# Necessary assumptions of the binomial distribution

• Which one of the following statement(s) is(are) not among the necessary assumptions of the binomial distribution?

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a

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Trials are independent of each other

b

The experiment consists of n identical trials

c

The probability of success remains constant from trial to trial

d

Sampling is with replacement

e

Each trial results in one of two mutually exclusive outcomes

• In a binomial distribution the random variable X is the result of counting.

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a

True

b

False

• The binomial experiment consists of n independent, identical trials, each of which results in either success or failure and the probability of success for different trials are different.

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a

True

b

False

he variance of the binomial distribution is np(1-p)

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a

True

b

False

• If the probability of success is 0.5 and the number of trials in a binomial distribution is 100, then its standard deviation is 25

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a

True

b

False

• The probability that a person catches a cold during the cold and flu season is 0.3. If 10 people are chosen at random, the sample variance for the number of persons catching cold is 1.45.

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a

True

b

False

• If a fair coin is tossed 20 times then the likelihood of exactly 10 Tails is more than 17 percent

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a

True

b

False

• A door-to-door salesperson for a Household appliance has learned from her past experience that out of twenty-four demonstrations of her appliance only six demonstrations result in actual sales (long run average). This week she needs to make at least 5 sales. At least how many demonstrations she needs to perform to ensure that the probability of meeting her target is at least 95 percent

• In a study conducted by a college, it was found that 25% of college freshmen support increased military spending. If 6 college freshmen are randomly selected, find the probability that more than 3 support increased military spending

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a

0.1318

b

0.0376

c

0.8682

d

0.9624

9)  Suppose in the most recent election, only 20% of all eligible college students voted. If a random sample of 20 eligible students were interviewed find the probability that less than five students in the sample voted in the election.

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a

0.1746

b

0.2182

c

0.6296

d

0.3704

e

0.4439

10) A fair die is rolled 10 times. What is the probability that an even number (2, 4 or 6) will occur 4 times or more?

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a

0.1719

b

0.1172

c

0.6230

d

0.9453

e

0.8281

11)  Suppose in the most recent election, only 20% of all eligible college students voted. If a random sample of 20 eligible students were interviewed find the probability that less than five students in the sample voted in the election.

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a

0.1746

b

0.2182

c

0.6296

d

0.3704

e

0.4439

12) Suppose a medical research study about prevalence of a disease defines X = 1 if test is positive and X = 0 if it is negative, then the variable in not continuous.

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a

True

b

False

13) We have P(X ≤ 50) = P(X < 50) for a continuous random variable X

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a

True

b

False

14) All continuous random variables are Normally distributed according to the Central Limit Theorem.

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a

True

b

False

15) The mean and variance are equal in the standard normal distribution

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a

True

b

False

16) If Z denotes the Standard Normal variable, P(1 ≤ Z ≤ 2) is _ P(2 ≤ Z ≤ 3)

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a

less than

b

greater than

c

the same as

d

A or B or C depending on the standard deviation

e

A or B or C depending on the Mean and the standard deviation

17) The mean and standard deviation for a random variable X are 40 and 10, respectively. The mean and variance of a new variable (X- 50)/10 will be

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a

50 and 10

b

1 and 0

c

0 and 1

d

0 and 100

e

50 and 100

18) A high blood pressure patient keeps record of blood pressure levels in the morning and is particularly concerned about his/her diastolic blood pressure levels. Suppose the morning diastolic blood pressure for this patient varies in a normal distribution pattern with Mean equal to 85 and Variance 16. What is the probability that, in a randomly selected morning, the diastolic pressure was between 80 and 90?

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a

0.2434

b

0.6217

c

0.8944

d

0.1056

e

0.7888

19) The weight of a consumer item is a normally distributed random variable with mean 910 lb. and variance 25. If one item is randomly selected from this population, the probability that the weight of the selected item will exceed 904 lb. is

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a

0.3849

b

0.8849

c

0.2051

d

0.1151

e

0.7949

20) Assume that the length an athlete throws a heavy metal object is a normal random variable with mean 60 feet and standard deviation 3 feet, what is the probability he throws it between 55.5 feet and 64.5 feet?

21) If x is a binomial random variable where n = 100 and p = 0.2, find the probability that x is less than or equal to 15 using the normal approximation to the binomial.

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