Deck 6: The Normal Distribution and Other Continuous Distributions

Question 1

The probability that a standard normal random variable,Z,is less than 50 is approximately 0.

Answer 1

False

Question 2

For some positive value of Z,the value of the cumulative standardised normal distribution is 0.8340.The value of Z is

A) 0.97.
B) 0.07.
C) 0.37.
D) 1.06.

Answer 2

A

Question 3

The normal distribution is related to the Central Limit Theorem.

Answer 3

True

Question 4

The value of the cumulative standardised normal distribution at Z is 0.6255.The value of Z is

A) 0.40.
B) 0.16.
C) 0.32.
D) 0.99.

Question 5

The probability that a standard normal random variable,Z,is between 1.50 and 2.10 is the same as the probability Z is between −2.10 and −1.50.

Question 6

Which of the following can be used to model the distribution of the values for a continuous random variable?

A) Exponential distribution.
B) Uniform distribution.
C) Normal distribution.
D) All of the above.

Question 7

The value of the cumulative standardised normal distribution at 1.5X is 0.9332.The value of X is

A) 0.10.
B) 1.00.
C) 1.50.
D) 0.50.

Question 8

Theoretically,the mean,median and mode are all equal for a normal distribution.

Question 9

The 'middle spread',that is the middle 50% of the normal distribution,is equal to one standard deviation.

Question 10

The probability that a standard normal random variable,Z,falls between −1.50 and 0.81 is 0.7242.

Question 11

The value of the cumulative standardised normal distribution at Z is 0.8770.The value of Z is

A) 0.18.
B) 1.16.
C) 0.81.
D) 1.47.

Question 12

If a particular batch of data is approximately normally distributed,we would find that approximately

A) 19 of every 20 observations would fall between ± 2 standard deviations around the mean.
B) 4 of every 5 observations would fall between ± 1.28 standard deviations around the mean.
C) 2 of every 3 observations would fall between ± 1 standard deviation around the mean.
D) All the above.

Question 13

The probability that a standard normal random variable,Z,is between 1.00 and 3.00 is 0.1574.

Question 14

For some value of Z,the probability that a standard normal variable is below Z is 0.2090.The value of Z is

A) −0.81.
B) −0.31.
C) 1.96.
D) 0.31.

Question 15

Which of the following about the normal distribution is NOT true?

A) About 2/3 of the observations fall within ± 1 standard deviation from the mean.
B) It is a discrete probability distribution.
C) Its parameters are the mean, μ, and standard deviation, σ.
D) Theoretically, the mean, median and mode are the same.

Question 16

In its standardised form,the normal distribution

A) has an area equal to 0.5.
B) has a mean of 1 and a variance of 0.
C) has a mean of 0 and a standard deviation of 1.
D) cannot be used to approximate discrete probability distributions.

Question 17

Any set of normally distributed data can be transformed to its standardised form.

Question 18

The probability that a standard normal random variable,Z,falls between −2.00 and −0.44 is 0.6472.

Question 19

A worker earns $15 per hour at a company and is told that only 2.5% of all workers make a higher wage.If the wage is assumed to be normally distributed and the standard deviation of wage rates is $5 per hour,the average wage for the company is $7.50 per hour.

Question 20

The probability that a standard normal random variable,Z,is below 1.96 is 0.4750.

Question 21

If we know that the length of time it takes a university student to find a parking space in the university car park follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute,find the probability that a randomly selected university student will find a parking space in the car park in less than 3 minutes.

A) 0.3085
B) 0.3551
C) 0.2674
D) 0.1915

Question 22

If we know that the length of time it takes a university student to find a parking space in the university car park follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute,find the probability that a randomly selected university student will take between 2 and 4.5 minutes to find a parking space in the car park.

