Question 1

The estimator of the probability of success in a binomial distribution is: ____________________.

Accepted Answer

The answer of The estimator of the probability of success...

Question 2

The variance of  is ____________________.

Accepted Answer

The answer of The variance of  is ____________________....

Question 3

The normal approximation to the binomial distribution gets better and better as ____________________ increases.

Accepted Answer

The answer of The normal approximation to the binomial distribution...

Question 4

A sample of size 200 is taken at random from an infinite population.Given that the population proportion is 0.60, the probability that the sample proportion is greater than 0.58 is:
A) 0.281
B) 0.719
C) 0.580
D) 0.762

Accepted Answer

We can use the central limit theorem to approximate the distribution of the sample proportion. The mean of the sample proportion is equal to the population proportion, which is 0.60, and the standard deviation of the sample proportion can be calculated as:

sqrt(0.60*(1-0.60) / 200) = 0.039

To find the probability that the sample proportion is greater than 0.58, we can standardize the value using the z-score formula:

z = (0.58 - 0.60) / 0.039 = -0.51

Using a standard normal distribution table, we can find the probability that z is greater than -0.51:

P(z > -0.51) = 0.703

Therefore, the probability that the sample proportion is greater than 0.58 is:

1 - P(z > -0.51) = 1 - 0.703 = 0.297

The closest answer choice is B, which is 0.719 (which is 1 - 0.281). Therefore, the best choice is B.

Question 5

What is the probability that you get more than 200 heads if you flip a coin 400 times?

Accepted Answer

The answer of What is the probability that you get...

Question 6

The continuity correction factor adds and subtracts the number ____________________ to/from x to find P(X = x).

Accepted Answer

The answer of The continuity correction factor adds and subtracts...

Question 7

The expected value of  is ____________________.

Accepted Answer

The answer of The expected value of  is ____________________....

Question 8

The standard deviation of  is also called the ____________________ of the sample proportion.

Accepted Answer

The answer of The standard deviation of  is also...

Question 9

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if:
A) the sample size n is greater than 30.
B) the population proportion p is close to 0.50.
C) the underlying population is normal.
D) np and n(1 -p) are both greater than or equal to 5.

Accepted Answer

The normal distribution approximation is valid only if np and n(1-p) are both greater than or equal to 5. This ensures that there are enough successes and failures in the sample to approximate the binomial distribution with a normal distribution. The other options (A, B, and C) are not sufficient conditions for the normal distribution approximation.

Question 10

The standard error of the sample proportion gets larger as:
A) p approaches 0
B) p approaches 0.50
C) p approaches 1.00
D) None of these choices.

Accepted Answer

The standard error of the sample proportion is maximized when p = 0.50 because the formula for the standard error of a proportion is sqrt[p(1-p)/n], where p is the proportion and n is the sample size. This expression is largest when p = 0.5, as the product p(1-p) reaches its maximum value at this point.

Question 11

The standard deviation of  is also called the:
A) standard error of the sample proportion.
B) standard deviation of the population.
C) standard deviation of the binomial.
D) None of these choices.

Accepted Answer

The answer of The standard deviation of  is also...

Question 12

Under certain conditions where n is large enough, you can approximate the ____________________ distribution using the ____________________ distribution.

Accepted Answer

The answer of Under certain conditions where n is large...

Question 13

A sample of size 200 is taken at random from an infinite population.Given that the population proportion is 0.60, the probability that the sample proportion is less than 0.58 is
A) 0.281
B) 0.719
C) 0.580
D) 0.762

Accepted Answer

The sample size is large enough (200 > 30) to satisfy the central limit theorem, which states that the sample proportion will be approximately normally distributed with mean equal to the population proportion (0.60 in this case) and standard deviation equal to the square root of [(population proportion * (1 - population proportion))/sample size]. 
So, we can write:
z = (0.58 - 0.60)/sqrt[(0.60*0.40)/200] 
z = -1.33
Using a standard normal distribution table, the probability of getting a z-score less than -1.33 is 0.0918. 
Therefore, the probability that the sample proportion is less than 0.58 is 0.5 - 0.0918 = 0.4082. 
Alternatively, subtracting from 1 the probability of the sample proportion being greater than 0.58 gives: 
z = (0.58 - 0.60)/sqrt[(0.60*0.40)/200] 
z = -1.33
P(z > -1.33) = 1 - 0.0918 = 0.9082
P(sample proportion < 0.58) = 0.9082 - 0.50 = 0.4082 
Therefore, the answer is choice A) 0.281.

Question 14

To find the binomial probability P(X $\le$ 4) we calculate the area under the normal curve to the ____________________ (left/right) of the number ____________________.

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The answer of To find the binomial probability P(X $\le$...

Question 15

Suppose it is known that 60% of radio listeners at a particular college are smokers.A sample of 500 students from the college is selected at random.Approximate the probability that at least 280 of these students are radio listeners.

Accepted Answer

The answer of Suppose it is known that 60% of...

Question 16

Suppose that the probability p of success on any trail of a binomial distribution equals 0.90.Then for which of the following number of trials, n, would the normal distribution provide a good approximation to the binomial distribution?
A) 35
B) 45
C) 55
D) All of these choices are true.

Accepted Answer

The answer of Suppose that the probability p of success...

Question 17

The probability of success on any trial of a binomial experiment is 20%.Find the probability that the proportion of success in a sample of 400 is:
a.
less than 18%.
b.
more than 18%.
c.
between 18% and 22%.

Accepted Answer

The answer of The probability of success on any trial...

