Chapter 7 Continuous Random Variable - Normal Distributions
QMS102-Business Statistics I
Chapter7
Chapter 7 Continuous Random Variable - Normal Distributions Outcomes: • Reveal the characteristics of the Normal Distribution • Recognize the Standard Normal Distribution • Calculate the normal probabilities Characteristic of Normal Distribution • bell-shaped and has a single peak. •
is symmetric about the mean.
•
is asymptotic, that is, the curve gets closer and closer to the x-axis but never actually touches it.
•
the total area under the curve is 1
•
is denoted by N(µ,σ)
The Standard Normal Distribution • with a mean of 0 and a standard deviation of 1, N(0, 1). •
Z-value or Z-score – the units marked on the horizontal axis of the standard normal curve and are called z values or z scores.
•
the z values on the right side of the mean are positive
•
the z values on the left side of the mean are negative
•
the standard normal distribution table
Fall2012
Page #1
QMS102-Business Statistics I
Chapter7
Standardizing a Normal Distribution, Z • If the random variable X has a normal distribution with mean µ and standard deviation σ, that are different from 0 and 1 respectively. • Convert the given normal distribution to the standard normal distribution • Convert the random variable X to the standard normal variable Z, using the equation x −µ σ Thus, P(X < x) = P(Z < z), where z is called the z-value. z=
•
Example1 Let x be a normal random variable with its mean equal to 40 and standard deviation equal to 5. Determine the following probabilities: a. P ( x ≥ 50 ) b. P ( x < 47 ) c. P ( 42 ≤ x ≤ 49)
Example2 Let x be a normal random variable with its mean equal to 12 and variance equal to 4. Determine the following probabilities: a. P ( x ≥ 12 ) b. P ( x < 12.8) c. P ( 9.24 ≤ x ≤ 11.8)
Fall2012
Page #2
QMS102-Business Statistics I
Chapter7
Example3 a. What is the area under the normal curve between z = –2.34 and z = –1.45? b. What is the area under the normal curve between z = -9.0 and z = 0? c. Determine i. P (−1.45 ≤ z ≤ 2.06)
ii. P( z ≥ −1.47 )
Example4 The monthly credit card bills for households in Toronto are normally distributed, with mean of $1476.30 and the standard deviation of $385.89. a. Define the random variable X b. Determine the probability that a randomly selected bill is i. less than $1600 ii. more than $1200 iii. between $1100 and $1700
Example5 The recent average starting salary for new college graduates in marketing is $38,000. Assume salaries are normally distributed with a standard deviation $3500. a. Define the random variable X b. Determine the probability of a new graduate receiving a salary of i.more than $39,000 ii.less than $45,000 iii.. between $35,000 and $40,000 iv.between $$38,000 and $46,000
Fall2012
Page #3
QMS102-Business Statistics I
Chapter7
Example6 Ryerson Trucking Company determined that on an annual basis the distance traveled per truck is normally distributed with a mean of 54.6 thousand miles and a standard deviation of 11.8 thousand miles. a. What percentage of trucks can be expected to travel either below 32 or above 65 thousand miles in the tear? b. How many miles will be traveled by at least 70% of the truck? c. How many miles will be traveled by at most 85% of the truck?
Fall2012
Page #4
QMS102-Business Statistics I
Chapter7
Example7 An orange juice producer buys all his oranges from a large grove. The amount of juice squeezed from each of these oranges is approximately normally distributed with a mean of 4.86 ounces and a standard deviation of 0.37 ounce. a. 85% of the oranges will contain at least how many ounces of juice? b. 78% of the oranges will contain at most how many ounces of juice? c. 90% of the oranges are between what two values (in ounces) symmetrically distributed around the population mean?
Fall2012
Page #5
QMS102-Business Statistics I
Chapter7
Example8 Matthew, an executive at Ryerson Inc. drives from his home in the suburbs near City A to his office in the center of the city. The driving times are normally distributed with a mean of 37 minutes and a standard deviation of 9 minutes. a. Determine the probability of the days that he will take i. 46 minutes or more to drive to work, ii. between 39 to 54 minutes to drive to work b. Some days there will be accidents or other delays, so the trip will take longer than usual. How long will the longest 6 percent of the trip take?
Fall2012
Page #6
QMS102-Business Statistics I
Chapter7
Example9 A survey indicates that a customer spent an average of 43 minutes with a variance of 100 minutes 2 in the store. The length of time spent in the store is normally distributed. a. What is the probability that the customer will be in the store between 35 minutes and 1 hour?
b. If there are 85 customers shop in the store, what is the probability that more than 12 customers will be in the store for less than half an hour?
c. If there are 100 customers shop in the store, how many would you expect for them to be in the store for more than 65 minutes?
Fall2012
Page #7
QMS102-Business Statistics I
Chapter7
Example10 Records indicate the mean amount spent by each customer at The Great Future Computer store is $150 with standard deviation of $24. Assume the amount spent by each customer at computer store is normally distributed. a. What is the probability that a randomly selected customer will spend less than $120 or more than $200 at The Great Future Computer Store?
b. According to this model, what is the probability of a randomly selected customer will spend within $20 of the expected amount spent at computer store?
c. If there are 80 customers who will go to The Great Future Computer Store this Friday, what is the probability that more than 80% of the customers will spend less than $170 at this store?
d. If there are 120 customers who will shop at The Great Future Computer store, how many of the customers would you expect them to spend more than $200 at this store?
