MATH 1680/ 317 Qs with Certified Ans/ 2025
What is at the "heart" of hypothesis testing in statistics? - Answer: Make an assumption about reality,
and collect sample evidence to determine whether it contradicts the assumption.
What is a hypothesis? - Answer: A statement regarding a characteristic of one or more populations.
Why do we test statements about a population parameter using sample data? - Answer: Because it is
usually impossible or impractical to gain access to the entire population.
State the definition of hypothesis testing. - Answer: A procedure based on sample evidence and
probability, used to test statements regarding a characteristic of one or more populations.
Constructing a Confidence Interval for a Population Proportion using StatCrunch - Answer: - Stat >
proportion stats > one sample > with summary
-Enter number of successes (x) and number of observations (n)
- Choose the confidence interval radio button, enter the level of confidence (leave method as Standard-
Wald)
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,- Click compute
As the sample size, n, increases, what happens to the margin of error? - Answer: As the sample size, n,
increases, the margin of error decreases. Therefore, larger sample sizes will result in narrower
confidence intervals.
If the sample size is quadrupled, the margin of error will be cut in half. - Answer: True
If the normality condition is not satisfied, how does the proportion of intervals that capture the
parameter compare to the level of confidence? - Answer: When the normality condition is not satisfied,
the proportion of intervals that capture the parameter is below the level of confidence.
If the normality requirement is not satisfied (that is, np(1−p) is not at least 10), then a 95% confidence
interval about the population proportion will include the population proportion in ________ 95% of the
intervals. - Answer: less than
What is the point estimate for a population mean mu? - Answer: The sample mean x-
What was the name of the brewery that Gosset worked for? What pseudonym did he choose to publish
his results about a model that accounts for the additional variability introduced by using s in place of
when determining margin of error? - Answer: The Guinness Brewery.
Chose Student as his pseudonym
State six properties of the t-distribution. - Answer: 1. The t distribution is different for different degrees
of freedom
2. The t distribution is centered at 0 and is symmetric about 0
3. The area under the curve is 1. The area under the curve to the right of 0 equals the area under the
curve to the left of 0 which equals 1/2
4. As t increases or decreases without bound, the graph approaches, but never equals 0
5. The area in the tails of the t-distribution is a little greater than the area in the tails of the standard
normal distribution, because we are using s as an estimate of sigma, thereby introducing further
variability into the t-statistic
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,6. As the sample size n increases, the density curve of t gets closer to the standard normal density curve.
This result occurs because, as the sample size increases, the values of s get closer to the value of sigma,
by the Law of Large Numbers
Put the following in order for the most area in the tails of the distribution.
(a) Standard Normal Distribution
(b) Student's t-Distribution with 15 degrees of freedom.
(c) Student's t-Distribution with 30 degrees of freedom. - Answer: b, c, a
What does tα represent? - Answer: The t value whose area under the t-distribution to the right of tα is
(α) alpha. The shape of the t-distribution depends on the sample size, n. Therefore, the value of tα
depends not only on α, but also on the degrees of freedom, n−1. In Table VII, the far left column gives
the degrees of freedom (df). The top row represents the area under the t-distribution to the right of
some t-value
List the three conditions required for constructing a confidence interval for a population mean μ -
Answer: 1. sample data come from a simple random sample or randomized experiment
2. sample size is small relative to the population size (n < 0.05N)
3. the data come from a population that is normally distributed with no outliers or the sample size is
large
What does it mean when we say that the procedure for constructing a confidence interval is robust? -
Answer: Notice that a confidence interval about μ can be computed for non-normal populations even
though Student's t-distribution requires a normal population. This is because the procedure for
constructing the confidence interval is robust—it is accurate despite minor departures from normality
Because the sample mean and sample standard deviation are not resistant to outliers, sample data
should always be inspected for serious departures from normality and for outliers. What tools can be
used to check for serious departures from normality and for outliers? - Answer: If a data set has outliers,
the confidence interval is not accurate because neither the sample mean nor the sample standard
deviation is resistant to outliers. Sample data should always be inspected for serious departures from
normality and for outliers. This is easily done with normal probability plots and boxplots.
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, The t-distribution gives a ________ critical value than the z-distribution, so the width of the confidence
interval is wider when it is constructed using Student's t-distribution. - Answer: larger
As the sample size increases, the margin of error decreases - Answer: True
As the level of confidence increases, the size of the interval increases - Answer: True
What happens to the proportion of intervals that capture the population mean as the sample size
increases? - Answer: They increase too.
When constructing 95% confidence intervals for the mean when the parent population is right skewed
and the sample size is small, the proportion of intervals that include the population mean is
_______ 0.95. - Answer: below
When constructing 95% confidence intervals for the mean when the parent population is right skewed
and the sample size is small, the proportion of intervals that include the population mean approaches
_______ as the sample size, n, increases. - Answer: 0.95
In a survey conducted by the Gallup Organization, 1100 adult Americans were asked how many hours
they worked in the previous week. Based on the results, a 95% confidence interval for the mean number
of hours worked had a lower bound of 42.7 and an upper bound of 44.5. Provide two recommendations
for decreasing the margin of error of the interval. - Answer: Decrease the confidence level and increase
the sample size
Explain why the t-distribution has less spread as the number of degrees of freedom increases. - Answer:
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.
