Question 1

Suppose 45% of voters favor candidate A.We can be 67% sure that out of 200 randomly selected voters, at least how many will favor candidate A?
A) 90
B) 87
C) 134
D) 93

Accepted Answer

To determine the minimum number of voters who favor candidate A with 67% confidence, we can use the normal approximation to the binomial distribution.

First, we calculate the expected value of the number of voters who favor candidate A:

E(X) = np = 200 * 0.45 = 90

Next, we calculate the standard deviation of the number of voters who favor candidate A:

SD(X) = sqrt(np(1-p)) = sqrt(200 * 0.45 * 0.55) ≈ 7.13

Using the normal approximation, we can now find the number of voters who favor candidate A with at least 67% confidence by finding the z-score that corresponds to a cumulative probability of 0.67 in the standard normal distribution.

z = invNorm(0.67) ≈ 0.44

Now we can use this z-score to calculate the minimum number of voters who favor candidate A:

X = E(X) + z * SD(X) = 90 + 0.44 * 7.13 ≈ 87

Therefore, we can be 67% sure that out of 200 randomly selected voters, at least 87 will favor candidate A.

Question 2

If the population is perfectly normal, the sampling distribution of  will be:
A) exactly normal for any sample size.
B) approximately normal for any sample size.
C) exactly normal only if the sample size is large enough.
D) approximately normal only if the sample size is large enough.

Accepted Answer

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

Question 3

If the sample size (n) is large (say, n ≥ 30), then it is safe to model the sampling distribution of  as a:
A) discrete distribution.
B) continuous distribution.
C) log-normal distribution.
D) normal distribution.

Accepted Answer

If the sample size is large (n ≥ 30), then the central limit theorem applies and the sampling distribution of the mean is approximately normal. Therefore, D is the best choice as it represents the normal distribution.

Question 4

The problems associated with sampling without replacement in the simple random sample process are negligible provided:
A) the sample is large relative to the population size.
B) the population is small relative to the sample size.
C) the sample and the population are of roughly the same size.
D) the population is large relative to the sample size.

Accepted Answer

The problems associated with sampling without replacement become negligible when the population is large relative to the sample size. This is because the odds of picking the same individual more than once become very small, making the sampling process approximate a simple random sample with replacement.

Question 5

We wish to survey high schools in a particular city.If the city has 22 high schools and we want to select 5 of them, how many unique samples of size 5 are possible?
A) 3,160,080
B) 5,153,632
C) 26,334
D) 2.384 × 1015

Accepted Answer

The number of unique samples of size 5 that can be selected from 22 high schools is calculated using the combination formula, which is $C(n, k) = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of items (22 high schools) and $k$ is the size of each sample (5 high schools). Plugging in the values, we get $C(22, 5) = \frac{22!}{5!(22-5)!} = \frac{22!}{5!17!} = 26,334$.

Question 6

A sample of 82 mice is observed from birth until they are one year old.The total amount of growth experienced by each mouse is recorded.The median amount of growth of these mice is 4.5 cm.In this example, 4.5 cm is a:
A) parameter.
B) sampling distribution.
C) statistic.
D) random variable.

Accepted Answer

The value of 4.5cm represents a calculated value from the observed sample of 82 mice, making it a statistic. A parameter would be a characteristic of the entire population, which is not given in this scenario. A sampling distribution would be a distribution of many possible statistics that could be obtained from repeated samples of the same size and from the same population, which is also not relevant in this case. A random variable would be a variable whose value is subject to randomness, which does not apply to the given scenario either.

Question 7

All statistics may be treated as:
A) characteristics of a population.
B) fixed quantities.
C) round variables.
D) random variables.

Accepted Answer

Statistics can be considered as either fixed or random variables. Fixed statistics are those that have a definite value, such as the mean of a sample or the population size. Random statistics, on the other hand, are those that vary and are subject to chance, such as the outcome of a coin flip or the number of customers who enter a store in a day. Therefore, D is the best choice.

Question 8

If n ≥ 30, we know that the sampling distribution of  is:
A) approximately normal.
B) exactly normal.
C) suitable for estimating the unknown parameter p.
D) unknown, because n ≥ 30 alone tells us nothing substantial concerning the sampling distribution of  .

Accepted Answer

The answer of If n ≥ 30, we know that...

Question 9

According to the Central Limit Theorem, as the sample size (n) from a non-normal population increases:
A) the mean of the sampling distribution of  becomes closer to μ, the population mean.
B) the sample standard deviation decreases.
C) the sampling distribution of  increasingly approximates a normal distribution.
D) there is less of a problem with nonrepresentative sample bias.

Accepted Answer

The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean increasingly approximates a normal distribution regardless of the shape of the population distribution. This means that the distribution of sample means becomes more symmetrical and bell-shaped with increasing sample size.

Question 10

If a population is skewed, based on the Central Limit Theorem, what is a good rule of thumb for assuming the sampling distribution of  is approximately normal?
A) np ≥ 5
B) n ≥ 30
C) μ ≥ 30
D) n ≤ 30

Accepted Answer

The Central Limit Theorem states that the sampling distribution of the means of any independent, random variable will be approximately normal if the sample size is sufficiently large (usually n ≥ 30), regardless of the shape of the population distribution. Therefore, choice B is the best option for assuming the sampling distribution of 11eb86f3_5f6e_e543_883a_838240c049d6_TB8195_11 is approximately normal.

Quiz 8: Sampling Distributions

Suppose 45% of voters favor candidate A.We can be 67% sure that out of 200 randomly selected voters, at least how many will favor candidate A?

If the population is perfectly normal, the sampling distribution of will be:

If the sample size (n) is large (say, n ≥ 30) , then it is safe to model the sampling distribution of as a:

The problems associated with sampling without replacement in the simple random sample process are negligible provided:

We wish to survey high schools in a particular city.If the city has 22 high schools and we want to select 5 of them, how many unique samples of size 5 are possible?

A sample of 82 mice is observed from birth until they are one year old.The total amount of growth experienced by each mouse is recorded.The median amount of growth of these mice is 4.5 cm.In this example, 4.5 cm is a:

All statistics may be treated as:

If n ≥ 30, we know that the sampling distribution of is:

According to the Central Limit Theorem, as the sample size (n) from a non-normal population increases:

If a population is skewed, based on the Central Limit Theorem, what is a good rule of thumb for assuming the sampling distribution of is approximately normal?