Fall2012
Page #8
Chapter7
Chapter 7 Continuous Random Variable - Normal Distributions Outcomes: • Reveal the characteristics of the Normal Distribution • Recognize the Standard Normal Distribution • Calculate the normal probabilities Characteristic of Normal Distribution • bell-shaped and has a single peak. •
is symmetric about the mean.
•
is asymptotic, that is, the curve gets closer and closer to the x-axis but never actually touches it.
•
the total area under the curve is 1
•
is denoted by N(µ,σ)
The Standard Normal Distribution • with a mean of 0 and a standard deviation of 1, N(0, 1). •
Z-value or Z-score – the units marked on the horizontal axis of the standard normal curve and are called z values or z scores.
•
the z values on the right side of the mean are positive
•
the z values on the left side of the mean are negative
•
the standard normal distribution table
Fall2012
Page #1
QMS102-Business Statistics I
Chapter7
Standardizing a Normal Distribution, Z • If the random variable X has a normal distribution with mean µ and standard deviation σ, that are different from 0 and 1 respectively. • Convert the given normal distribution to the standard normal distribution • Convert the random variable X to the standard normal variable Z, using the equation x −µ σ Thus, P(X < x) = P(Z < z), where z is called the z-value. z=
•
Example1 Let x be a normal random variable with its mean equal to 40 and standard deviation equal to 5. Determine the following probabilities: a. P ( x ≥ 50 ) b. P ( x < 47 ) c. P ( 42 ≤ x ≤ 49)
Example2 Let x be a normal random variable with its mean equal to 12 and variance equal to 4. Determine the following probabilities: a. P ( x ≥ 12 ) b. P ( x < 12.8) c. P ( 9.24 ≤ x ≤ 11.8)
Fall2012
Page #2
QMS102-Business Statistics I
Chapter7
Example3 a. What is the area under the normal curve between z = –2.34 and z = –1.45? b. What is the area under the normal curve between z = -9.0 and z = 0? c. Determine i. P (−1.45 ≤ z ≤ 2.06)
ii. P( z ≥ −1.47 )
Example4 The monthly credit card bills for households in Toronto are normally distributed, with mean of $1476.30 and the standard deviation of $385.89. a. Define the random variable X b. Determine the probability that a randomly selected bill is i. less than $1600 ii. more than $1200 iii. between $1100 and $1700
Example5 The recent average starting salary for new college graduates in marketing is $38,000. Assume salaries are normally distributed with a standard deviation $3500. a. Define the random variable X b. Determine the probability of a new graduate receiving a salary of i.more than $39,000 ii.less than $45,000 iii.. between $35,000 and $40,000 iv.between $$38,000 and $46,000
Fall2012
Page #3
QMS102-Business Statistics I
Chapter7
Example6 Ryerson Trucking Company determined that on an annual basis the distance traveled per truck is normally distributed with a mean of 54.6 thousand miles and a standard deviation of 11.8 thousand miles. a. What percentage of trucks can be expected to travel either below 32 or above 65 thousand miles in the tear? b. How many miles will be traveled by at least 70% of the truck? c. How many miles will be traveled by at most 85% of the truck?
Fall2012
Page #4
QMS102-Business Statistics I
Chapter7
Example7 An orange juice producer buys all his oranges from a large grove. The amount of juice squeezed from each of these oranges is approximately normally distributed with a mean of 4.86 ounces and a standard deviation of 0.37 ounce. a. 85% of the oranges will contain at least how many ounces of juice? b. 78% of the oranges will contain at most how many ounces of juice? c. 90% of the oranges are between what two values (in ounces) symmetrically distributed around the population mean?
Fall2012
Page #5
QMS102-Business Statistics I
Chapter7
Example8 Matthew, an executive at Ryerson Inc. drives from his home in the suburbs near City A to his office in the center of the city. The driving times are normally distributed with a mean of 37 minutes and a standard deviation of 9 minutes. a. Determine the probability of the days that he will take i. 46 minutes or more to drive to work, ii. between 39 to 54 minutes to drive to work b. Some days there will be accidents or other delays, so the trip will take longer than usual. How long will the longest 6 percent of the trip take?
Fall2012
Page #6
QMS102-Business Statistics I
Chapter7
Example9 A survey indicates that a customer spent an average of 43 minutes with a variance of 100 minutes 2 in the store. The length of time spent in the store is normally distributed. a. What is the probability that the customer will be in the store between 35 minutes and 1 hour?
b. If there are 85 customers shop in the store, what is the probability that more than 12 customers will be in the store for less than half an hour?
c. If there are 100 customers shop in the store, how many would you expect for them to be in the store for more than 65 minutes?
Fall2012
Page #7
QMS102-Business Statistics I
Chapter7
Example10 Records indicate the mean amount spent by each customer at The Great Future Computer store is $150 with standard deviation of $24. Assume the amount spent by each customer at computer store is normally distributed. a. What is the probability that a randomly selected customer will spend less than $120 or more than $200 at The Great Future Computer Store?
b. According to this model, what is the probability of a randomly selected customer will spend within $20 of the expected amount spent at computer store?
c. If there are 80 customers who will go to The Great Future Computer Store this Friday, what is the probability that more than 80% of the customers will spend less than $170 at this store?
d. If there are 120 customers who will shop at The Great Future Computer store, how many of the customers would you expect them to spend more than $200 at this store?
Fall2012
Page #8
