ELEMENTARY STATISTICS EXAM 3 ACTUAL
TEST QUESTIONS AND EXPERT REVIEWED
SOLUTIONS
●● How does a prediction interval differ from a confidence interval?
Answer: A confidence interval estimates a population parameter (like p,
µ, or σ), whereas a prediction interval estimates a single predicted value
of y.
●● What is the primary requirement for the distribution of y-values in a
prediction interval?
Answer: For each fixed value of x, the corresponding sample values of y
must be normally distributed about the regression line with the same
variance.
●● What are the three main reasons why prediction intervals are often
wide?
Answer: Individual outcomes vary more than averages, small sample
sizes increase uncertainty, and predicting far from the center of x
increases uncertainty.
●● What does the margin of error (E) in a prediction interval formula
account for?
,Answer: It accounts for the uncertainty in the regression estimate, the
sample size, and the distance of the given x-value from the mean of x.
●● What degrees of freedom are used for the t-score in a prediction
interval calculation?
Answer: n - 2 degrees of freedom.
●● What does the standard error of estimate (se) measure in regression?
Answer: It measures the dispersion of the observed y-values about the
regression line.
●● How do you interpret a 95% prediction interval?
Answer: We are 95% confident that the actual value of y for a specific x
will fall between the calculated lower and upper limits.
●● What is the definition of dependent samples in statistics?
Answer: Samples are dependent when the data consist of matched pairs,
such as before/after measurements from the same subjects.
●● Why is using matched pairs generally preferred in experimental
design?
Answer: It reduces variability by controlling for subject-to-subject
differences, as each subject acts as their own control.
,●● How is a matched-pairs problem converted into a one-sample t-test?
Answer: By calculating the difference (d) for each pair and analyzing the
list of differences as a single sample.
●● What is the null hypothesis for a matched-pairs t-test?
Answer: H0: µd = 0, meaning there is no mean difference between the
two measurements.
●● How is the individual difference (d) calculated for matched pairs?
Answer: d = (first measurement) - (second measurement).
●● What does µd represent in matched-pairs notation?
Answer: The mean value of the differences (d) for the population of all
matched pairs.
●● What are the three requirements for performing inferences on
matched pairs?
Answer: The samples must be dependent (matched pairs), they must be a
simple random sample, and the sample size must be large (n > 30) or the
differences must be approximately normally distributed.
●● Are matched-pairs t-test methods sensitive to departures from
normality?
, Answer: No, these methods are robust, meaning the normality
requirement is loose.
●● What is the purpose of a confidence interval for matched pairs?
Answer: To estimate the mean of the population of all differences (µd)
between the two dependent measurements.
●● What does 'sd' represent in the context of matched pairs?
Answer: The standard deviation of the differences (d) for the paired
sample data.
●● What is the effect of a small sample size on a prediction interval?
Answer: It increases the uncertainty, resulting in a wider interval.
●● Why does predicting far from the mean of x (x-bar) increase interval
width?
Answer: Because the regression line is most accurate near the center of
the data; as you move away from the mean, the uncertainty in the
prediction increases.
●● What is the difference between a one-sample t-test and a matched-
pairs t-test?
Answer: Mathematically, they are identical; the matched-pairs test is
simply a one-sample t-test performed on the list of differences.
TEST QUESTIONS AND EXPERT REVIEWED
SOLUTIONS
●● How does a prediction interval differ from a confidence interval?
Answer: A confidence interval estimates a population parameter (like p,
µ, or σ), whereas a prediction interval estimates a single predicted value
of y.
●● What is the primary requirement for the distribution of y-values in a
prediction interval?
Answer: For each fixed value of x, the corresponding sample values of y
must be normally distributed about the regression line with the same
variance.
●● What are the three main reasons why prediction intervals are often
wide?
Answer: Individual outcomes vary more than averages, small sample
sizes increase uncertainty, and predicting far from the center of x
increases uncertainty.
●● What does the margin of error (E) in a prediction interval formula
account for?
,Answer: It accounts for the uncertainty in the regression estimate, the
sample size, and the distance of the given x-value from the mean of x.
●● What degrees of freedom are used for the t-score in a prediction
interval calculation?
Answer: n - 2 degrees of freedom.
●● What does the standard error of estimate (se) measure in regression?
Answer: It measures the dispersion of the observed y-values about the
regression line.
●● How do you interpret a 95% prediction interval?
Answer: We are 95% confident that the actual value of y for a specific x
will fall between the calculated lower and upper limits.
●● What is the definition of dependent samples in statistics?
Answer: Samples are dependent when the data consist of matched pairs,
such as before/after measurements from the same subjects.
●● Why is using matched pairs generally preferred in experimental
design?
Answer: It reduces variability by controlling for subject-to-subject
differences, as each subject acts as their own control.
,●● How is a matched-pairs problem converted into a one-sample t-test?
Answer: By calculating the difference (d) for each pair and analyzing the
list of differences as a single sample.
●● What is the null hypothesis for a matched-pairs t-test?
Answer: H0: µd = 0, meaning there is no mean difference between the
two measurements.
●● How is the individual difference (d) calculated for matched pairs?
Answer: d = (first measurement) - (second measurement).
●● What does µd represent in matched-pairs notation?
Answer: The mean value of the differences (d) for the population of all
matched pairs.
●● What are the three requirements for performing inferences on
matched pairs?
Answer: The samples must be dependent (matched pairs), they must be a
simple random sample, and the sample size must be large (n > 30) or the
differences must be approximately normally distributed.
●● Are matched-pairs t-test methods sensitive to departures from
normality?
, Answer: No, these methods are robust, meaning the normality
requirement is loose.
●● What is the purpose of a confidence interval for matched pairs?
Answer: To estimate the mean of the population of all differences (µd)
between the two dependent measurements.
●● What does 'sd' represent in the context of matched pairs?
Answer: The standard deviation of the differences (d) for the paired
sample data.
●● What is the effect of a small sample size on a prediction interval?
Answer: It increases the uncertainty, resulting in a wider interval.
●● Why does predicting far from the mean of x (x-bar) increase interval
width?
Answer: Because the regression line is most accurate near the center of
the data; as you move away from the mean, the uncertainty in the
prediction increases.
●● What is the difference between a one-sample t-test and a matched-
pairs t-test?
Answer: Mathematically, they are identical; the matched-pairs test is
simply a one-sample t-test performed on the list of differences.
