%.

STEP 9

We also know that 99.7% of serves are within three standard deviations of the mean.
Since 133 mph is three standard deviations above the mean, the percentage of serves between 115 mph and 133 mph is half of 99.7%, which is 49.85%.

STEP 10

To find the percentage of serves between 109 mph and 133 mph, we add the percentages we just calculated: 34%+49.85%=83.85%34\% + 49.85\% = 83.85\%.

STEP 11

(a) Approximately **16%** of Rafa's first serves were faster than 121 mph. (b) Approximately **83.85%** of Rafa's first serves were between 109 and 133 mph.

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