Continuous Probability and the Normal Distribution

�Discrete versus Continuous Random Variables

Discrete Random Variables

• Have outcomes that can be counted
• Number of times heads comes up on a coin flip out of 10 total flips
• Number of cars arriving at a parking garage in one hour
• Number of participants falling within a category

Continuous Random Variables

• Have [theoretically] infinitely many outcomes
• Temperatures
• Commute times
• Machine time to failure

Continuous Random Variables and �Probability Densities

• Continuous random variables can take on infinitely many values
• Because of this, the probability of observing exactly one specified outcome is 0
• Continuous random variables have probability distributions defined by densities rather than masses

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b

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��The Normal Distribution(s)

��The Normal Distribution(s)

��The Normal Distribution(s)

�The Center of a Normal Distribution

�The Spread of a Normal Distribution

�Probabilities and the Normal Distribution

Note: It is also worth knowing that the Normal Distribution is symmetric

Probabilities and the Normal Distribution: The Empirical Rule for Estimation

For normally distributed data, the Empirical Rule states that…

1. About 68% of observations fall within one standard deviation of the mean
2. About 95% of observations are within two standard deviations of the mean
3. About 99.7% (nearly all) are within three standard deviations of the mean

�Examples with the Empirical Rule

Scenario: The wing lengths of Great Blue Herons in Everglades National Park follow an approximately normal distribution. The average wingspan is 167 cm, with a standard deviation of 9 cm. Use the Empirical Rule to answer each of the following.

• What proportion of Great Blue Herons in the Everglades have wingspans between 149 cm and 185 cm?
• Construct an interval of wingspans within which about 68% of Great Blue Herons fall.
• What proportion of Great Blue Herons in Everglades National Park have wingspans between 140 cm and 185 cm?

�Normal, Binomial, or Neither

Determine, if possible, which of the following scenarios are well-modeled by a normal distribution, binomial distribution, or neither

Scenario 1: The time it takes runners to complete a marathon is approximately normally distributed with a mean of 4.5 hours and a standard deviation of 0.75 hours.

Scenario 2: You roll a fair six-sided die repeatedly until a six appears, and you want to know how many rolls it takes.

Scenario 3: A factory has a 2% defect rate. Each day, 200 items are produced, and the number of defective items is counted.

Scenario 4: The lifespan of a certain smartphone battery is approximately normally distributed with a mean of 18 months and a standard deviation of 3 months.

Scenario 5: The number of cars passing through a toll booth in a 10-minute period is recorded. On average, 50 cars pass through every 10 minutes.

�Finding Probabilities Using a Normal Distribution

�Calculating Probability: A Completed Example, Part I

�Calculating Probability: A Completed Example, Part I

�Calculating Probability: A Completed Example, Part II

�Calculating Probability: A Completed Example, Part II

�Calculating Probability: A Completed Example, Part III

�Calculating Probability: A Completed Example, Part III

�Examples: Smartphone Battery Lifespan

�Calculating Percentiles/Quantiles

Sometimes, rather than looking for the probability of an event, we’re more interested in finding the event corresponding to a probability

Example: The manufacturer wants to put a warranty on their batteries, but they want to replace no more than 3% of batteries via warranty. What is the cutoff for the lifespan of these shortest lasting batteries?

Solution. The answer here will be a lifespan in hours rather than a probability. Let’s start with a picture.

�Calculating Percentiles/Quantiles

Sometimes, rather than looking for the probability of an event, we’re more interested in finding the event corresponding to a probability

Example: The manufacturer wants to put a warranty on their batteries, but they want to replace no more than 3% of batteries via warranty. What is the cutoff for the lifespan of these shortest lasting batteries?

Solution. The answer here will be a lifespan in hours rather than a probability. Let’s start with a picture.

�Some Advice on Approaching Problems

�Examples: Marathon Runners

Scenario: The time it takes runners to complete a marathon is approximately normally distributed with a mean of 4.5 hours and a standard deviation of 0.75 hours.

• What is the probability that a randomly selected runner finishes the marathon in less than 4 hours?
• What is the probability that a randomly selected runner finishes the marathon in 3 and a half hours or more?
• What is the probability that a randomly selected runner finishes the marathon between 3 hours and 5 hours?
• At what finishing time do the slowest 20% of runners finish the marathon?

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�Examples: Apple Orchard

Scenario: The weight of apples grown in an orchard is approximately normally distributed with a mean of 150 grams and a standard deviation of 20 grams.

• What is the probability that a randomly selected apple weighs less than 100 grams?
• What is the probability that a randomly selected apple weighs more that 175 grams?
• What is the probability that a randomly selected apple weighs between 160 and 195 grams?
• What is the cutoff for the lightest 5% of apples?
• What is the cutoff for the heaviest 1% of apples?

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Exit Ticket

Navigate to our MAT240 Exit Ticket Form, answer the questions, and complete the task below.

Note. Today’s discussion is listed as 6. Normal Distribution

Task: The time in minutes it takes a city bus to complete a particular route during mid-day traffic is approximately normally distributed with a mean of 42 minutes and a standard deviation of 6 minutes. Discuss your approach to each of the following.

1. Find the probability that a randomly selected bus takes more than 48 minutes to complete the route.
2. Identify the cutoff for the longest 5% of route times.

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�Summary

�Next Time…

• What we’ll be doing…
• Probability workshop

Homework: Review our recent discussions on probability and begin preparing for Exam I.