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Math 1040 Study Guide/Lecture Notes (Ch. 7.1–7.2)
Section 7.1—Properties of the Normal Distribution and Uniform Distribution When computing probabilities for discrete random variable we usually substitute the value of the random variable into a formula. This is not easy for continuous random variables because there is an infinite number of outcomes. To resolve this we will compute probabilities of continuous random variables over an interval of values. Definitions:
• A ___________________________________ is an equation, table, or graph used to describe reality.
• A _________________________________________________________ (pdf) is an equation used to compute probabilities of continuous random variables. Properties of pdfs 1. Total area under the graph of the equation over all possible values of the random
variable must equal _________.
2. Height of the graph must be greater than or equal to _________ for all possible values of the random variable.
• The __________________________________________________________ is a distribution where any two
intervals of equal length are equally likely.
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Example: The graph to the right is a uniform probability distribution function for the distance that a group of toddlers can through a ball.
a. Find the probability (6 ≤ x ≤ 9)
b. Find the P(at least 8) =
c. There is a 25% probability that x is within _______ feet from 2 feet.
• The __________________________________ distribution of a continuous random variable whose relative frequency distribution has the shape of a normal curve (bell-shaped)
Properties of the Normal Distribution 1. Symmetric about its mean __________
2. Mean = median = mode, so it has a single _________________ that occurs at x = __________
3. Inflection points at _______________ and _______________
4. Area under curve is __________
5. Area to the right of µ equals the area to the left of μ, which equals __________
6. As x gets larger and larger or more and more negative (you move further out from μ), the graph approaches, but never reaches, the ___________________________________________
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The Empirical Rule: Gives an approximation to the area under the normal curve
• Approximately _________% of the area under the normal curve is between µ–σ and
µ+σ
• Approximately _________% of the area under the normal curve is between µ–2σ and µ+2σ
• Approximately _________% of the area under the normal curve is between µ–3σ and
µ+3σ
Effect of the mean and standard deviation on the normal distribution
The center and the spread of the normal distribution is described by µ and σ
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Example: Consider the lifespan of a refrigerators are normally distributed with 14µ = and 2.5σ =
a. Draw the normal curve and label the horizontal axis.
b. What would happen to the model if the mean were larger?
c. Shade the proportion greater than 17 years.
d. Suppose the area to the right of 17 is 0.1151. Provide two interpretations:
i. The probability that a randomly selected refrigerator will last more than ___________ years is ______________. ii. The proportion of refrigerators that last more than ________ years is _____________.
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Example: Elena conducts an experiment in which she fills up the gas tank on her Toyota 40 times and records the miles per gallon for each fill-up. She determines the mean mph of her car is 24.6 miles per gallon and a standard deviation of 3.2 miles per gallon.
a. Draw the normal curve and label the horizontal axis.
b. Shade the portion between 18 and 21.
c. Provide two interpretation if the area between 18 and 21 is 0.1107. i. The probability that a randomly selected tank of gas will last between _____________ and ___________ is ______________. ii. The proportion of times the gas will that last between ___________ and _____________ is _______________.
Section 7.2— Applications of the Normal Distribution
Standardizing a Normal Random Variable Computing a z-score for any normal random variable X (on any scale) puts it on a scale where μ = 0 and σ = 1. This is very useful for comparing values from different scales (i.e., ACT and SAT scores or comparing genders).
Z = X − μσ
is normally distributed with mean µ = ________ and standard deviation σ = ________.
The random variable Z is said to have the __________________________________________ distribution.
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Example: Roberto finishes a triathlon in 63.2 minutes. Among all men in the race, the mean finishing time was 69.4 minutes with a standard deviation of 8.9 minutes. Zandra finishes the same triathlon in 79.3 minutes. Among all woman in the race, the mean finishing time was 84.7 minutes with a standard deviation of 7.4 minutes. Who did better in relation to their gender?
The consequences of this is powerful. If a normal random variable X has a mean different from 0 or a standard deviation different from 1, we can transform X into a standard normal random variable Z whose mean is 0 and standard deviation of 1. IMPORTANT CONSEQUENCE: Areas under the curve corresponding to a data value, X, are the same as the area under the curve corresponding to a standardized score, Z, computed from X. 2 Methods for computing areas under the normal distribution
1. Convert the normal random variable X to a standardized normal random variable Z and use table 5.
2. Compute areas using the TI-graphing calculator (this will be our method)
Notice that the only difference between the two graphs is the scale. Standardizing by computing z-scores slides the graph so that its mean becomes 0 and stretches
or compresses it so that its standard deviation becomes 1.
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Example: A soda can dispenser has a mean fill volume of 12 ounces with a standard deviation of 0.12 ounces. What is the proportion of cans that have fewer than 11.9 ounces?
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Example: Determine the area under the standard normal curve for each of the following using the TI-Calculator: a. left of 1.52Z = − b. right of 1.52Z = − c. right of 1.98Z = d. between 0.5Z = − and 1.2Z =
Example: The number of chocolate chips in an 18-ounce bag of Chips Ahoy chocolate chip
cookies is approximately normally distributed with a mean of 1262 chips with a standard deviation of 118 chips.
a. What is the probability that a randomly selected 18-ounce bag contains between 1000 and 1400 chocolate chips inclusive?
b. What is the probability that a randomly selected bag contains fewer than 1000 chips?
c. What proportion of 18-ounce bags contains more than 1200 chips?
d. Would it be unusual to select a bag that contains fewer than 900 chips? Why?
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Example: The height of American adult women is distributed almost exactly as a normal distribution. The mean height of adult American women is 63.5 inches with a standard deviation of 2.5 inches
a. What is the probability that a randomly selected woman is 70 inches or taller?
b. What is the probability that a randomly selected woman is shorter than 62 inches? c. Would it be unusual to select a woman at random that is less than 62 inches? Why?
The notation zα is defined to be the z-score such that the area under the standardized normal curve to the right of zα is α . Given the probability we use the invNorm function on your calculator to find zα Example: Find the value of 0.10z
Example: Find the z-score that separate the top 20% of the distribution from the area.
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Example: The manager of an In-N-Out Burger has determined that the wait times in the drive thru are normally distributed with a mean of 2.3 minutes with a standard deviation of 0.42 minutes. a. What is the probability that a customer’s wait time is greater than 2.9 minutes?
b. What is the probability that a customer’s wait time is less than 2 minutes? c. The shortest 40% of wait times are shorter then what time? d. The longest 25% of the wait times are longer then what time?
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Example: Consider the IQ scores for a certain high school have a μ = 100, and σ = 15
a. A highly selective school only accepts applicants that are above the 75th percentile? Draw the area and find the cut score.
b. What scores correspond to the middle 50%?
c. How many students does a school need to enroll to have 500 with IQs above 120?