Question 1

An unbiased estimator will have a value, on average across samples, equal to the population parameter value.

Accepted Answer

An unbiased estimator is defined as an estimator whose expected value is equal to the true population parameter. Therefore, on average across samples, it will have a value equal to the population parameter value.

Question 2

The question "Is your household income last year somewhere in between $25,000 and $35,000?" will most likely result in coverage error.

Accepted Answer

Coverage error occurs when some members of the population are inadequately represented in the sample. The question itself does not cause coverage error; it is the sampling method that might lead to such an error.

Question 3

The only reliable way a researcher can make statistical inferences from a sample to a population is to use nonprobability sampling methods.

Accepted Answer

The statement is false. Probability sampling methods are considered more reliable for making statistical inferences from a sample to a population because they allow for random selection and an unbiased representation of the population. Nonprobability sampling methods may introduce sampling bias and limit the generalizability of the results.

Question 4

You stand at the main entrance to a departmental store and pick the first 20 customers that enter the store after it has opened its door for business on a single day. This is an example of a systematic sample.

Accepted Answer

This is an example of a convenience sample because the selection of the first 20 customers is based on ease of access and timing, rather than a systematic method of selecting every nth customer.

Question 5

When participants are allowed to self-select into the sample, you have a nonprobability sample.

Accepted Answer

Nonprobability sampling is a sampling technique where participants are not selected randomly and there is no way to determine the likelihood of any particular person being included in the sample. Self-selection is a type of nonprobability sampling where people choose to participate in the study on their own, rather than being randomly selected.

Question 6

Sampling error can be completely eliminated by taking larger sample sizes.

Accepted Answer

While larger sample sizes can reduce the likelihood of sampling error, they cannot completely eliminate it. Sampling error is inherent in any sample and can only be minimized through proper sampling techniques and statistical analysis.

Question 7

If the amount of gasoline purchased per car at a large service station has a population mean of $15 and a population standard deviation of $4 and a random sample of 64 cars is selected, there is approximately a 95.44% chance that the sample mean will be between $14 and $16.

Accepted Answer

This statement is true. Given a normal population with known mean and standard deviation, the distribution of sample means will also be normal with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size. The confidence interval for the population mean can be calculated using the formula:

sample mean +/- (critical value) * (population standard deviation / sqrt(sample size))

For a 95% confidence interval, the critical value is approximately 1.96. Plugging in the given values, we get:

$15 +/- 1.96 * ($4 / sqrt(64))
$15 +/- $0.98

This means that there is a 95% chance that the sample mean will fall between $14.02 and $15.98, which includes the range of $14 to $16. Therefore, the statement is true.

Question 8

If the amount of gasoline purchased per car at a large service station has a population mean of $15 and a population standard deviation of $4 and a random sample of 4 cars is selected, there is approximately a 68.26% chance that the sample mean will be between $13 and $17.

Accepted Answer

We can use the Central Limit Theorem to approximate the sampling distribution of the sample mean. The mean of the sampling distribution will also be $15, but the standard error of the mean will be $\frac{4}{\sqrt{4}}=2$.

To find the probability that the sample mean is between $13$ and $17$, we need to standardize the values:

$Z_1=\frac{13-15}{2}=-1$

$Z_2=\frac{17-15}{2}=1$

Using a standard normal distribution table or calculator, we can find the probability of $P(-1<Z<P1)\approx0.6827$. However, this is the probability of getting a sample mean between $13$ and $17$ for an infinite number of samples. We cannot say for certain that a particular sample mean will fall within this range with 68.26% probability. Therefore, the answer is False.

Question 9

The amount of bleach a machine pours into bottles has a mean of 36 oz. with a standard deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this machine. The sampling distribution of the sample mean has a mean of 36 oz.

Accepted Answer

This statement is true, as the mean of the sampling distribution of the sample mean is equal to the population mean.

Question 10

As the sample size increases, the standard error of the mean increases.

Accepted Answer

As the sample size increases, the standard error of the mean actually decreases, as the sample becomes more representative of the population.

Question 11

A sampling distribution is a distribution for a statistic.

Accepted Answer

The answer of A sampling distribution is a distribution for...

Question 12

If the population distribution is symmetric, the sampling distribution of the mean can be approximated by the normal distribution if the samples contain 15 observations.

Accepted Answer

The central limit theorem allows for the approximation of the sampling distribution of the mean with a normal distribution for samples of any size, as long as the population distribution is normal or approximately normal. However, when the population distribution is symmetric and not too skewed, even smaller samples (such as 15 observations) can be used for this approximation.

Question 13

As the size of the sample is increased, the standard deviation of the sampling distribution of the sample mean for a normally distributed population will stay the same.

Accepted Answer

As the size of the sample is increased, the standard deviation of the sampling distribution of the sample mean, also known as the standard error, decreases. This is because the standard error is inversely related to the square root of the sample size.

Question 14

The amount of bleach a machine pours into bottles has a mean of 36 oz. with a standard deviation of 0.15 oz. Suppose we take a random sample of 36 bottles filled by this machine. The sampling distribution of the sample mean has a standard error of 0.15.

Accepted Answer

The standard error of the sampling distribution of the sample mean can be calculated as the standard deviation of the population divided by the square root of the sample size. In this case, the standard error would be 0.15/√36 = 0.025. Therefore, the given standard error of 0.15 is incorrect and the answer is false.

Question 15

A sample of size 25 provides a sample variance of 400. The standard error, in this case equal to 4, is best described as the estimate of the standard deviation of means calculated from samples of size 25.

Accepted Answer

The standard error is the estimate of the standard deviation of the sample means, which is calculated from samples of a specific size, in this case 25. The standard deviation of the sample is different from the standard error, the latter being a measure of the variability of the sample means. Thus, the statement is true.

Question 16

Coverage error can become an ethical issue if a particular group is intentionally excluded from the frame.

Accepted Answer

Coverage error occurs when a particular group is excluded from the frame, which can lead to biased or inaccurate conclusions. If this exclusion is intentional, it can become an ethical issue because it could perpetuate systemic inequality or discrimination against that group.

Question 17

If a simple random sample is chosen with replacement, each individual has the same chance of selection on every draw.

Accepted Answer

With replacement, each individual remains in the pool of potential selections for every draw, and therefore has the same chance of being selected on each draw.

Question 18

Suppose µ = 80 and ?² = 400 for a population. In a sample where n = 100 is randomly taken, 95% of all possible sample means will fall above 76.71.

Accepted Answer

This statement is true based on the central limit theorem and the calculation of a 95% confidence interval for the sample mean using the given information.

Question 19

Suppose µ = 50 and ?² = 100 for a population. In a sample where n = 100 is randomly taken, 90% of all possible sample means will fall between 49 and 51.

Accepted Answer

The standard deviation of the sample mean is 1 (since the population standard deviation is 10 and n = 100), so 90% of sample means will fall within approximately 1.645 standard deviations from the mean, which is not between 49 and 51.

Question 20

As a population becomes large, it is usually better to obtain statistical information from the entire population.

Accepted Answer

As the population becomes large, it may become impractical or impossible to obtain statistical information from the entire population. In such cases, a sample of the population is taken and statistical inferences are made based on that sample. This process is known as "sampling."

Quiz 7: Sampling Distributions and Sampling