Question 1

What is the interpretation of a 96% confidence level?
A)There's a 96% chance that the given interval includes the true value of the population parameter.
B)Approximately 96 out of 100 such intervals would include the true value of the population parameter.
C)There's a 4% chance that the given interval does not include the true value of the population parameter.
D)The interval contains 96% of all sample means.

Accepted Answer

A 96% confidence level means that if we were to repeat this study many times and construct a 96% confidence interval each time, approximately 96 out of 100 intervals would include the true value of the population parameter. It does not mean that there is a 96% chance that the given interval we constructed includes the true value, nor does it mean that the interval contains 96% of all sample means.

Question 2

The distribution of Student's t has _________.
A)A mean of zero and a standard deviation of one
B)A mean of one and a standard deviation of one
C)A mean of zero and a standard deviation that depends on the sample size
D)A mean that depends on the sample size and a standard deviation of one

Accepted Answer

The distribution of Student's t has a mean of zero and a standard deviation that depends on the sample size. The standard deviation is equal to the square root of the degrees of freedom (which is the sample size minus one). Therefore, option C is the correct choice.

Question 3

How is the t distribution similar to the standard z distribution?
A)Both are discrete distributions.
B)Both are skewed distributions.
C)Both are families of distributions.
D)Both are continuous distributions.

Accepted Answer

Both the t distribution and the standard z distribution are continuous distributions. The t distribution is actually a family of distributions, similar to the normal distribution family of the standard z distribution. The t distribution is also similar to the standard z distribution in that they are both used in hypothesis testing and confidence interval estimation. However, the t distribution is more appropriate when the sample size is small or the population standard deviation is unknown, while the standard z distribution is appropriate when the sample size is large and the population standard deviation is known.

Question 4

A sample standard deviation is the best point estimate of the ___________.
A)Population range
B)Population skewness
C)Population mode
D)Population standard deviation

Accepted Answer

A sample standard deviation is a measure of the variation within a sample, but it is also considered the best point estimate of the population standard deviation, meaning it provides a good estimate of the variability in the entire population being studied. Therefore, the correct choice is D, population standard deviation.

Question 5

Knowing the population standard deviation, a 95% confidence interval infers that the population mean ___________.
A)Is between 0 and 100%
B)Is within ±1.96 standard deviations of the sample mean
C)Is within ±1.96 standard errors of the sample mean
D)Is too large

Accepted Answer

A 95% confidence interval for the population mean, when the population standard deviation is known, is calculated as the sample mean plus or minus 1.96 times the standard error of the mean.

Question 6

When we use a confidence interval to reach a conclusion about the population mean, we are applying a type of reasoning or logic called __________.
A)Descriptive statistics
B)The normal distribution
C)Statistical inference
D)Graphics

Accepted Answer

The application of confidence interval to reach a conclusion about the population mean involves making an inference or conclusion about an entire population based on a sample, which is a form of statistical inference. Descriptive statistics deals with summarizing and describing data, the normal distribution is a statistical distribution often used in inferential statistics, and graphics are visual representations of data.

Question 7

Which statement(s) is/are correct about the t distribution?
A)The mean is zero.
B)Its shape is symmetric.
C)Its dispersion is based on degrees of freedom.
D)All apply.

Accepted Answer

The t distribution is known for having a mean of zero, being symmetric in shape, and its dispersion varies with the degrees of freedom, making all the given statements correct.

Question 8

What kind of distribution is the t distribution?
A)Continuous
B)Discrete
C)Subjective
D)A z distribution

Accepted Answer

The t distribution is a continuous probability distribution.

Question 9

When using Student's t to compute an interval estimate, ___________.
A)We assume that the samples are collected from populations that are uniformly distributed
B)We estimate the population mean based on the sample mean
C)We use the z distribution
D)We assume that the samples are collected from normally distributed populations

Accepted Answer

When using Student's t to compute an interval estimate, we assume that the samples are collected from normally distributed populations. This is because the t-distribution is used for small sample sizes (n<30) and only provides accurate results if the underlying population distribution is normal.

Question 10

A local company wants to evaluate their quality of service by surveying their customers. Their budget limits the number of surveys to 100. What is their maximum error of the estimated mean quality for a 95% level of confidence and an estimated standard deviation of 5?
A)0.9604
B)0.98
C)1.96
D)5%

Accepted Answer

The answer of A local company wants to evaluate their...

Question 11

A random sample of 20 items is selected from a population. When computing a confidence interval for the population mean, what number of degrees of freedom should be used to determine the appropriate t-value?
A)20
B)19
C)21
D)25

Accepted Answer

When computing a confidence interval for the population mean, the appropriate t-value should be determined using (n-1) degrees of freedom, where n is the sample size. In this case, n=20, so the appropriate number of degrees of freedom is (20-1) = 19. Therefore, the best choice is B.

Question 12

Suppose 1,600 of 2,000 registered voters sampled said they planned to vote for the Republican candidate for president. Using the 0.95 degree of confidence, what is the interval estimate for the population proportion (to the nearest 10^th of a percent)? A)78.2% to 81.8% B)69.2% to 86.4% C)76.5% to 83.5% D)77.7% to 82.3%

Accepted Answer

The answer of Suppose 1,600 of 2,000 registered voters sampled...