A) 0.4938
B) 0.0919
C) 0.2255
D) 0.7745

Question 23

The probability that a particular brand of smoke alarm will function properly and sound an alarm in the presence of smoke is 0.8.A batch of 100,000 such alarms was produced by independent production lines.Which of the following distributions would you use to figure out the probability that at least 90,000 of them will function properly in case of a fire?

A) Binomial distribution.
B) Poisson distribution.
C) Normal distribution.
D) None of the above.

Question 24

The owner of a fish market determined that the average weight for a rainbow trout is 3.2 kilograms with a standard deviation of 0.8 kilogram.Assuming the weights of rainbow trout are normally distributed,the probability that a randomly selected rainbow trout will weigh more than 4.4 kilograms is_________.

Question 25

Instruction 6.1
The number of column centimetres of classified advertisements appearing on Mondays in a certain daily newspaper is normally distributed with population mean 320 and population standard deviation 20 cm.
Referring to Instruction 6.1,for a randomly chosen Monday,what is the probability there will be less than 340 column centimetres of classified advertisement?

Question 26

Scientists in Indonesia are trying to find a cure for a deadly disease that is attacking the rain forests on Borneo and Sumatra.One of the variables that the scientists have been measuring involves the diameter of the trunk of the trees that have been affected by the disease.Scientists have calculated that the average diameter of the diseased trees is 42 centimetres.They also know that approximately 95% of the diameters fall between 32 and 52 centimetres and almost all of the diseased trees have diameters between 27 and 57 centimetres.When modelling the diameters of diseased trees,which distribution should the scientists use?

A) Exponential distribution.
B) Normal distribution.
C) Uniform distribution.
D) Binomial distribution.

Question 27

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-The probability that Z is more than 0.77 is _________.

Question 28

If we know that the length of time it takes a university student to find a parking space in the university car park follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute,find the point in the distribution in which 75.8% of the university students exceed when trying to find a parking space in the car park.

A) 4.2 minutes.
B) 3.2 minutes.
C) 3.4 minutes.
D) 2.8 minutes.

Question 29

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-The probability that Z is less than 1.15 is _________.

Question 30

The weight of a randomly selected cookie from a production line can most likely be modelled by which of the following distributions?

A) Binomial distribution.
B) Poisson distribution.
C) Normal distribution.
D) None of the above.

Question 31

The owner of a fish market determined that the average weight for a rainbow trout is 3.2 kilograms with a standard deviation of 0.8 kilogram.Assuming the weights of rainbow trout are normally distributed,the probability that a randomly selected rainbow trout will weigh between 3 and 5 kilograms is _________

Question 32

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-The probability that Z is less than -2.20 is _________.

Question 33

Instruction 6.1
The number of column centimetres of classified advertisements appearing on Mondays in a certain daily newspaper is normally distributed with population mean 320 and population standard deviation 20 cm.
Referring to Instruction 6.1,a single Monday is chosen at random.State in which of the following ranges the number of column centimetres of classified advertisement is most likely to be

A) 310-330.
B) 330-350.
C) 320-340.
D) 300-320.

Question 34

Instruction 6.1
The number of column centimetres of classified advertisements appearing on Mondays in a certain daily newspaper is normally distributed with population mean 320 and population standard deviation 20 cm.
Referring to Instruction 6.1,for a randomly chosen Monday the probability is 0.1 that there will be less than how many column centimetres of classified advertisements?

Question 35

The owner of a fish market determined that the average weight for a rainbow trout is 3.2 kilograms with a standard deviation of 0.8 kilogram.A citation rainbow trout should be one of the top 2% in weight.Assuming the weights of rainbow trout are normally distributed,at what weight (in kilograms)should the citation designation be established?

A) 4.84 kilograms.
B) 5.20 kilograms.
C) 7.36 kilograms.
D) 1.56 kilograms.

Question 36

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-The probability that Z is more than -0.98 is _________

Question 37

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-The probability that Z is between -2.33 and 2.33 is _________.

Question 38

The probability that a standard normal variable Z is positive is_________.

