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Wize University Statistics Textbook > Continuous Probability Distributions

Normal Distribution

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Normal Distribution

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

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

Note
We denote this as X ∼ N(μ, σ)\boxed{X\ \sim\ N\left(\mu,\ \sigma\right)}

Wize Tip
Suppose you plot continuous data in a histogram and draw a smooth curve connecting the tops of the bars. If this curve forms a symmetrical bell-shape, then we say that the data is normally distributed or follows the normal model.

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


Portions of information contained in this publication/book are printed with permission of Minitab, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved.





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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 XX takes on a value between μ−σ\mu-\sigma and μ+σ\mu+\sigma is approximately 68%
  • 68% of the data lies within 1 standard deviation away from the mean

95% Rule
  • The probability that XX takes on a value between μ−2σ\mu-2\sigma and μ+2σ\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 XX takes on a value between μ−3σ\mu-3\sigma and μ+3σ\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?
Extra Practice
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
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
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
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 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: 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
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
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
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?
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
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?
Hypothesis Testing: Correlation
For the following statements, select true or false. Enter T for True, F for False.
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?
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
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
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 and standardizing
The random variable X has a normal distribution. If μ=20μ=20 and σ=4σ=4 find the value of aa such that P(16<X<a)=0.7357P(16<X<a)=0.7357. [Round aa to the nearest whole number.]
Normal Distribution and Standardizing
If X = 200, we get z = 2.00; and
if X = 115, we get z = –1.40
Solve for μ\mu and σ\sigma.
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: 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?
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?
Central Limit Theorem
Which of the following statements best describes the Central Limit Theorem?
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)
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?