Question 13

Which of the following is NOT necessary to determine how large a sample to select from a population?
A)The level of confidence in estimating the population parameter
B)The size of the population
C)The maximum allowable error in estimating the population parameter
D)An estimate of the population variation

Accepted Answer

Although the size of the population can influence sample size calculations, it is not a necessary piece of information to determine sample size. The other factors listed (level of confidence, maximum allowable error, and estimate of population variation) are all crucial in determining the appropriate sample size.

Question 14

Of the following characteristics, the t distribution and z distribution are the same in all BUT one. Which one is it?
A)Continuous
B)Symmetrical
C)Bell-shaped
D)Mean = 0, and standard deviation = 1

Accepted Answer

The t distribution has a mean of 0 but a standard deviation greater than 1, whereas the z distribution has a mean of 0 and a standard deviation of 1.

Question 15

A sample mean is the best point estimate of _______.
A)The population standard deviation
B)The population median
C)The population mean
D)The sample standard deviation

Accepted Answer

The sample mean is the best point estimate of the population mean, as the sample mean is a good representative of the central tendency of the sample, and assuming the sample is a good representation of the population, it should also be a good estimate of the population mean.

Question 16

A random sample of 42 college graduates revealed that they worked an average of 5.5 years on the job before being promoted. The sample standard deviation was 1.1 years. Using the 0.99 degree of confidence, what is the confidence interval for the population mean?
A)5.04 and 5.96
B)5.06 and 5.94
C)2.67 and 8.33
D)4.40 and 6.60

Accepted Answer

To calculate the confidence interval for the population mean with a 99% confidence level, we use the formula: $\bar{x} \pm z\frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $z$ is the z-score corresponding to the confidence level. For a 99% confidence level, the z-score is approximately 2.576. Plugging in the values: $5.5 \pm 2.576\frac{1.1}{\sqrt{42}}$, we get the interval $5.5 \pm 0.46$, which rounds to the interval (5.04, 5.96).

Question 17

The distribution of Student's t is ________.
A)Symmetrical
B)Negatively skewed
C)Positively skewed
D)A discrete probability distribution

Accepted Answer

Student's t distribution is symmetric, bell-shaped, and centered at zero.

Question 18

Mileage tests were conducted on a randomly selected sample of 100 newly developed automobile tires. The results showed that the mean tread life was 50,000 miles, with a standard deviation of 3,500 miles. What is the best estimate of the mean tread life in miles for the entire population of these tires?
A)50,000
B)3,500
C)500
D)35

Accepted Answer

The best estimate of the mean tread life in miles for the entire population of these tires is the sample mean, which is 50,000 miles. This is because the sample mean is an unbiased estimator of the population mean and is the best estimate based on the information given.

Question 19

A confidence interval for a population mean __________.
A)Estimates the population range
B)Estimates a likely interval for a population mean
C)Estimates likelihood or probability
D)Estimates the population standard deviation

Accepted Answer

A confidence interval provides a range of values within which the true population mean is likely to lie. It does not estimate the population range, probability, or standard deviation.

Question 20

When a confidence interval for a population mean is constructed from sample data, __________.
A)We can conclude that the population mean is in the interval
B)We can conclude that the population mean is not in the interval
C)We can conclude, for an infinite number of samples and corresponding confidence intervals, that the population mean is in a stated percentage of the intervals
D)We cannot make any inferences

Accepted Answer

C is correct because when constructing confidence intervals for a population mean from sample data, the interpretation is that if we were to take an infinite number of samples and construct confidence intervals in the same way, a stated percentage of these intervals would contain the population mean. This does not guarantee that the population mean is in any specific interval constructed from a single sample.

Quiz 9: Estimation and Confidence Intervals

What is the interpretation of a 96% confidence level?

The distribution of Student's t has _________.

How is the t distribution similar to the standard z distribution?

A sample standard deviation is the best point estimate of the ___________.

Knowing the population standard deviation, a 95% confidence interval infers that the population mean ___________.

When we use a confidence interval to reach a conclusion about the population mean, we are applying a type of reasoning or logic called __________.

Which statement(s) is/are correct about the t distribution?

What kind of distribution is the t distribution?

When using Student's t to compute an interval estimate, ___________.

A local company wants to evaluate their quality of service by surveying their customers. Their budget limits the number of surveys to 100. What is their maximum error of the estimated mean quality for a 95% level of confidence and an estimated standard deviation of 5?

A random sample of 20 items is selected from a population. When computing a confidence interval for the population mean, what number of degrees of freedom should be used to determine the appropriate t-value?

Suppose 1,600 of 2,000 registered voters sampled said they planned to vote for the Republican candidate for president. Using the 0.95 degree of confidence, what is the interval estimate for the population proportion (to the nearest 10^th of a percent) ?

Which of the following is NOT necessary to determine how large a sample to select from a population?

Of the following characteristics, the t distribution and z distribution are the same in all BUT one. Which one is it?

A sample mean is the best point estimate of _______.

A random sample of 42 college graduates revealed that they worked an average of 5.5 years on the job before being promoted. The sample standard deviation was 1.1 years. Using the 0.99 degree of confidence, what is the confidence interval for the population mean?

The distribution of Student's t is ________.

A confidence interval for a population mean __________.

When a confidence interval for a population mean is constructed from sample data, __________.