2025 MATH 1680 EXAMS PRACTICE QUESTIONS WITH
ANSWERS GRADED A+ 2025/2026
Explain why the t-distribution has less spread as the number of degrees of freedom
increases. - The t-distribution has less spread as the degrees of freedom increase
because, as n increases, s becomes closer to
σ
by the law of large numbers.
What type of data are needed to construct a confidence interval for a population
proportion, p? - Qualitative with 2 outcomes
Besides the fact that the sample must be obtained by simple random sampling or
through a randomized experiment, list the two conditions that must be met when
constructing a confidence interval for a population proportion, p. - np^ (1-p^) > 10 and
n<0.05N
What type of data are needed to construct a confidence interval for a population mean,
? - Quantitative
Besides the facts that the sample must be obtained by simple random sampling or
through a randomized experiment and that the sample size must be small relative to the
size of the population, what other condition must be satisfied? - n > 30 (good to go)
n < 30 we create a box plot/
Statistics are _________ variables because the value of a statistic varies from sample
to sample. - random
Remember, when we describe a distribution, we do so in terms of its ___________ -
shape, center, and spread
What is the sampling distribution of a statistic? - The sampling distribution of a statistic
is a probability distribution for all possible values of the statistic computed from a
sample of size n.
What is the sampling distribution of the sample mean ? - The sampling distribution of
the sample mean x- is the probability distribution of all possible values of the random
variable x- computed from a sample of size n from a population with mean μ and
standard deviation σ.
List the three steps for determining the sampling distribution of the sample mean. - Step
1: Obtain a simple random sample of size n
Step 2: Compute the sample mean
Step 3: Assuming that we are sampling from a finite population, repeat Steps 1 and 2
until all distinct simple random samples of size n have been obtained.
,Note: Once a particular sample is obtained, it cannot be obtained a second time
Describe the shape of the distribution of the sample mean as the sample size increases.
- As the sample size increases, the shape of the distribution becomes approximately
normal.
What does the mean of the distribution of the sample mean, x bar, equal? - The mean
of the distribution of the sample mean will equal the mean of the parent population.
As the sample size n increases, what happens to the standard deviation of the
distribution of the sample mean? - The standard deviation decreases. The standard
deviation of the distribution of the sample mean is less than the standard deviation of
the population and the larger the sample size, n, the smaller the standard deviation of
the distribution of the sample mean.
What is the standard error of the mean? - The standard deviation of the sampling
distribution of the mean
State the Central Limit Theorem. - The shape of the distribution of the sample mean
becomes approximately normal as the sample size n increases, regardless of the shape
of the underlying population.
How large does the sample size need to be before we can say that the sampling
distribution of is approximately normal? - The answer depends on the shape of the
distribution of the underlying population. Distributions that are highly skewed will require
a larger sample size for the distribution of to become approximately normal.
State the rule of thumb for invoking the Central Limit Theorem. - If the distribution of the
population is unknown or not normal, then the distribution of the sample mean is
approximately normal provided that the sample size is greater than or equal to 30
To cut the standard error of the mean in half, the sample size must be doubled - False.
The sample size must be increased by a factor of four to cut the standard error in half.
Define the sample proportion, . - Suppose that a random sample of size n is obtained
from a population in which each individual either does or does not have a certain
characteristic. The sample proportion, denoted pˆ (read "p-hat"), is given by
pˆ=x/n
where x is the number of individuals in the sample with the specified characteristic. The
sample proportion, pˆ, is a statistic that estimates the population proportion, p.
Under what conditions is the shape of the sampling distribution of p^ approximately
normal? - The shape of the sampling distribution of p^ is approximately normal provided
np(1-p) >_ 10
, We discuss a _________ distribution to see the relation between area and probability. -
uniform
Things are not as easy for continuous random variables. - Because an infinite number
of outcomes are possible for continuous random variables, the probability of observing
one particular value is zero.
A probability density function (pdf) is an equation used to compute probabilities of
continuous random variables. It must satisfy the following two properties: - 1. The total
area under the graph of the equation over all possible values of the random variable
must equal 1
2. The height of the graph of the equation must be greater than or equal to 0 for all
possible values of the random variable. That is, the graph of the equation must lie on or
above the horizontal axis for all possible values of the random variable.
If the possible values of a uniform density function go from 0 through n, what is the
height of the rectangle? - 1/n
What does the area under the graph of a probability density function over an interval
represent? - The probability of observing a value of the random variable in that interval.
Not all continuous random variables follow a uniform distribution. - For example,
continuous random variables such as IQ scores and birth weights of babies have
distributions that are symmetric and bell-shaped.
What does it mean to say that a continuous random variable is normally distributed? - A
continuous random variable is normally distributed or has a normal probability
distribution, if its relative frequency histogram has the shape of a normal curve
What value of x is associated with the peak of a normal curve? - the mean
What values of x are associated with the inflection points of a normal curve? - mean +
standard deviation and mean - standard deviation
What happens to the graph as the standard deviation increases? What happens to the
graph as the standard deviation decreases? - As the standard deviation increases, the
curve gets flatter
As the standard deviation decreases, the peak value of f(x) increases aka the midpoint
gets higher
What happens to the graph as the mean increases? What happens to the graph as the
mean decreases? - -As the mean increases, the graph of the normal curve slides right
and x increases
-As the mean decreases, the graph of the normal curve slides left and x decreases
ANSWERS GRADED A+ 2025/2026
Explain why the t-distribution has less spread as the number of degrees of freedom
increases. - The t-distribution has less spread as the degrees of freedom increase
because, as n increases, s becomes closer to
σ
by the law of large numbers.
