Deck 5: The Standard Deviation As a Ruler and the Normal Model

Question 1

A vendor offers various t-shirts for sale on an Internet commerce site.Some of his t-shirts sell quite well, others have yet to find an interested customer.The summary $$\begin{array} { | l | c | } \hline \bar { x } & 23 \\\hline \text { Median } & 15 \\\hline \text { Min } & 0 \\\hline \text { Q1 } & 3 \\\hline \text { Q3 } & 21 \\\hline \text { Max } & 82 \\\hline \text { sd } & 7.84 \\\hline\end{array}$$
statistics are listed on right, based on the number of sales
per month for each of 80 different t-shirt designs.
A.What is the shape of this distribution? Explain how you know.
B.If the vendor spends $2 per t-shirt in shipping costs, what is his average monthly costs
per t-shirt sold? What is the standard deviation of this cost? Show work.
c.As time passes, the vendor has simplified his profit calculation.He estimates that he has
monthly flat costs of $35, with a gain of $3.50 per t-shirt sold.What is the median of his
profit? The IQR?
d.What are the mean and standard deviation of his profit?
e.What is more appropriate in this case: using mean and standard deviation, or using
median and IQR?

Answer 1

a. The distribution is clearly skewed to the right. The mean is much larger than the median. The max is to be an outlier and is likely a part of a long tail of the most popular t-shirts. The min and Q1 are very close together, while the spread increases between the median, Q3, and the max.
b. $\text { mean } = 23 * \$ 2 = \$ 46 ; \quad \mathrm { sd } = 7.84 * \$ 2 = \$ 15.68$

c. $\text { median } = 3.5 * 15 - 35 = \$ 17.50 ; \quad \mathrm { IQR } = 3.5 * ( 21 - 3 ) = \$ 63$

d. $\text { mean } = 3.5 ^ { * } 23 - 35 = \$ 45.50 ; \mathrm { sd } = 3.5 ^ { * } 7.84 = \$ 27.44$

e.
The distribution is strongly right skewed, thus mean and standard deviation are not appropriate.

Question 2

The distribution of the diameters of a particular variety of grapes is approximately normal With a standard deviation of 0.3 cm.How does the diameter of a grape at the 67th percentile Compare with the mean diameter?

A)0.201 cm below the mean
B)0.132 cm below the mean
C)0.132 cm above the mean
D)0.201 cm above the mean

Answer 2

0.132 cm above the mean

Question 3

Adult female Dalmatians weigh an average of 50 pounds with a standard deviation of 3.3
pounds.Adult female Boxers weigh an average of 57.5 pounds with a standard deviation of 1.7 pounds.One statistics teacher owns an underweight Dalmatian and an underweight
Boxer.The Dalmatian weighs 45 pounds, and the Boxer weighs 52 pounds.Which dog is
more underweight? Explain.

Answer 3

Dalmatian: $\mathrm { z } _ { \mathrm { D } } = ( 45 - 50 ) / 3.3 \equiv - 1.52$
Boxer: $\mathrm { Z } _ { \mathrm { B } } = ( 52 - 57.5 ) / 1.7 \cong - 3.24$
The Dalmatian is 1.52 standard deviations underweight, while the Boxer is 3.24 standard deviations underweight.So, the Boxer is more underweight.

Question 4

Owners of a minor league baseball team believe that a Normal model is useful in projecting
the number of fans who will attend home games.They use a mean of 8500 fans and a
standard deviation of 1500 fans.Draw and clearly label this model.

Question 5

A soft drink dispenser can be adjusted to deliver any fixed number of centiliters.If the Machine is operating with a standard deviation in delivery equal to 0.3 centiliter, what should Be the mean setting so that a 12-centilitre cup will overflow less than 1% of the time?
Assume a normal distribution for centiliters delivered.

A)11.23
B)11.30
C)11.70
D)12.00
E)12.70

Question 6

Question 7

At a local manufacturing plant, employees must complete new machine set ups within 30 Minutes.New machine set-up times can be described by a normal model with a mean of 22 Minutes and a standard deviation of four minutes.The typical worker needs five minutes to Adjust to his or her surroundings before beginning duties.What percent of new machine set Ups are completed within 25 minutes to allow for this?

A)77.3%
B)27.3%
C)22.7%
D)72.7%
E)none of the above

Question 8

Suppose that the weights of vans delivering goods to inner city shops are normally Distributed.If 70% of the vans weigh more than 1200 kg, and 80% weigh more than 1000 kg, What are the mean and standard deviation for the weights of vans serving the inner city
(round to tens)?

A) $\mu = 1510 \mathrm {~kg} ; \sigma = 620 \mathrm {~kg}$
B) $\mu = 1530 \mathrm {~kg} ; \sigma = 630 \mathrm {~kg}$ ..
C) $\mu = 1550 \mathrm {~kg} ; \sigma = 640 \mathrm {~kg}$
D) $\mu = 1530 \mathrm {~kg} ; \sigma = 630 \mathrm {~kg}$ .
E) There is not enough information given to determine the mean and standard deviation.

