According to the National Health Survey, the heights of...

Name: Problem 14, Chapter 8: Calculus: Early Transcendentals
Uploaded: 2020-01-17T14:42:31-08:00
Duration: 7 min 41 s
Channel: Gatjang Deng
Description: According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. (a) What is the probability that an adult male chosen at random is between 65 inches and 73 inches tall? (b) What percentage of the adult male population is more than 6 feet tall?

According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. (a) What is the probability that an adult male chosen at random is between 65 inches and 73 inches tall? (b) What percentage of the adult male population is more than 6 feet tall?

According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches.
(a) What is the probability that an adult male chosen at random is between 65 inches and 73 inches tall?
(b) What percentage of the adult male population is more than 6 feet tall?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve. It is defined by its mean and standard deviation, and many natural phenomena, including human characteristics such as height, approximately follow this pattern. The shape of the curve indicates that data near the mean are more frequent in occurrence than data far from the mean.

Mean and Standard Deviation

The mean and standard deviation are parameters that describe the center and spread of a normal distribution, respectively. The mean indicates the average value of the data distribution, while the standard deviation provides a measure of the variability or dispersion around the mean. These values are crucial for determining probabilities of outcomes within a normal distribution.

Standardization and Z-Scores

Standardization is the process of converting data points from a normal distribution to a standard normal distribution using Z-scores. A Z-score represents the number of standard deviations a data point is from the mean. This transformation enables the use of a standard normal table or computing software to determine probabilities associated with specific ranges of the variable.

Probability Calculations

Probability calculations for normally distributed variables often involve computing the area under the curve, which represents the likelihood of a variable falling within a given range. By using the standard normal distribution and Z-scores, one can calculate the probability that an observation lies between two values or exceeds a certain threshold.

Unit Conversion

Unit conversion is the process of changing measurements from one unit system to another, which can be necessary when comparing data. In probability problems, particularly those involving physical dimensions such as height, it is important to convert all measurements to the same units before applying standardization and probability techniques.

Transcript

00:04 Okay, so for this problem, we are told that the, according to the health survey, the height of adult males in the united states are normally distributed, which means of 69 inches and 28 inches.

00:27 So we're given mean, so it means we don't use a good better meal, because of 69 inches.

00:35 And then you're given this standard deviation is 28th.

00:46 The standard deviation we would use sigma equals 2 .8 inches.

00:55 So that's how you define.

01:03 So the survey is normal distributed.

01:07 So that means it can be modeled by f of x is equal to 1 over and then the square root 2 pi times sigma.

01:17 Times e to minus 1 half, x minus mu over sigma squared.

01:33 And this is true for all x.

01:37 This is true for all x greater than equals zero.

01:43 So that means f of x is zero everywhere else in that domain.

01:48 You can plug in mu equals 69 and sigma equals 2 .8 inches to this expression here.

01:57 Now the question we would like to answer is to determine the probability that an adult male is chosen random.

02:08 Chosen a random will be between 65 inches and 73 inches tall.

02:17 So a is similar to saying what is p of x is between 65 and 73 inches tall.

02:34 So mathematically, we can translate this into an interval, we will become.

02:41 So our limits of integration are going to be from 65 to 73, f of x, dx...

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