What type of data are needed to construct a confidence interval for a population
proportion, p? - Qualitative with 2 outcomes
Besides the fact that the sample must be obtained by simple random sampling or
through a randomized experiment, list the two conditions that must be met when
constructing a confidence interval for a population proportion, p. - np^ (1-p^) > 10 and
n<0.05N
What type of data are needed to construct a confidence interval for a population mean,
? - Quantitative
Besides the facts that the sample must be obtained by simple random sampling or
through a randomized experiment and that the sample size must be small relative to the
size of the population, what other condition must be satisfied? - n > 30 (good to go)
n < 30 we create a box plot/
Statistics are _________ variables because the value of a statistic varies from sample
to sample. - random
Remember, when we describe a distribution, we do so in terms of its ___________ -
shape, center, and spread
What is the sampling distribution of a statistic? - The sampling distribution of a statistic
is a probability distribution for all possible values of the statistic computed from a
sample of size n.
What is the sampling distribution of the sample mean ? - The sampling distribution of
the sample mean x- is the probability distribution of all possible values of the random
variable x- computed from a sample of size n from a population with mean μ and
standard deviation σ.
List the three steps for determining the sampling distribution of the sample mean. - Step
1: Obtain a simple random sample of size n
Step 2: Compute the sample mean
Step 3: Assuming that we are sampling from a finite population, repeat Steps 1 and 2
until all distinct simple random samples of size n have been obtained.
,Note: Once a particular sample is obtained, it cannot be obtained a second time
Describe the shape of the distribution of the sample mean as the sample size increases.
- As the sample size increases, the shape of the distribution becomes approximately
normal.
What does the mean of the distribution of the sample mean, x bar, equal? - The mean
of the distribution of the sample mean will equal the mean of the parent population.
As the sample size n increases, what happens to the standard deviation of the
distribution of the sample mean? - The standard deviation decreases. The standard
deviation of the distribution of the sample mean is less than the standard deviation of
the population and the larger the sample size, n, the smaller the standard deviation of
the distribution of the sample mean.
What is the standard error of the mean? - The standard deviation of the sampling
distribution of the mean
State the Central Limit Theorem. - The shape of the distribution of the sample mean
becomes approximately normal as the sample size n increases, regardless of the shape
of the underlying population.
How large does the sample size need to be before we can say that the sampling
distribution of is approximately normal? - The answer depends on the shape of the
distribution of the underlying population. Distributions that are highly skewed will require
a larger sample size for the distribution of to become approximately normal.
State the rule of thumb for invoking the Central Limit Theorem. - If the distribution of the
population is unknown or not normal, then the distribution of the sample mean is
approximately normal provided that the sample size is greater than or equal to 30
To cut the standard error of the mean in half, the sample size must be doubled - False.
The sample size must be increased by a factor of four to cut the standard error in half.
Define the sample proportion, . - Suppose that a random sample of size n is obtained
from a population in which each individual either does or does not have a certain
characteristic. The sample proportion, denoted pˆ (read "p-hat"), is given by
pˆ=x/n
where x is the number of individuals in the sample with the specified characteristic. The
sample proportion, pˆ, is a statistic that estimates the population proportion, p.
Under what conditions is the shape of the sampling distribution of p^ approximately
normal? - The shape of the sampling distribution of p^ is approximately normal provided
np(1-p) >_ 10
, We discuss a _________ distribution to see the relation between area and probability. -
uniform
Things are not as easy for continuous random variables. - Because an infinite number
of outcomes are possible for continuous random variables, the probability of observing
one particular value is zero.
A probability density function (pdf) is an equation used to compute probabilities of
continuous random variables. It must satisfy the following two properties: - 1. The total
area under the graph of the equation over all possible values of the random variable
must equal 1
2. The height of the graph of the equation must be greater than or equal to 0 for all
possible values of the random variable. That is, the graph of the equation must lie on or
above the horizontal axis for all possible values of the random variable.
If the possible values of a uniform density function go from 0 through n, what is the
height of the rectangle? - 1/n
What does the area under the graph of a probability density function over an interval
represent? - The probability of observing a value of the random variable in that interval.
Not all continuous random variables follow a uniform distribution. - For example,
continuous random variables such as IQ scores and birth weights of babies have
distributions that are symmetric and bell-shaped.
What does it mean to say that a continuous random variable is normally distributed? - A
continuous random variable is normally distributed or has a normal probability
distribution, if its relative frequency histogram has the shape of a normal curve
What value of x is associated with the peak of a normal curve? - the mean
What values of x are associated with the inflection points of a normal curve? - mean +
standard deviation and mean - standard deviation
What happens to the graph as the standard deviation increases? What happens to the
graph as the standard deviation decreases? - As the standard deviation increases, the
curve gets flatter
As the standard deviation decreases, the peak value of f(x) increases aka the midpoint
gets higher
What happens to the graph as the mean increases? What happens to the graph as the
mean decreases? - -As the mean increases, the graph of the normal curve slides right
and x increases
-As the mean decreases, the graph of the normal curve slides left and x decreases