Question 9

Light bulbs are measured in lumens (light output), watts (energy used), and hours (life).Astandard white light bulb has a mean life of 675 hours and a standard deviation of 50 hours.A soft white light bulb has a mean life of 700 hours and a standard deviation of 35 hours.In atest at a local science competition, both light bulbs lasted 750 hours.Which light bulb's life
span was better? Explain.

Question 10

At a local manufacturing plant, employees must complete new machine set ups within 30 Minutes.New machine set-up times can be described by a normal model with a mean of 22 Minutes and a standard deviation of four minutes.What percent of new machine set ups take More than 30 minutes?

A)97.72%
B)47.72%
C)2.28%
D)52.28%
E)none of the above

Question 11

Based on data collected from its production processes, Crosstiles InC.determines that the Breaking strength of its most popular porcelain tile is normally distributed with a mean of 400 pounds per square inch and a standard deviation of 12.5 pounds per square inch.Based On the 68-95-99.7 Rule, about what percent of its popular porcelain tile will have breaking Strengths greater than 412.5 pounds per square inch?

A)95%
B)68%
C)16%
D)32%
E)47.5%

Question 12

Although most of us buy milk by the quart or gallon, farmers measure daily production in
pounds.Guernsey cows average 39 pounds of milk a day with a standard deviation of 8
pounds.For Jerseys the mean daily production is 43 pounds with a standard deviation of 5
pounds.When being shown at a state fair a champion Guernsey and a champion Jersey each
gave 54 pounds of milk.Which cow's milk production was more remarkable? Explain.

Question 13

During a budget meeting, local school board members decided to review class size information to determine if budgets were correct. Summary statistics are shown in the table
$$\begin{array} { | l | c | } \hline \bar { x } & 33.39 \text { students } \\\hline s & 5.66 \text { students } \\\hline \text { min } & 17 \\\hline \text { Q1 } & 29 \\\hline \text { median } & 33 \\\hline \text { Q3 } & 40 \\\hline \text { max } & 40 \\\hline\end{array}$$
a.Notice that the third quartile and maximum class sizes are the same. Explain how this can be.
b.The school district declares that classes with enrollments fewer than 20 students are "too small." Would you consider a class of 20 students to be unusually small? Explain.
c.The school district sets the office supply budgets of their high schools on the enrollment of students.The district budgets each class $12 plus $0.75 per student, so a class with one student receives $12.75 and the classes with 40 students receive 12 + 0.75(40) = $42.
What is the median class budget for office supplies? And the IQR?
d.What are the mean and standard deviation of the class office supply budgets?

Question 14

Both SAT and ACT are well-known placement tests that most US colleges require from Prospective students to be admitted in their programs.Scores in the SAT test are Approximately normally distributed with a mean of 500 and a standard deviation of 100.
Scores in the ACT test are approximately normally distributed with a mean of 18 and a Standard deviation of 6.What would be the score in the SAT test to get the same z-score as The admission requirement of an ACT score of 27?

A)500
B)550
C)600
D)650
E)700

Question 15

Students taking an intro stats class reported the number of credit hours
that they were taking that quarter.Summary statistics are shown in the
table.
$$\begin{array} { | l | c | } \hline \bar { x } & 16.65 \\\hline s & 2.96 \\\hline \min & 5 \\\hline \text { Q1 } & 15 \\\hline \text { median } & 16 \\\hline \text { Q3 } & 19 \\\hline \max & 28 \\\hline\end{array}$$
A.Suppose that the college charges $73 per credit hour plus a flat
student fee of $35 per quarter.For example, a student taking 12
credit hours would pay $35 + $73(12) = $911 for that quarter.
i.What is the mean fee paid? ii.What is the standard deviation for the
fees paid?
iii.What is the median fee paid? iv.What is the IQR for the fees paid?
B.Twenty-eight credit hours seems like a lot.Would you consider 28 credit hours to be
unusually high? Explain.

Question 16

Adult female Dalmatians weigh an average of 50 pounds with a standard deviation of 3.3 Pounds.Adult female Boxers weigh an average of 57.5 pounds with a standard deviation of 1.7 pounds.One statistics teacher owns an underweight Dalmatian and an underweight oxer.The Dalmatian weighs 45 pounds, and the Boxer weighs 52 pounds.Which dog is More underweight?

A)The Boxer because it is 9.6% underweight.
B)The Dalmatian because it is 9.6% underweight.
C)The Boxer because it is 5.5 pounds underweight.
D)The Dalmatian because it is 5 pounds underweight.
E)The Boxer because it is 3.24 standard deviations underweight.

Question 17

Based on data collected from its production processes, Crosstiles InC.determines that the Breaking strength of its most popular porcelain tile is normally distributed with a mean of 400 pounds per square inch and a standard deviation of 12.5 pounds per square inch.Based On the 68-95-99.7 Rule, about what percent of its popular porcelain tile will have breaking Strengths between 375 and 425 pounds per square inch?

A)95%
B)68%
C)84%
D)32%
E)47.5%

Question 18

Scores on the Wechsler Adult Intelligence Scale - Revised (WAIS-R) follow a Normal
model with mean 100 and standard deviation 15.Draw and clearly label this model.

