Question 1

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 2

As the number of throws of a fair die increases, the probability that the sample mean is close to (the number) ____________________ increases.

Accepted Answer

The answer of As the number of throws of a...

Question 3

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) 0.4314
B) 0.0320
C) 0.0200
D) None of these choices.

Accepted Answer

The sample proportion, denoted by p-hat, follows a normal distribution with a mean of the population proportion, denoted by p, and a standard deviation of sqrt((p(1-p))/n), where n is the sample size. 
We are given that p = 0.60 and n = 200, so the standard deviation is sqrt((0.6(1-0.6))/200) = 0.039. 
Using the z-score formula, we get z1 = (0.58 - 0.60)/0.039 = -0.51 and z2 = (0.62 - 0.60)/0.039 = 0.51. 
Using a standard normal distribution table, the probability of getting a z-score between -0.51 and 0.51 is approximately 0.4314. Therefore, the answer is A).

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 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 5

The width of the sampling distribution of  gets ____________________ as the sample size increases.

Accepted Answer

The answer of The width of the sampling distribution of...

Question 6

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 7

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 8

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) $\le$  5
B) np $\ge$ 5 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 9

As n gets larger, the sampling distribution of  becomes increasingly bell shaped. This phenomenon is due to the ____________________.

Accepted Answer

The answer of As n gets larger, the sampling distribution...

Question 10

The accuracy of the approximation of  with a normal distribution depends on the probability distribution of the ____________________ and on the sample ____________________.

Accepted Answer

The answer of The accuracy of the approximation of ...

Question 11

If the population is normal, then  is normally distributed for __________________ values of n.

Accepted Answer

The answer of If the population is normal, then ...

Question 12

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 sample proportion as normal distribution with mean = population proportion = 0.60 and standard deviation = sqrt((0.6*0.4)/200) = 0.0498.
Then, we can calculate the z-score: z = (0.58 - 0.60)/0.0498 = -0.402
Using standard normal distribution table or calculator, the probability of z-score less than -0.402 is 0.281. Therefore, the probability of sample proportion greater than 0.58 is 1 - 0.281 = 0.719.

Question 13

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 14

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 15

The finite population correction is not needed if the population size is large relative to the sample size. As a rule of thumb, we will treat any population that is at least ____________________ times larger than the sample size as large.

Accepted Answer

The answer of The finite population correction is not needed...

Question 16

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...

Question 17

A randomly selected value of  is likely to be ____________________ to(than) the mean of the population than is a randomly selected value of X.

Accepted Answer

The answer of A randomly selected value of  is...

Question 18

The variance of the sampling distribution of  is ____________________ the variance of the population we're sampling from, for any n > 1.

Accepted Answer

The answer of The variance of the sampling distribution of...

Question 19

Because the value of the ____________________ varies from sample to sample, we can regard it as a new random variable, created by sampling.

Accepted Answer

The answer of Because the value of the ____________________ varies...

Question 20

As n gets ____________________, the shape of the sampling distribution of  becomes increasingly bell shaped.

Accepted Answer

The answer of As n gets ____________________, the shape of...

Quiz 8: Sampling Distributions

The standard deviation of is also called the:

As the number of throws of a fair die increases, the probability that the sample mean is close to (the number) ____________________ 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 between 0.58 and 0.62 is:

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 width of the sampling distribution of gets ____________________ as the sample size increases.

The standard error of the sample proportion gets larger as:

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?

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

As n gets larger, the sampling distribution of becomes increasingly bell shaped. This phenomenon is due to the ____________________.

The accuracy of the approximation of with a normal distribution depends on the probability distribution of the and on the sample .

If the population is normal, then is normally distributed for __________________ values of n.

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 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.

The finite population correction is not needed if the population size is large relative to the sample size. As a rule of thumb, we will treat any population that is at least ____________________ times larger than the sample size as large.

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 randomly selected value of is likely to be ____________________ to(than) the mean of the population than is a randomly selected value of X.

The variance of the sampling distribution of is ____________________ the variance of the population we're sampling from, for any n > 1.

Because the value of the ____________________ varies from sample to sample, we can regard it as a new random variable, created by sampling.

As n gets ____________________, the shape of the sampling distribution of becomes increasingly bell shaped.