Question 1

A local planning commission is interested in finding out what proportion of its city's residents is opposed to constructing a new baseball stadium in the downtown area. A random sample of 1870 residents is obtained, and 41.2 percent of them are opposed to the stadium.
The commission is 95 percent confident that between _____ of the city's residents are opposed to constructing a new baseball stadium in the downtown area.

Accepted Answer

We can construct a 95% confidence interval for the true proportion of residents opposed to the stadium using the formula:
sample proportion ± z* standard error
where z* is the z-score corresponding to the desired level of confidence, and the standard error is given by:
sqrt[(sample proportion * (1 - sample proportion)) / n]
Plugging in the given values, we get:
0.412 ± 1.96 * sqrt[(0.412 * 0.588) / 1870]
Simplifying this expression, we get:
0.412 ± 0.016
Therefore, the 95% confidence interval for the true proportion of residents opposed to the stadium is:
0.396 to 0.428
This corresponds to answer choice C.

Question 2

A recent survey of 174,745 randomly selected U.S. adults studied the importance of religion to Americans in each state.
The survey found that nationwide 40 percent said that they were very religious. But in New Hampshire, the least religious state, only 20 percent answered that they were "very religious." The sample included 1000 respondents from New Hampshire. The margin of error in a 95 percent confidence interval for the proportion of all adults who are very religious is:

Accepted Answer

The margin of error for a proportion in a sample is influenced by the sample size and the proportion itself. Given the same confidence level, the margin of error is larger for smaller samples (1000 in New Hampshire vs. 174,745 nationwide) and when the proportion is closer to 50%. Since the sample size for New Hampshire is significantly smaller than the nationwide sample, the margin of error for New Hampshire's estimate will be larger.

Question 3

A local planning commission is interested in finding out what proportion of its city's residents is opposed to constructing a new baseball stadium in the downtown area. A random sample of 1870 residents is obtained, and 41.2 percent of them are opposed to the stadium.
A 95 percent confidence interval was created for the proportion of the city's residents who are opposed to constructing a new baseball stadium in the downtown area. Decreasing the sample size will:

Accepted Answer

Decreasing the sample size will result in a wider interval because the margin of error will increase. The margin of error is directly proportional to the square root of the sample size. Therefore, decreasing the sample size will increase the margin of error, resulting in a wider confidence interval.

Question 4

A local planning commission is interested in finding out what proportion of its city's residents is opposed to constructing a new baseball stadium in the downtown area. A random sample of 1870 residents is obtained, and 41.2 percent of them are opposed to the stadium.
In this sample, the proportion of the city's residents opposed to constructing a new baseball stadium in the downtown area is:

Accepted Answer

The sample proportion of 41.2 percent represents the proportion of the 1870 residents in the sample who are opposed to the stadium. It does not represent the proportion of the entire city's residents. Therefore, we cannot conclude anything about the proportion of the city's residents from this sample proportion.

Question 5

What is the most important issue facing the next president? In a January 2016 survey of 1022 adults, the Gallup organization found that Republicans and Democrats agreed on the top four issues, and terrorism was reported as the top challenge by 17 percent of respondents, with a margin of error of plus or minus 4 percentage points at 95 percent confidence level.
In this sample, the proportion of Americans who feel that the most important issue facing the next president is terrorism is:

Accepted Answer

The proportion of Americans who feel that the most important issue facing the next president is terrorism is 17 percent, as reported in the Gallup survey. Option A is an incorrect calculation, as it multiplies the percentage by the sample size, which does not provide the proportion. Option B is incorrect, as it refers to the confidence level and not the proportion. Option D is also incorrect, as it refers to the margin of error and not the proportion.

Question 6

A local planning commission is interested in finding out what proportion of its city's residents is opposed to constructing a new baseball stadium in the downtown area. A random sample of 1870 residents is obtained, and 41.2 percent of them are opposed to the stadium.
In this sample, the margin of error is

Accepted Answer

The margin of error for a proportion can be calculated using the formula: 
Margin of error = z*(sqrt(p*(1-p)/n))
where z is the critical value from the standard normal distribution corresponding to the level of confidence, p is the sample proportion, and n is the sample size. Assuming a 95% level of confidence (corresponding to a z-value of 1.96), we can plug in the values from the problem: 
Margin of error = 1.96*(sqrt(0.412*(1-0.412)/1870)) = 0.023 or 2.3%
Therefore, the best choice is B, which gives the correct margin of error as a percentage. A represents an incorrect calculation (770 is not a reasonable margin of error), C is not a valid option as it simply repeats the sample proportion, and D gives the margin of error as a decimal instead of a percentage.

Question 7

_____ in a sampling method describes how spread out the values of the sample statistic are when one takes repeated samples.

Accepted Answer

Variability in a sampling method refers to how spread out the values of the sample statistic are when one takes repeated samples. It is an important consideration when drawing conclusions from a single sample, as a high level of variability can indicate that the results are not very reliable or generalizable to the larger population.