Question 19

The Postmaster of a city's Post Office believes that a Normal model is useful in projecting
the number of letters which will be mailed during the day.They use a mean of 20,000 letters
and a standard deviation of 250 letters.Draw and clearly label this model.

Question 20

The ASQ (American Society for Quality) regularly conducts a salary survey of its Membership, primarily quality management professionals.Based on the most recently Published mean and standard deviation, a quality control specialist calculated the z-score Associated with his own salary and found it was -2.50.This tells him that his salary is

A)2 and a half times more than the average salary.
B)2 and a half times less than the average salary.
C)is 2.5 standard deviations above the average salary.
D)is 2.5 standard deviations below the average salary.
E)much higher than the average salary.

Question 21

The time it takes to process phone orders in a small florist/gift shop is normally distributed With a mean of 6 minutes and a standard deviation of 1.24 minutes.What cutoff values Would separate the 16% of orders that take the least time to process?

A)3.52 minutes
B)4.76 minutes
C)8.48 minutes
D)10.01 minutes
E)11.98 minutes

Question 22

Suppose that a Normal model describes fuel economy (miles per gallon) for automobiles and That a Toyota Corolla has a standardized score (z-score) of +2.2.This means that Corollas…

A)get 2.2 miles per gallon.
B)have a standard deviation of 2.2 mpg.
C)get 2.2 mpg more than the average car.
D)get 2.2 times the gas mileage of the average car.
E)achieve fuel economy that is 2.2 standard deviations better than the average car.

Question 23

A small flower shop takes orders by phone and then one of the staff florists is assigned to Prepare the arrangement.The time it takes to process phone orders is normally distributed With a mean of 6 minutes and a standard deviation of 2.5 minutes.The time it takes for an Arrangement to be completed is normally distributed with a mean of 35 minutes and a Standard deviation of 8.6 minutes.What is the standard deviation for the total time to process a phone order and complete the floral arrangement at this flower shop (assuming times are Independent)?

A)8.96 minutes
B)41 minutes
C)80.28 minutes2
D)4.87 minutes
E)12.43 minutes

Question 24

A company's manufacturing process uses 500 gallons of water at a time.A "scrubbing" Machine then removes most of a chemical pollutant before pumping the water into a nearby Lake .To meet federal regulations the treated water must not contain more than 80 parts per Million (ppm.) of the chemical.The machine's output can be described by a normal model With standard deviation 4.2 ppm.The company's lawyers insist that not more than 2% of the Treated water should be over the limit.To achieve this, to what mean should the company set The scrubbing machine?

A)80 ppm.
B)75 ppm.
C)71.374 ppm.
D)88.626 ppm.
E)69.459 ppm.

Question 25

Light bulbs are measured in lumens (light output), watts (energy used), and hours (life).A Standard white light bulb has a mean life of 675 hours and a standard deviation of 50 hours.
A soft white light bulb has a mean life of 700 hours and a standard deviation of 35 hours.At
A local science competition, both light bulbs lasted 750 hours.Which light bulb's life span
Was better?

A)The standard white light bulb because it is 75 hours beyond the mean.
B)The soft white light bulb because it is 50 hours beyond the mean.
C)The standard white light bulb because it is 11% beyond the mean.
D)The soft white light bulb because it is 7% beyond the mean.
E)The standard light bulb because it lasted 1.5 standard deviations above the mean life.

Question 26

The time it takes to process phone orders in a small florist/gift shop is normally distributed With a mean of 6 minutes and a standard deviation of 1.24 minutes.What cutoff value would Separate the 2.5% of orders that take the most time to process?

A)3.52 minutes
B)4.76 minutes
C)8.48 minutes
D)10.01 minutes
E)11.98 minutes

Question 27

Based on the Normal model for yearly snowfall in cm in a certain town N(57, 8), how many Cm's of snow would represent the 80th percentile approximately?

A)64.5 cm.
B)63.7 cm.
C)63.3 cm.
D)62.7 cm.
E)61.7 cm.

Question 28

A small flower shop takes orders by phone and then one of the staff florists is assigned to Prepare the arrangement.The time it takes to process phone orders is normally distributed With a mean of 6 minutes and a standard deviation of 2.5 minutes.The time it takes for an Arrangement to be completed is normally distributed with a mean of 35 minutes and a Standard deviation of 8.6 minutes.What is the probability that it will take more than 50 Minutes to process a phone order and complete the floral arrangement at this flower shop?

A)0.8413
B)0.3413
C)0.2167
D)0.1587
E)0.6843

Question 29

A company's manufacturing process uses 500 gallons of water at a time.A "scrubbing" Machine then removes most of a chemical pollutant before pumping the water into a nearby Lake .To meet federal regulations the treated water must not contain more than 80 parts per Million (ppm) of the chemical.Because a fine is charged if regulations are not met, the Company sets the machine to attain an average of 75 ppm in the treated water.The Machine's output can be described by a normal model with standard deviation 4.2 ppm.What Percent of the batches of water discharged exceed the 80 ppm standard?

A)11.7%
B)1.17%
C)88.3%
D)8.83%