Question 1

Each "building block" in a sampling distribution of sample means is

Accepted Answer

Each "building block" in a sampling distribution of sample means is a sample mean, not a population mean or a set of sample means. It also does not represent the difference between the sample mean and the population mean.

Question 2

It is common that a sample mean will not exactly equal the mean of the population from which the sample was drawn. This phenomenon is referred to as:

Accepted Answer

Sampling error refers to the natural variation that occurs in a sample, which leads to differences between the sample mean and the population mean. The standard error is a measure of the variability of sample means, while the standard deviation measures the spread of individual data points around the mean. Sampling bias occurs when the selection of a sample is not representative of the population, leading to inaccurate results.

Question 3

Jake takes many samples from a population, calculates the mean for each sample on a variable of interest, then places these means on a graph. The graph Jake has created is called:

Accepted Answer

The graph Jake has created is a sampling distribution, which shows the distribution of all the possible sample means from a population. This is important for inferential statistics, as it allows us to make inferences about the population based on the sample means.

Question 4

According to the Central Limit Theorem, if you take decently sized samples from the highly skewed distribution:

Accepted Answer

The Central Limit Theorem states that regardless of the shape of the original distribution, if the sample size is large enough, the sampling distribution of the mean will be normal (or approximately normal if the sample size is not very large) 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. Therefore, even if the original distribution is highly skewed, the resulting sampling distribution of the mean will be normal.

Question 5

Gail takes many samples from a highly skewed distribution, and shows that the resulting sampling distribution is normally shaped (bell-shaped). Gail is illustrating a key aspect of:

Accepted Answer

This is an example of the Central Limit Theorem, which states that the sampling distribution of the mean of an independent and identically distributed random variable approaches normality as the sample size increases, regardless of the underlying distribution of the population.

Question 6

What two pieces of information do you need to find the estimate of the standard error of the mean?

Accepted Answer

To find the estimate of the standard error of the mean, we need to know the sample standard deviation (which measures the spread of the sample data) and the sample size (which determines how well the sample represents the population).

Question 7

With regard to a claim about a population mean, what is the correct order:

Accepted Answer

The correct order is: number of standard errors, actual distance, percentage probability. This is because the number of standard errors tells us how far away the sample mean is from the population mean, the actual distance tells us the actual difference between the sample mean and the population mean, and the percentage probability tells us the probability of getting a sample mean as far away from the population mean as the one we observed, assuming that the null hypothesis is true.

Question 8

Which of the following is the best interpretation of a 95% confidence interval?

Accepted Answer

A 95% confidence interval means that if the same population is sampled many times and interval estimates are made each time, 95% of those intervals would contain the true population mean. Therefore, we are 95% confident that the population mean is within this interval. Option A is incorrect as we cannot determine the proportion of the population that falls within the confidence interval based on the given information. Option B is incorrect as we cannot make any statement about the sample mean being accurate based solely on the confidence interval. Option D is also incorrect as the interval implies a level of confidence, not a probability.

Question 9

Which of the following is true?

Accepted Answer

A higher confidence level requires a wider interval to capture the true population parameter with more certainty. Therefore, a 99% confidence interval will always be wider than a 95% confidence interval using the same sample statistics. Option A is incorrect because the opposite is true. Option C is incorrect because it is rare for the two intervals to have the same width. Option D is incorrect because a 100% confidence interval would be the entire range of possible parameter values, which is not informative for making inferential conclusions.

Question 10

With regard to building a confidence interval, what is the correct order:

Accepted Answer

The correct order for building a confidence interval is first determining the percentage probability (confidence level), then calculating the number of standard errors (margin of error), and finally determining the actual distance (range of values) around the sample statistic.

Question 11

In the 2008GSS, the 61 Hispanic respondents' average age at the birth of their first child was 22.95, with a sample standard deviation of 4.88. What is the 95% confidence interval for the population mean?

Accepted Answer

Estimate of standard error is: .63, and we multiply this by 2.00 standard errors to get 1.26, which we subtract and add to the sample mean.

Question 12

A researcher finds that the 13 Muslims surveyed in the 2008GSS had a mean of 2.38 children with a standard deviation of 2.10. With this information, she builds a confidence interval for the Muslim population's number of children: 1.05 ≤ μ ≤ 3.71. Which confidence interval is she using?

Accepted Answer

Working backwards, we know that the researcher added and subtracted 1.33 to and from the sample mean of 2.38. The value of the estimate of the standard error is .61 (2.10 divided by the square root of 12). So, .61 multiplied by some number is 1.33. This number must be 2.18, and in the t-table, with 12 degrees of freedom, this matches up to .025. .025 + .025 = .05, therefore it is a 95% confidence interval.

Question 13

All other things being equal, a confidence interval based on a larger sample will be ________ a confidence interval based on a smaller sample.

Accepted Answer

A larger sample size reduces the standard error, leading to a narrower confidence interval. This is because a larger sample provides a more precise estimate of the population parameter.

Question 14

What two pieces of information do you need to find the estimate of the standard error of the proportion?

Accepted Answer

To find the estimate of the standard error of the proportion, we need the sample size and the sample proportion. The formula for the standard error of the proportion is SEp = sqrt[p(1-p)/n], where p is the sample proportion and n is the sample size.

Question 15

All other things being equal, which proportion will produce the widest confidence interval?

Accepted Answer

The widest confidence interval occurs when the proportion is closest to 0.5, as the standard error is maximized at this point.

Question 16

Of the 219 white GSS2008 respondents in their 20s, 63 of them claim the ability to speak a language other than English. With 99% confidence, what is the upper limit of the population proportion based on these statistics?

Accepted Answer

the sample proportion is 63/219 = .29. The square root of (.29)(.71)/219 is .03. Multiply this by 1.96 standard errors to get .06. .29 + .06 = .35

Question 17

Which proportion represents the largest amount of variation:

Accepted Answer

The answer of Which proportion represents the largest amount of...

Question 18

If we are building a confidence interval, and our sample size is 25, how many degrees of freedom do we have?

Accepted Answer

The degrees of freedom for a confidence interval based on a sample size (n) is calculated as n - 1. Therefore, with a sample size of 25, the degrees of freedom would be 25 - 1 = 24.

Question 19

What media does Hughey study in his article on "magical negros"?

Accepted Answer

Hughey specifically references films in his article on "magical negros," analyzing the portrayal of black characters as mystical or supernatural helpers in popular movies.

Question 20

What method does Hughey use in his article on "magical negros"?

Accepted Answer

In his article on "magical negros," Hughey uses content analysis to analyze the portrayal of Black characters in American films. He examines the ways in which Black characters are depicted as "magical negroes," or stereotypes that perpetuate a myth of mystical powers and self-sacrifice for the benefit of white characters. This type of analysis involves closely examining the content of texts or media to identify patterns and themes.

Quiz 5: Using a Sample Mean or Proportion to Talk About a Population: Confidence Intervals