Question 1

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 standard error of the sampling distribution of the sample proportion is calculated using the formula: SE = sqrt[(p(1-p))/n], where p is the population proportion and n is the sample size. Substituting the given values: SE = sqrt[(0.25(1-0.25))/250] = 0.0274.

Question 2

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

Accepted Answer

The answer of To find the binomial probability P(X ≤...

Question 3

Given that X is a binomial random variable with very large n,the binomial probability P(X ≤ 5)is approximated by the area under a normal curve to the left of
A)5
B)−5
C)5.5
D)4.5

Accepted Answer

When approximating a binomial distribution with a normal distribution for a binomial random variable X with a very large n, we use the continuity correction factor. To approximate P(X ≤ 5), we use 5.5 as the cutoff point on the normal curve to include all values up to and including 5.

Question 4

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 ≤ 5 and n(1 − p)≤ 5
B)np ≥ 5 and np(1 − p)≥ 5
C)np ≥ 5 and n(1 − p)≥ 5
D)None of these choices.

Accepted Answer

The normal approximation to the binomial distribution is considered adequate if both np ≥ 5 and n(1 − p) ≥ 5, ensuring that there are enough expected successes and failures.

Question 5

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

First, we need to find the standard error of the sample proportion:
SE = sqrt(p(1-p)/n) = sqrt(0.6(1-0.6)/200) = 0.0387

Next, we need to standardize the sample proportion using the equation:
z = (p̂ - p) / SE

We want to find the probability that the sample proportion (p̂) is between 0.58 and 0.62, which can be rewritten as the probability that the standardized z-score is between:

z1 = (0.58 - 0.6) / 0.0387 = -0.515
z2 = (0.62 - 0.6) / 0.0387 = 0.515

Using a standard normal distribution table or calculator, we can find the area under the curve between these two z-scores:
P(-0.515 < z < 0.515) = 0.8779 - 0.4399 = 0.4380

Therefore, the answer is choice A) 0.4380.

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 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 8

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 9

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 10

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 proportion follows the normal distribution with mean equal to population proportion i.e 0.60 and standard deviation equal to sqrt((0.60*0.40)/200) = 0.0499 (using formula for standard deviation of sample proportion).
Now, we need to calculate the z-score for sample proportion less than 0.58:
z = (0.58 - 0.60) / 0.0499 = -0.401
Using standard normal distribution table, the area/probability to the left of z=-0.401 is 0.281 (approximately).
Therefore, the probability that the sample proportion is less than 0.58 is 0.281.

Question 11

The expected value of  is ____________________.

Accepted Answer

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

Question 12

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 13

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 14

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 15

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 16

The variance of  is ____________________.

Accepted Answer

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

Question 17

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 best choice(s) is D because it is the essential condition for using a normal approximation for the sampling distribution of the sample proportion. When np and n(1-p) both greater than or equal to 5, the sampling distribution of the sample proportion can be approximated using a normal distribution, regardless of the sample size or the population proportion.

Question 18

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 19

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

Accepted Answer

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

Question 20

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...

Quiz 9: Sampling Distributions

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:

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

Given that X is a binomial random variable with very large n,the binomial probability P(X ≤ 5) is approximated by the area under a normal curve to the left of

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:

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

The standard error of the sample proportion gets larger as:

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

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

The expected value of is ____________________.

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

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.

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:

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%.

The variance of is ____________________.

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

The standard deviation of is also called the:

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

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?