Question 8

A recent survey of 174,745 randomly selected U.S. adults studied the importance of religion to Americans in each state.
Of those questioned in the poll, 29 percent said that they were "moderately religious." The number 29 percent is a:

Accepted Answer

The number 29 percent represents a statistical measure calculated from the survey data, making it a statistics. It is not a margin of error, which is a range of values used to estimate the true population value, nor is it a parameter, which is a characteristic of a population. Confidence level refers to the level of certainty in the accuracy of the survey results, and is not represented by the number 29 percent.

Question 9

A local planning commission is interested in finding out what proportion of its city's residents is opposed to constructing a new baseball stadium in the downtown area. A random sample of 1870 residents is obtained, and 41.2 percent of them are opposed to the stadium.
A 95 percent confidence interval was created for the proportion of the city's residents who are opposed to constructing a new baseball stadium in the downtown area. A "95 percent confidence" means that:

Accepted Answer

A 95 percent confidence interval means that if we were to repeat the sampling process many times and construct 95 percent confidence intervals using the same method, then we would expect 95 percent of those intervals to contain the true proportion of residents who oppose constructing a new stadium in the downtown area. It does not mean that 95 percent of all residents oppose constructing a new stadium, nor is it related to the beliefs of educated adults.

Question 10

For a simple random sample of size 4761, the margin of error for 95 percent confidence is about:

Accepted Answer

The formula for the margin of error is: 
margin of error = z* (standard deviation / sqrt(sample size))
For a 95% confidence interval, the z-value is 1.96.
Since the sample size is very large (n=4761), we can assume that the standard deviation is equal to the population standard deviation (or use a conservative estimate for the standard deviation).

Therefore, the margin of error is approximately: 
1.96 * (standard deviation / sqrt(sample size)) 
= 1.96 * (1 / sqrt(4761)) 
= 1.45%
So the best choice is B, ±1.45%.

Question 11

What is the most important issue facing the next president? In a January 2016 survey of 1022 adults, the Gallup organization found that Republicans and Democrats agreed on the top four issues, and terrorism was reported as the top challenge by 17 percent of respondents, with a margin of error of plus or minus 4 percentage points at 95 percent confidence level.
In this sample, the margin of error is:

Accepted Answer

The margin of error in the sample is 4 percent, as stated in the question. The other answer choices are not correct interpretations of the information provided.

Question 12

What is the most important issue facing the next president? In a January 2016 survey of 1022 adults, the Gallup organization found that Republicans and Democrats agreed on the top four issues, and terrorism was reported as the top challenge by 17 percent of respondents, with a margin of error of plus or minus 4 percentage points at 95 percent confidence level.
There is a 95 percent confidence that between 13 percent and 21 percent of Americans feel that the most important issue facing the next president is terrorism. Increasing the sample size will:

Accepted Answer

Increasing the sample size will decrease the margin of error, resulting in a narrower interval. This is because a larger sample size reduces the variability in the sample mean and provides a more precise estimate of the population mean.

Question 13

What is the most important issue facing the next president? In a January 2016 survey of 1022 adults, the Gallup organization found that Republicans and Democrats agreed on the top four issues, and terrorism was reported as the top challenge by 17 percent of respondents, with a margin of error of plus or minus 4 percentage points at 95 percent confidence level.
There is a 95 percent confidence that between 13 percent and 21 percent of Americans feel that the most important issue facing the next president is terrorism. For a 99 percent confidence, the range of points would be:

Accepted Answer

A higher confidence level requires a larger margin of error to account for the increased certainty, thus widening the range.

Question 14

What is the most important issue facing the next president? In a January 2016 survey of 1022 adults, the Gallup organization found that Republicans and Democrats agreed on the top four issues, and terrorism was reported as the top challenge by 17 percent of respondents, with a margin of error of plus or minus 4 percentage points at 95 percent confidence level.
With these results, one may say, "With 95 percent confidence, one can say that between 13 percent and 21 percent of Americans feel that the most important issue facing the next president is terrorism." What "95 percent confidence" means is that:

Accepted Answer

"95 percent confidence" means that if the survey were conducted 100 times using the same method, 95 out of those 100 times, the range obtained from the survey would contain the true value for the entire population. It does not mean that 95 percent of all Americans will say terrorism is the most important issue or that one can be certain about a specific percentage range. Option A and B are incorrect for these reasons. Option D is incorrect as the statement is not about just educated adults, but rather the entire population. Therefore, the best choice is C.

Question 15

In an SRS of students, 23 percent said they smoked, while another 16 percent said they "hardly ever" smoked. The numbers 23 percent and 16 percent are:

Accepted Answer

These percentages are statistics because they are measures that describe a characteristic of a sample drawn from a population.

Quiz 3: What Do Samples Tell Us

_____ in a sampling method describes how spread out the values of the sample statistic are when one takes repeated samples.

A recent survey of 174,745 randomly selected U.S. adults studied the importance of religion to Americans in each state. Of those questioned in the poll, 29 percent said that they were "moderately religious." The number 29 percent is a:

For a simple random sample of size 4761, the margin of error for 95 percent confidence is about:

In an SRS of students, 23 percent said they smoked, while another 16 percent said they "hardly ever" smoked. The numbers 23 percent and 16 percent are: