2025 MATH 1680 ACTUAL EXAMS QUESTIONS WITH
ANSWERS GRADED A+ 2025/2026
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
What happens to the graph of the normal curve as the standard deviation decreases? -
The graph of the normal curve compresses and becomes steeper
State the seven properties of the normal density curve. - 1. The normal curve is
symmetric about its mean.
2. Because mean = median = mode, the normal curve has a single peak and the
highest point occurs at x = μ.
3. The normal curve has inflection points at μ - σ and μ+σ
4. The area under the normal curve is 1
5. The area under the normal curve to the right of μ equals the area under the normal
curve to the left of μ which equals 1/2
6. As x increases without bound (gets larger and larger), the graph approaches, but
never reaches, the horizontal axis. As x decreases without bound (gets more and more
negative), the graph approaches, but never reaches, the horizontal axis.
7. The Empirical Rule: Approximately 68% of the area under the normal curve is
between x=μ−σ and x=μ+σ, Approximately 95% of the area is between x=μ−2σ and
x=μ+2σ, and Approximately 99.7% of the area is between x=μ−3σ and x=μ+3σ.
Suppose that a random variable X is normally distributed with mean μ and standard
deviation Give two representations for the area under the normal curve for any interval
of values of the random variable X. - 1. The proportion of the population with the
characteristic described by the interval of values
,2. The probability that a randomly selected individual from the population will have the
characteristic described by the interval of values.
Explain how to find the area to the left of x for a normally distributed random variable X,
using Table V. - If a normal random variable X has a mean different from 0 or a
standard deviation different from 1, we can transform X into a standard normal random
variable Z whose mean is 0 and standard deviation is 1. Then we can use Table V to
find the area to the left of a specified z-score, z, as shown in Figure 5, which is also the
area to the left of the value of x in the distribution of X. The graph in Figure 5 is called
the standard normal curve.
What does the notation za represent? - (pronounced z sub alpha) is the z score such
that the area under the standard normal curve to the right of zα is a.
For any continuous random variable, what is the probability of observing a specific value
of the random variable? - 0
Since the probability of observing a specific value of a continuous random variable is 0,
the following probabilities are equivalent: - P(a < X < b) = P(a <_ X < b) = P(a < X <_ b)
= P(a <_ X <_ b)
What is a normal score? - The expected z score of the data value, assuming that the
distribution of the random variable is normal. The expected z score of an observed
value depends on the number of observations in the data set.
What is a normal probability plot? - A normal probability plot is a graph that plots
observed data versus normal scores.
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
ANSWERS GRADED A+ 2025/2026
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
What happens to the graph of the normal curve as the standard deviation decreases? -
The graph of the normal curve compresses and becomes steeper
State the seven properties of the normal density curve. - 1. The normal curve is
symmetric about its mean.
2. Because mean = median = mode, the normal curve has a single peak and the
highest point occurs at x = μ.
3. The normal curve has inflection points at μ - σ and μ+σ
4. The area under the normal curve is 1
5. The area under the normal curve to the right of μ equals the area under the normal
curve to the left of μ which equals 1/2
6. As x increases without bound (gets larger and larger), the graph approaches, but
never reaches, the horizontal axis. As x decreases without bound (gets more and more
negative), the graph approaches, but never reaches, the horizontal axis.
7. The Empirical Rule: Approximately 68% of the area under the normal curve is
between x=μ−σ and x=μ+σ, Approximately 95% of the area is between x=μ−2σ and
x=μ+2σ, and Approximately 99.7% of the area is between x=μ−3σ and x=μ+3σ.
Suppose that a random variable X is normally distributed with mean μ and standard
deviation Give two representations for the area under the normal curve for any interval
of values of the random variable X. - 1. The proportion of the population with the
characteristic described by the interval of values
,2. The probability that a randomly selected individual from the population will have the
characteristic described by the interval of values.
Explain how to find the area to the left of x for a normally distributed random variable X,
using Table V. - If a normal random variable X has a mean different from 0 or a
standard deviation different from 1, we can transform X into a standard normal random
variable Z whose mean is 0 and standard deviation is 1. Then we can use Table V to
find the area to the left of a specified z-score, z, as shown in Figure 5, which is also the
area to the left of the value of x in the distribution of X. The graph in Figure 5 is called
the standard normal curve.
What does the notation za represent? - (pronounced z sub alpha) is the z score such
that the area under the standard normal curve to the right of zα is a.
For any continuous random variable, what is the probability of observing a specific value
of the random variable? - 0
Since the probability of observing a specific value of a continuous random variable is 0,
the following probabilities are equivalent: - P(a < X < b) = P(a <_ X < b) = P(a < X <_ b)
= P(a <_ X <_ b)
What is a normal score? - The expected z score of the data value, assuming that the
distribution of the random variable is normal. The expected z score of an observed
value depends on the number of observations in the data set.
What is a normal probability plot? - A normal probability plot is a graph that plots
observed data versus normal scores.
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
