Wize University Statistics Textbook > Continuous Probability Distributions

Normal Distribution

Normal Distribution

A continuous random variable $X$ follows a normal distribution if its probability distribution curve has the following properties:

It is symmetric about the mean
It has a bell-shape (mean=median=mode)
The total area under the curve is equal to 1
$P\left(a<X<b\right)$ is the same as the area underneath the bell-curve, between the values $a$ and $b$
The horizontal axis measures the possible value of the continuous random variable
The vertical axis measures the frequency or %’s

Note

We denote this as $\boxed{X\ \sim\ N\left(\mu,\ \sigma\right)}$

Examples

The height of a randomly selected student from a particular University
The weight of an apple that is picked from an apple tree

Shape of a Normal Distribution

Two normal distributions can have the same mean but different standard deviations:

Example:

Two normal distributions can have different means and the same standard deviation:

Example:

Image illustrating Normal Distribution

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Empirical Rule (68 - 95 - 99.7% Rule)

The Empirical Rule states that almost all of the data fall within three standard deviations for a normal distribution.

68% Rule

The probability that $X$ takes on a value between $\mu-\sigma$ and $\mu+\sigma$ is approximately 68%
68% of the data lies within 1 standard deviation away from the mean

95% Rule

The probability that $X$ takes on a value between $\mu-2\sigma$ and $\mu+2\sigma$ is approximately 95%
95% of the data lies within 2 standard deviations away from the mean

99.7% Rule

The probability that $X$ takes on a value between $\mu-3\sigma$ and $\mu+3\sigma$ is approximately 99.7%
99.7% of the data lies within 3 standard deviations away from the mean

Car batteries are designed to last on average 5 years with standard deviation of 1.3 years. Assume that battery life is normally distributed.

a) Using the 68-95-99.7% rule, what percentage of batteries will last between 2.4 and 3.7 years?

b) Using the 68-95-99.7% rule, what proportion of batteries will last under 2.4 years or more than 6.3 years?

a) Enter the percentage as a decimal.b) Enter the percentage as a decimal.

I don't know

Extra Practice

Normal Distribution: Standardizing

When calculating students' marks out of 25, their z-scores are also calculated. The average score was 17 and the standard deviation was 1.4. Test scores are normally distributed. Eric tells John "my teacher said I scored higher than 60% of the class". What's the lowest mark Eric could have had?

Normal Distribution

Typically users of credit cards don’t pay on time. The number of days a payment is made past the due date is normal with mean 2.5 days and standard deviation 0.30 days.

According to the empirical rule,

a) what percent of users pay more than 3.1 days late? (write % in 2 decimal places)

Central Limit Theorem

Which of the following statements best describes the Central Limit Theorem?

Normal distribution

A random skills test was administered to a large group of students at a university. Suppose that the time the students took to complete the test is approximately normal with a mean of 40 minutes and a standard deviation of 10 minutes. If the top 20% of the students who completed the skills test the fastest get a prize, how fast must a candidate complete the test to get a prize?

Normal distribution: Standardizing

The average tips servers receive is normally distributed with a mean of 13.5% and a standard deviation of 4.5%. What is the probability of Trevor receiving more than 15% on a given bill?

Standardizing: Normal Distribution

The exam grades in Hugh’s statistics class are normally distributed. If his z-score is 0.00, which of the following must be true?

Normal Distribution and Standardizing

If X = 200, we get z = 2.00; and

if X = 115, we get z = –1.40

Solve for $\mu$ and $\sigma$ .

Normal distribution and standardizing

The random variable X has a normal distribution. If $μ=20$ and $σ=4$ find the value of $a$ such that $P(16<X<a)=0.7357$ . [Round $a$ to the nearest whole number.]

Normal distribution

A farmer needs to remove apples that weigh less than 80 grams. His trees grow apples that weight an average of 100 grams. About 3.75% of apples are too small. Assuming the weight of apples is normally distributed, what is the standard deviation of the weight of apples?

Normal distribution

95% of all Koka-Kola cans have between 150 ml and 200 ml of soda pop. According to the 68-95-99.7% rule, find the values of the mean and standard deviation of this distribution.

Normal Distribution

Mean average employee salary at CLT (Cash Loving Tycoons) Corp is $89,450; but the median salary is $71,600. If someone randomly samples 7 employees, which of the following is true?

Normal Distribution

CLT (Cats Living Together) Agency has hundreds of cats up for adoption. The time it takes for a cat to get adopted is 25 days on average and is Normally distributed. If someone randomly samples 50 cats, which of the following is true?

Hypothesis Testing: Correlation

For the following statements, select true or false. Enter T for True, F for False.

Normal Distribution

About 95% of all transactions with a bank teller at HSBC take between 2 and 16 minutes, but the distribution is skewed to the right. We took a random sample of 101 customers at HSBC. Which of the following is true?

Empirical Rule

Car batteries are designed to last on average 5 years with standard deviation of 1.3 years. Assume that battery life is normally distributed.

a) Using the 68-95-99.7% rule, what percentage of batteries will last between 2.4 and 3.7 years?

b) Using the 68-95-99.7% rule, what proportion of batteries will last under 2.4 years or more than 6.3 years?

Normal Distribution

Frite Alors is a Montreal fast food chain specializing in burgers and poutine. If the amount of grams of cheese in a poutine follows a normal distribution:

50% of poutines have more than 230 grams of cheese, and 15.46% have less than 180 grams.

What is the mean of the amount of cheese?

Normal Distribution

Typically users of credit cards don’t pay on time. The number of days a payment is made past the due date is normal with mean 2.5 days and standard deviation 0.30 days.

According to the empirical rule, what percent of users pay more than 3.1 days late?

Normal Distribution

Typically users of credit cards don’t pay on time. The number of days a payment is made past the due date is normal with mean 2.5 days and standard deviation 0.30 days.

According to the empirical rule, what percent of users pay between 1.6 and 2.8 days late?

Normal Distribution: Empirical Practice Problem

The number of calories consumed by an athlete varies according to a normal distribution with mean 2500 and standard deviation 100.

(a) What % of the days does this athlete consume under 2300 or between 2600 and 2700 calories?

(b) In a full year (365 days), approximately how many days do we expect him to consume between 2200 and 2300 calories?

Normal Distribution

The weights of 10 week old puppies at a certain dog training school have a normal distribution with mean 12 lbs and a standard deviation of 4 lbs. The proportion of puppies that are heavier than 16 lbs is

Normal Distribution

The volume of juice boxes produced by a certain company follows the normal distribution with mean 125ml and standard deviation 2ml. Suppose the company will only ship out juice boxes with less 126ml. If the company wants to ship out 97.5% of juice boxes, by how much should the mean juice box volume be lowered in order for the company to achieve this shipping goal?

Normal Distribution

The weight of an apple from a certain apple tree is a random variable with mean 25g and variance 4g. What is the variance of the average weight of 25 randomly chosen apples?

Normal Distribution

IQ scores for 15 year olds are approximately normally distributed with mean 120 and standard deviation 16.

According to the 68-95-99.7% empirical rule, what proportion of 15 year olds have an IQ score between 88 and 152?

Normal Distribution

Normal Distribution

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