Question 1

The t-distribution for developing a confidence interval for a mean has degrees of freedom.
A) n + 2
B) n +1
C) n
D) n - 1
E) n - 2

Accepted Answer

The correct formula for the degrees of freedom for a t-distribution when developing a confidence interval for a mean is n - 1. This is because we are estimating the population standard deviation using the sample standard deviation, which introduces an additional source of variability and reduces the effective sample size by one.

Question 2

When we replace $\sigma$
With the sample standard deviation (s),we introduce a new source of variability and the sampling distribution becomes the:
A) t -distribution
B) F- distribution
C) chi-square distribution
D) normal distribution

Accepted Answer

When the population standard deviation ($\sigma$) is unknown and is replaced by the sample standard deviation (s), the sampling distribution of the sample mean follows a t-distribution, especially when the sample size is small.

Question 3

For a given confidence level,the procedure for controlling interval length usually begins with the specification of
A) the point estimate
B) the population standard deviation,s
C) the sample standard deviation,s
D) the interval half-length,B

Accepted Answer

The interval half-length, B, is specified to control the precision of the confidence interval, determining how wide or narrow the interval will be.

Question 4

The general form of a confidence interval is:
A) Point Estimate = Multiple × Standard Error
B) Point Estimate = Multiple +Standard Error
C) Point Estimate ± Multiple × Standard Error
D) Point Estimate = Multiple ± Standard Error

Accepted Answer

The general form of a confidence interval is Point Estimate ± Multiple × Standard Error, where the Point Estimate is the calculated statistic (such as the mean or proportion), the Standard Error is a measure of the variability of the statistic, and the Multiple is determined by the desired level of confidence and the sample size. Therefore, the correct choice is C.

Question 5

The shape of a chi-square distribution
A) is symmetric
B) is skewed to the left
C) is skewed to the right
D) depends on the sample data

Accepted Answer

The shape of a chi-square distribution is skewed to the right, and its shape depends on the degrees of freedom, not the sample data. As the degrees of freedom increase, the distribution becomes less skewed and more symmetric.

Question 6

A parameter such as $\sigma$
Is sometimes referred to as a ________ parameter,because many times we need its value even though it is not the parameter of primary interest.
A) special
B) random
C) nuisance
D) independent
E) dependent

Accepted Answer

A nuisance parameter is a parameter that is not of direct interest but must be accounted for in the analysis of other parameters. In this context, $\sigma$ (standard deviation) is often considered a nuisance parameter.

Question 7

As the sample size increases,the t-distribution becomes more similar to the ________ distribution.
A) normal
B) exponential
C) multinominal
D) chi-square
E) binomial

Accepted Answer

As the sample size increases, the t-distribution approximates more closely to the normal distribution. This is due to the central limit theorem, which states that as the sample size increases, the distribution of the sample mean approaches a normal distribution.

Question 8

When the samples we want to compare are paired in some natural way,such as pretest/posttest for each person or husband/wife pairs,a more appropriate form of analysis is to not compare two separate variables,but their .
A) difference
B) sum
C) ratio
D) total
E) product

Accepted Answer

When the samples are paired, it is more appropriate to compare the difference between the paired values, as they are likely to be more similar and have less variation than comparing two separate variables. Therefore, the best choice is to compare their differences.

Question 9

There are,generally speaking,two types of statistical inference.They are:
A) sample estimation and population estimation
B) confidence interval estimation and hypothesis testing
C) interval estimation for a mean and point estimation for a proportion
D) independent sample estimation and dependent sample estimation
E) None of these choices

Accepted Answer

The two types of statistical inference are confidence interval estimation and hypothesis testing. Confidence interval estimation involves constructing a range of values that is likely to contain the true population parameter with a certain degree of confidence. Hypothesis testing involves testing a claim or hypothesis about the population parameter using sample data. Therefore, option B is the best choice.

Question 10

Two independent samples of sizes 50 and 50 are randomly selected from two populations to test the difference between the population means, $\mu _ { 1 } - \mu _ { 2 }$
)The sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$
Is:
A) normally distributed
B) approximately normal
C) t - distributed with 98 degrees of freedom
D) chi-squared distributed with 99 degrees of freedom

Accepted Answer

The sampling distribution of the difference between two independent sample means is approximately normal if the sample sizes are large enough (typically n > 30), according to the Central Limit Theorem.

Question 11

When you calculate the sample size for a proportion,you use an estimate for the population proportion;namely $P _ { \text {est } }$
)A conservative value for n can be obtained by using $P _ { \text {est } }$
=
A) 0.01
B) 0.05
C) 0.10
D) 0.50
E) 1.00

Accepted Answer

Using $P _ { \text {est } } = 0.50$ provides the maximum variability, leading to the largest sample size, which is a conservative estimate.

Question 12

If you increase the confidence level,the confidence interval:
A) decreases
B) increases
C) stays the same
D) may increase or decrease,depending on the sample data

Accepted Answer

When you increase the confidence level, the confidence interval widens or increases. This is because a higher confidence level means you are more certain about the accuracy of your sample data and you need a larger margin of error to accommodate for any possible inaccuracies.

Question 13

Confidence intervals are a function of which of the following three things?
A) The population,the sample,and the standard deviation
B) The sample,the variable of interest,and the degrees of freedom
C) The data in the sample,the confidence level,and the sample size
D) The sampling distribution,the confidence level,and the degrees of freedom
E) The mean,median,and mode

Accepted Answer

Confidence intervals are a function of the data in the sample, the confidence level chosen by the researcher, and the sample size. The population and standard deviation are used to calculate the standard error, which is then used to construct the interval. However, they are not directly a function of the confidence interval. The variable of interest and degrees of freedom are also important concepts in statistics but are not directly related to constructing a confidence interval. The sampling distribution is used to calculate the standard error but is not part of the actual construction of the interval. The mean, median, and mode are measures of central tendency and are not directly related to constructing a confidence interval.

Quiz 8: Confidence Interval Estimation

The t-distribution for developing a confidence interval for a mean has degrees of freedom.

When we replace $\sigma$ With the sample standard deviation (s) ,we introduce a new source of variability and the sampling distribution becomes the:

For a given confidence level,the procedure for controlling interval length usually begins with the specification of

The general form of a confidence interval is:

The shape of a chi-square distribution

A parameter such as $\sigma$ Is sometimes referred to as a ________ parameter,because many times we need its value even though it is not the parameter of primary interest.

As the sample size increases,the t-distribution becomes more similar to the ________ distribution.

When the samples we want to compare are paired in some natural way,such as pretest/posttest for each person or husband/wife pairs,a more appropriate form of analysis is to not compare two separate variables,but their .

There are,generally speaking,two types of statistical inference.They are:

Two independent samples of sizes 50 and 50 are randomly selected from two populations to test the difference between the population means, $\mu _ { 1 } - \mu _ { 2 }$ ) The sampling distribution of the sample mean difference $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ Is:

When you calculate the sample size for a proportion,you use an estimate for the population proportion;namely $P _ { \text {est } }$ ) A conservative value for n can be obtained by using $P _ { \text {est } }$ =

If you increase the confidence level,the confidence interval:

Confidence intervals are a function of which of the following three things?