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What is at the "heart" of hypothesis testing in statistics? - Answer: Make an assumption about reality,
and collect sample evidence to determine whether it contradicts the assumption.
What is a hypothesis? - Answer: A statement regarding a characteristic of one or more populations.
Why do we test statements about a population parameter using sample data? - Answer: Because it is
usually impossible or impractical to gain access to the entire population.
State the definition of hypothesis testing. - Answer: A procedure based on sample evidence and
probability, used to test statements regarding a characteristic of one or more populations.
Constructing a Confidence Interval for a Population Proportion using StatCrunch - Answer: - Stat >
proportion stats > one sample > with summary
-Enter number of successes (x) and number of observations (n)
- Choose the confidence interval radio button, enter the level of confidence (leave method as Standard-
Wald)
Page 1 of 41
,- Click compute
As the sample size, n, increases, what happens to the margin of error? - Answer: As the sample size, n,
increases, the margin of error decreases. Therefore, larger sample sizes will result in narrower
confidence intervals.
If the sample size is quadrupled, the margin of error will be cut in half. - Answer: True
If the normality condition is not satisfied, how does the proportion of intervals that capture the
parameter compare to the level of confidence? - Answer: When the normality condition is not satisfied,
the proportion of intervals that capture the parameter is below the level of confidence.
If the normality requirement is not satisfied (that is, np(1−p) is not at least 10), then a 95% confidence
interval about the population proportion will include the population proportion in ________ 95% of the
intervals. - Answer: less than
What is the point estimate for a population mean mu? - Answer: The sample mean x-
What was the name of the brewery that Gosset worked for? What pseudonym did he choose to publish
his results about a model that accounts for the additional variability introduced by using s in place of
when determining margin of error? - Answer: The Guinness Brewery.
Chose Student as his pseudonym
State six properties of the t-distribution. - Answer: 1. The t distribution is different for different degrees
of freedom
2. The t distribution is centered at 0 and is symmetric about 0
3. The area under the curve is 1. The area under the curve to the right of 0 equals the area under the
curve to the left of 0 which equals 1/2
4. As t increases or decreases without bound, the graph approaches, but never equals 0
5. The area in the tails of the t-distribution is a little greater than the area in the tails of the standard
normal distribution, because we are using s as an estimate of sigma, thereby introducing further
variability into the t-statistic
Page 2 of 41
,6. As the sample size n increases, the density curve of t gets closer to the standard normal density curve.
This result occurs because, as the sample size increases, the values of s get closer to the value of sigma,
by the Law of Large Numbers
Put the following in order for the most area in the tails of the distribution.
(a) Standard Normal Distribution
(b) Student's t-Distribution with 15 degrees of freedom.
(c) Student's t-Distribution with 30 degrees of freedom. - Answer: b, c, a
What does tα represent? - Answer: The t value whose area under the t-distribution to the right of tα is
(α) alpha. The shape of the t-distribution depends on the sample size, n. Therefore, the value of tα
depends not only on α, but also on the degrees of freedom, n−1. In Table VII, the far left column gives
the degrees of freedom (df). The top row represents the area under the t-distribution to the right of
some t-value
List the three conditions required for constructing a confidence interval for a population mean μ -
Answer: 1. sample data come from a simple random sample or randomized experiment
2. sample size is small relative to the population size (n < 0.05N)
3. the data come from a population that is normally distributed with no outliers or the sample size is
large
What does it mean when we say that the procedure for constructing a confidence interval is robust? -
Answer: Notice that a confidence interval about μ can be computed for non-normal populations even
though Student's t-distribution requires a normal population. This is because the procedure for
constructing the confidence interval is robust—it is accurate despite minor departures from normality
Because the sample mean and sample standard deviation are not resistant to outliers, sample data
should always be inspected for serious departures from normality and for outliers. What tools can be
used to check for serious departures from normality and for outliers? - Answer: If a data set has outliers,
the confidence interval is not accurate because neither the sample mean nor the sample standard
deviation is resistant to outliers. Sample data should always be inspected for serious departures from
normality and for outliers. This is easily done with normal probability plots and boxplots.
Page 3 of 41
, The t-distribution gives a ________ critical value than the z-distribution, so the width of the confidence
interval is wider when it is constructed using Student's t-distribution. - Answer: larger
As the sample size increases, the margin of error decreases - Answer: True
As the level of confidence increases, the size of the interval increases - Answer: True
What happens to the proportion of intervals that capture the population mean as the sample size
increases? - Answer: They increase too.
When constructing 95% confidence intervals for the mean when the parent population is right skewed
and the sample size is small, the proportion of intervals that include the population mean is
_______ 0.95. - Answer: below
When constructing 95% confidence intervals for the mean when the parent population is right skewed
and the sample size is small, the proportion of intervals that include the population mean approaches
_______ as the sample size, n, increases. - Answer: 0.95
In a survey conducted by the Gallup Organization, 1100 adult Americans were asked how many hours
they worked in the previous week. Based on the results, a 95% confidence interval for the mean number
of hours worked had a lower bound of 42.7 and an upper bound of 44.5. Provide two recommendations
for decreasing the margin of error of the interval. - Answer: Decrease the confidence level and increase
the sample size
Explain why the t-distribution has less spread as the number of degrees of freedom increases. - Answer:
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.
Page 4 of 41
