Math  /  Data & Statistics

QuestionLEKS 4.5: Statistics \& Unit 4 R ALEKS Quiz 4 (Statistics) ALEKS - Jorge Mandujano - Lea Wow! www-awu.aleks.com/alekscgi/x/sl.exe/1o_u-lgNslkasNW8D8A9PVfh-8q3gyGHU3RuYLjKQkQoiOk1A8mqLVs492_RhY1of4ZFuDMOAwRJjpunuqw-1ivxImx3RJ_LHHK0VaRofAbM8-9D... Stalisties 3/53 / 5 Jorge Finding a probability given a normal distribution: Basic Español
Suppose that resting pulse rates among healthy adults are normally distributed with a mean of 72 beats per minute and a standard deviation of 25 beats per minute. Use this table or the ALEKS calculator to find the percentage of healthy adults who have resting pulse rates that are more than 109 beats per minute. For your intermediate computations, use four or more decimal places. Give your final answer to two decimal places (for example 98.23%98.23 \% ). \square 【% \%

Studdy Solution

STEP 1

What is this asking? What percentage of healthy adults have resting pulse rates above 109 beats per minute, given that the pulse rates follow a normal distribution with a mean of 72 and a standard deviation of 25? Watch out! Don't forget to convert the pulse rate to a *z*-score before using the *z*-table or calculator!
Also, make sure to look for the percentage *above* the given value, not below it.

STEP 2

1. Calculate the *z*-score.
2. Find the corresponding probability.
3. Calculate the final percentage.

STEP 3

Alright, let's **start** by calculating the *z*-score!
The *z*-score tells us how many standard deviations a particular value is away from the mean.
It's like a universal translator for normal distributions!

STEP 4

The formula for the *z*-score is: z=xμσ z = \frac{x - \mu}{\sigma} Where xx is the **value** we're interested in (109 beats per minute), μ\mu is the **mean** (72 beats per minute), and σ\sigma is the **standard deviation** (25 beats per minute).

STEP 5

Let's plug in the **numbers**: z=1097225 z = \frac{109 - 72}{25}

STEP 6

z=3725 z = \frac{37}{25}

STEP 7

z=1.48 z = 1.48 So, a pulse rate of 109 beats per minute is **1.48 standard deviations** above the mean.

STEP 8

Now, we need to find the probability of having a *z*-score greater than **1.48**.
We can use a *z*-table or a calculator for this.
The *z*-table gives us the probability of a *z*-score being *less than* a given value.
Since we want the probability of it being *greater*, we'll need to do a little trick!

STEP 9

Let's say our *z*-table tells us that the probability of a *z*-score being less than 1.48 is **0.9306** (you can find this value using a *z*-table or calculator).

STEP 10

Since the total probability under the normal distribution curve is **1**, the probability of a *z*-score being *greater* than 1.48 is: 10.9306 1 - 0.9306

STEP 11

0.0694 0.0694

STEP 12

Finally, to express this probability as a **percentage**, we multiply by 100: 0.0694100 0.0694 \cdot 100

STEP 13

6.94 6.94 So, approximately **6.94%** of healthy adults have resting pulse rates above 109 beats per minute.

STEP 14

6.94% of healthy adults have resting pulse rates more than 109 beats per minute.

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