Question 39

Suppose that past history shows that 60% of university students prefer Pepsi .A sample of 10,000 students is to be selected.Which of the following distributions would you use to figure out the probability that at least half of them will prefer Pepsi ?

A) Binomial distribution.
B) Poisson distribution.
C) Normal distribution.
D) None of the above.

Question 40

The owner of a fish market determined that the average weight for a rainbow trout is 3.2 kilograms with a standard deviation of 0.8 kilogram.Assuming the weights of rainbow trout are normally distributed,the probability that a randomly selected rainbow trout will weigh less than 2.2 kilograms is _________

Question 41

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.The probability is _________that a product is assembled in between 14 and 16 minutes.

Question 42

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-So 96% of the possible Z values are between _________and _________ (symmetrically distributed about the mean).

Question 43

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

- The probability is _________that a product is assembled in between 15 and 21 minutes.

Question 44

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-The probability is _________that a product is assembled in more than 19 minutes.

Question 45

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-The probability that Z is between -0.88 and 2.29 is _________.

Question 46

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-So,90% of the products require more than _________minutes for assembly.

Question 47

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-So,15% of the products require more than_________minutes for assembly.

Question 48

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-The probability that Z values are larger than _________is 0.3483.

Question 49

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-The probability that Z values are larger than _________is 0.6985.

Question 50

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-So,60% of the products would be assembled within _________ and _________ minutes (symmetrically distributed about the mean).

Question 51

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-The probability that Z is between -2.89 and -1.03 is _________.

Question 52

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-So,17% of the products would be assembled within _________ minutes.

Question 53

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-So 27% of the possible Z values are smaller than _________.

Question 54

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-The probability is _________ that a product is assembled in between 10 and 12 minutes.

Question 55

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-So 50% of the possible Z values are between _________and _________ (symmetrically distributed about the mean).

Question 56

Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1.

-So 85% of the possible Z values are smaller than _________.

Question 57

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-The probability is _________that a product is assembled in less than 20 minutes.

Question 58

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.The probability is _________ that a product is assembled in less than 12 minutes.

Question 59

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-The probability is _________that a product is assembled in between 16 and 21 minutes.

Question 60

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-The probability is _________ that a product is assembled in more than 11 minutes.

Question 61

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes.

-So,70% of the products would be assembled within _________ minutes.

Question 62

The amount of pyridoxine (in grams)in a multiple vitamin is normally distributed with $\mu$ = 110 grams and $\sigma$ 25 grams.

-What is the probability that a randomly selected vitamin will contain less than 100 grams or more than 120 grams of pyridoxine?

Question 63

Given that X is a normally distributed random variable with a mean of 50 and a standard deviation of 2,find the probability that X is between 47 and 54.

Question 64

A food processor packages green olives in small jars.The weights of the filled jars are approximately normally distributed with a mean of 336 grams and a standard deviation of 9.6 grams.

-Find the proportion of all jars packaged by this process that have weights that fall below 348 grams.

Question 65

The true length of boards cut at a mill with a listed length of 3 metres is normally distributed with a mean of 303 cm and a standard deviation of 1 cm

-What proportion of the boards will be over 305 cm in length?

Question 66

A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan.Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years.

-What proportion of the plan recipients would receive payments beyond age 75?

Question 67

The true length of boards cut at a mill with a listed length of 3 metres is normally distributed with a mean of 303 cm and a standard deviation of 1 cm.

-What proportion of the boards will be less than 304 cm?

Question 68

The amount of pyridoxine (in grams)in a multiple vitamin is normally distributed with $\mu$ = 110 grams and $\sigma$ 25 grams.

-What is the probability that a randomly selected vitamin will contain between 82 and 100 grams of pyridoxine?