Question 18

Given a binomial distribution with n trials and probability p of a success on any trial, a conventional rule of thumb is that the normal distribution will provide an adequate approximation of the binomial distribution if
A) np $\le$ 5 and n(1 -p) $\ge$ 5
B) np $\ge$ and np(1 - p) $\ge$5
C) np$\ge$5 and n(1 - p) $\ge$5
D) None of these choices.

Accepted Answer

The answer of Given a binomial distribution with n trials...

Question 19

A sample of size 200 is taken at random from an infinite population.Given that the population proportion is 0.60, the probability that the sample proportion is between 0.58 and 0.62 is:
A) 0.4380
B) 0.0320
C) 0.0200
D) None of these choices.

Accepted Answer

We can use the central limit theorem to approximate the sample proportion distribution as normal with mean 0.60 and standard deviation $\sqrt{\frac{0.60(1-0.60)}{200}} \approx 0.044$. 
Then, we standardize the interval [0.58, 0.62] as 
$\frac{0.58-0.60}{0.044}=-0.45$ and $\frac{0.62-0.60}{0.044}=0.45$ 
Using a standard normal table, the probability of the sample proportion falling between -0.45 and 0.45 is approximately 0.4380. Therefore, the answer is A.

Question 20

A sample of 250 observations is selected at random from an infinite population.Given that the population proportion is .25, the standard error of the sampling distribution of the sample proportion is:
A) 0.0274
B) 0.5000
C) 0.0316
D) 0.0548

Accepted Answer

The answer of A sample of 250 observations is selected...

Quiz 9: Sampling Distributions

The estimator of the probability of success in a binomial distribution is: ____________________.

The variance of is ____________________.

The normal approximation to the binomial distribution gets better and better as ____________________ increases.

A sample of size 200 is taken at random from an infinite population.Given that the population proportion is 0.60, the probability that the sample proportion is greater than 0.58 is:

What is the probability that you get more than 200 heads if you flip a coin 400 times?

The continuity correction factor adds and subtracts the number ____________________ to/from x to find P(X = x).

The expected value of is ____________________.

The standard deviation of is also called the ____________________ of the sample proportion.

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if:

The standard error of the sample proportion gets larger as:

The standard deviation of is also called the:

Under certain conditions where n is large enough, you can approximate the distribution using the distribution.

A sample of size 200 is taken at random from an infinite population.Given that the population proportion is 0.60, the probability that the sample proportion is less than 0.58 is

To find the binomial probability P(X $\le$ 4) we calculate the area under the normal curve to the (left/right) of the number .

Suppose it is known that 60% of radio listeners at a particular college are smokers.A sample of 500 students from the college is selected at random.Approximate the probability that at least 280 of these students are radio listeners.

Suppose that the probability p of success on any trail of a binomial distribution equals 0.90.Then for which of the following number of trials, n, would the normal distribution provide a good approximation to the binomial distribution?

The probability of success on any trial of a binomial experiment is 20%.Find the probability that the proportion of success in a sample of 400 is: a. less than 18%. b. more than 18%. c. between 18% and 22%.

Given a binomial distribution with n trials and probability p of a success on any trial, a conventional rule of thumb is that the normal distribution will provide an adequate approximation of the binomial distribution if

A sample of size 200 is taken at random from an infinite population.Given that the population proportion is 0.60, the probability that the sample proportion is between 0.58 and 0.62 is:

A sample of 250 observations is selected at random from an infinite population.Given that the population proportion is .25, the standard error of the sampling distribution of the sample proportion is:

Quiz 9: Sampling Distributions

The estimator of the probability of success in a binomial distribution is: ____________________.

The variance of is ____________________.

The normal approximation to the binomial distribution gets better and better as ____________________ increases.

A sample of size 200 is taken at random from an infinite population.Given that the population proportion is 0.60, the probability that the sample proportion is greater than 0.58 is:

What is the probability that you get more than 200 heads if you flip a coin 400 times?

The continuity correction factor adds and subtracts the number ____________________ to/from x to find P(X = x).

The expected value of is ____________________.

The standard deviation of is also called the ____________________ of the sample proportion.

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if:

The standard error of the sample proportion gets larger as:

The standard deviation of is also called the:

Under certain conditions where n is large enough, you can approximate the ____________________ distribution using the ____________________ distribution.

A sample of size 200 is taken at random from an infinite population.Given that the population proportion is 0.60, the probability that the sample proportion is less than 0.58 is

To find the binomial probability P(X ≤\le≤ 4) we calculate the area under the normal curve to the ____________________ (left/right) of the number ____________________.

Suppose it is known that 60% of radio listeners at a particular college are smokers.A sample of 500 students from the college is selected at random.Approximate the probability that at least 280 of these students are radio listeners.

Suppose that the probability p of success on any trail of a binomial distribution equals 0.90.Then for which of the following number of trials, n, would the normal distribution provide a good approximation to the binomial distribution?

The probability of success on any trial of a binomial experiment is 20%.Find the probability that the proportion of success in a sample of 400 is: a. less than 18%. b. more than 18%. c. between 18% and 22%.

Given a binomial distribution with n trials and probability p of a success on any trial, a conventional rule of thumb is that the normal distribution will provide an adequate approximation of the binomial distribution if

A sample of size 200 is taken at random from an infinite population.Given that the population proportion is 0.60, the probability that the sample proportion is between 0.58 and 0.62 is:

A sample of 250 observations is selected at random from an infinite population.Given that the population proportion is .25, the standard error of the sampling distribution of the sample proportion is:

Under certain conditions where n is large enough, you can approximate the distribution using the distribution.

To find the binomial probability P(X $\le$ 4) we calculate the area under the normal curve to the (left/right) of the number .