Question 69

Instruction 6.2
John has two jobs. For daytime work at a jewellery store he is paid $15,000 per month, plus a commission. His monthly commission is normally distributed with mean $10,000 and standard deviation $2,000. At night he works as a waiter, for which his monthly income is normally distributed with mean $1,000 and standard deviation $300. John's income levels from these two sources are independent of each other.
Referring to Instruction 6.2,for a given month,what is the probability that John's commission from the jewellery store is at least $12,000?

Question 70

The amount of pyridoxine (in grams)in a multiple vitamin is normally distributed with $\mu$ = 110 grams and $\sigma$ 25 grams.

-What is the probability that a randomly selected vitamin will contain less than 100 grams of pyridoxine?

Question 71

The owner of a fish market determined that the average weight for a rainbow trout is 3.2 kilograms with a standard deviation of 0.8 kilogram.Assuming the weights of rainbow trout are normally distributed,above what weight (in kilograms)do 89.80% of the weights occur?

Question 72

The amount of pyridoxine (in grams)in a multiple vitamin is normally distributed with $\mu$ = 110 grams and $\sigma$ 25 grams.

-What is the probability that a randomly selected vitamin will contain between 100 and 110 grams of pyridoxine?

Question 73

A food processor packages green olives in small jars.The weights of the filled jars are approximately normally distributed with a mean of 336 grams and a standard deviation of 9.6 grams

-Find the proportion of all jars packaged by this process that have weights that fall above 350.4 grams.

Question 74

A company that sells annuities must base the annual payout on the probability distribution of the length of life of the participants in the plan.Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years.

-Find the age at which payments have ceased for approximately 86% of the plan participants.

Question 75

The amount of pyridoxine (in grams)in a multiple vitamin is normally distributed with $\mu$ = 110 grams and $\sigma$ 25 grams.

-What is the probability that a randomly selected vitamin will contain at least 100 grams of pyridoxine?

Question 76

Instruction 6.2
John has two jobs. For daytime work at a jewellery store he is paid $15,000 per month, plus a commission. His monthly commission is normally distributed with mean $10,000 and standard deviation $2,000. At night he works as a waiter, for which his monthly income is normally distributed with mean $1,000 and standard deviation $300. John's income levels from these two sources are independent of each other.
Referring to Instruction 6.2,for a given month,what is the probability that John's commission from the jewellery store is between $11,000 and $12,000?

Question 77

Instruction 6.2
John has two jobs. For daytime work at a jewellery store he is paid $15,000 per month, plus a commission. His monthly commission is normally distributed with mean $10,000 and standard deviation $2,000. At night he works as a waiter, for which his monthly income is normally distributed with mean $1,000 and standard deviation $300. John's income levels from these two sources are independent of each other.
Referring to Instruction 6.2,for a given month,what is the probability that John's commission from the jewellery store is less than $13,000?

Question 78

Instruction 6.2
John has two jobs. For daytime work at a jewellery store he is paid $15,000 per month, plus a commission. His monthly commission is normally distributed with mean $10,000 and standard deviation $2,000. At night he works as a waiter, for which his monthly income is normally distributed with mean $1,000 and standard deviation $300. John's income levels from these two sources are independent of each other.
Referring to Instruction 6.2,for a given month,what is the probability that John's commission from the jewellery store is no more than $8,000?

Question 79

Instruction 6.2
John has two jobs. For daytime work at a jewellery store he is paid $15,000 per month, plus a commission. His monthly commission is normally distributed with mean $10,000 and standard deviation $2,000. At night he works as a waiter, for which his monthly income is normally distributed with mean $1,000 and standard deviation $300. John's income levels from these two sources are independent of each other.
Referring to Instruction 6.2,for a given month,what is the probability that John's commission from the jewellery store is more than $9,500?

Question 80

The amount of pyridoxine (in grams)in a multiple vitamin is normally distributed with $\mu$ = 110 grams and $\sigma$ 25 grams.

-What is the probability that a randomly selected vitamin will contain between 100 and 120 grams of pyridoxine?