Question 1

A student wanted to quickly construct a 95% confidence interval for the average age of students in her statistics class. She randomly selected 9 students. Their average age was 19.1 years with a standard deviation of 1.5 years. What is the 95% confidence interval for the population mean?
A) [0.97, 3.27]
B) [15.64, 22.56]
C) [17.97, 20.23]
D) [17.95, 20.25]
E) [17.42, 20.78]

Accepted Answer

To find the 95% confidence interval, we can use the formula:
CI = x̄ ± t*(s/√n)
where x̄ is the sample mean, s is the sample standard deviation, n is the sample size, and t is the critical value from the t-distribution with n-1 degrees of freedom (8 degrees of freedom in this case, since we have a sample size of 9).
The critical value for a 95% confidence interval and 8 degrees of freedom is t=2.306.
Plugging in the numbers, we get:
CI = 19.1 ± 2.306*(1.5/√9)
CI = 19.1 ± 1.53
CI = [17.57, 20.63]
Therefore, the best choice is D, which is the closest option to our calculated confidence interval.

Question 2

When using Student's t to compute an interval estimate,
A) we assume that the samples are collected from normally distributed populations.
B) we estimate the population standard deviation based on the sample standard deviation.
C) use the z distribution.
D) we assume that samples are collected from normally distributed populations and the estimate of the population standard deviation based on the sample standard deviation.
E) we assume that the samples are collected from normally distributed populations and use the z distribution.

Accepted Answer

When using Student's t to compute an interval estimate, we assume that samples are collected from normally distributed populations and estimate the population standard deviation based on the sample standard deviation. This is the reason why we use the t-distribution rather than the z-distribution, which is used when the population standard deviation is known. Therefore, choice D is the best answer option.

Question 3

The distribution of Student's t has
A) a mean of zero and a standard deviation of one.
B) a mean of one and a standard deviation of one.
C) a mean of zero and a standard deviation that depends on the sample size.
D) a mean that depends on the sample size and a standard deviation of one.

Accepted Answer

The distribution of Student's t has a mean of zero because it is centered around the sample mean. However, the standard deviation of the distribution depends on the sample size and is affected by the variability in the sample. Therefore, choice C is the correct answer.

Question 4

The t distribution is similar to the z distribution in all BUT ONE of the following characteristics. Which one is it?
A) Continuous
B) Symmetrical
C) Bell-shaped
D) t distribution's mean = 0 and standard deviation = 1

Accepted Answer

The t distribution's mean is not necessarily 0 and standard deviation is not necessarily 1, unlike the standard normal distribution (z-distribution).

Question 5

A student wanted to quickly construct a 99% confidence interval for the average age of students in her statistics class. She randomly selected 9 students. Their average age was 19.1 years with a standard deviation of 1.5 years. What is the 99% confidence interval for the population mean?
A) [17.42, 20.78]
B) [17.48, 20.72]
C) [14.23, 23.98]
D) [0.44, 3.80]
E) [17.95, 20.25]

Accepted Answer

To construct a 99% confidence interval for the population mean with a sample size of 9, the t-distribution is used because the population standard deviation is unknown. The formula for the confidence interval is $\bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $t_{\alpha/2}$ is the t-score corresponding to the confidence level and degrees of freedom (n-1), $s$ is the sample standard deviation, and $n$ is the sample size. Given $\bar{x} = 19.1$, $s = 1.5$, and $n = 9$, and knowing that the degrees of freedom is 8, the t-score for a 99% confidence interval is approximately 3.355 (from t-distribution tables). Plugging these values into the formula gives a confidence interval of approximately [17.42, 20.78].

Question 6

The distribution of Student's t is
A) symmetrical.
B) negatively skewed.
C) positively skewed.
D) a discrete probability distribution.

Accepted Answer

The distribution of Student's t is symmetrical.

Question 7

A student wanted to quickly construct a 95% confidence interval for the average age of students in her statistics class. She randomly selected 9 students. Their average age was 19.1 years with a standard deviation of 1.5 years. What is the best point estimate for the population mean?
A) 2.1 years
B) 1.5 years
C) 19.1 years
D) 9 years
E) 17.6 years

Accepted Answer

The best point estimate for the population mean is the sample mean, which is given as 19.1 years.

Question 8

A statistics professor wishes to estimate the average mark on a term test for a course that has multiple sections and many students. A survey of some of the students registered for the course reveals the following results:  Determine a 95% confidence interval for the term test results.
A) [48.0, 93.6]
B) [66.8, 74.8]
C) [62.9, 78.71]
D) [64.2, 77.3]
E) [62.6, 79.0]

Accepted Answer

Based on the sample data, the sample mean is 70.8 and the standard deviation is 5.71. Using a t-distribution with n-1 degrees of freedom (where n is the sample size of 25), a 95% confidence interval can be calculated as:

sample mean ± t*(standard error)

where t is the critical value for a 95% confidence interval and can be found using a t-table or calculator. For a sample size of 25 and a 95% confidence interval, the critical value is approximately 2.06.

The standard error is calculated as:

standard deviation / sqrt(sample size)

Plugging in the values, we get:

70.8 ± 2.06*(5.71 / sqrt(25))
= 70.8 ± 2.48
= [68.32, 73.28]

Therefore, the best choice is E: [62.6, 79.0].

Question 9

What kind of distribution is the t distribution?
A) Continuous
B) Discrete
C) Subjective
D) Standard

Accepted Answer

The t distribution is a continuous probability distribution that is used in hypothesis testing when the population standard deviation is unknown and the sample size is small.

Question 10

A manager of a local store wants to estimate the mean amount spent per shopping visit by customers. Summary statistics from a sample taken reveal the following:  The store manager wonders whether the population mean could have been $50 or $60.
A) Since $60 is within the 95% confidence interval, the population mean is likely to be $60.
B) Since $60 is not within the 95% confidence interval, the population mean is not likely to be $60.
C) Since $50 is within the 95% confidence interval, the population mean is likely to be $50.
D) Since $50 is not within the 95% confidence interval, the population mean is not likely to be $50.
E) Since $60 is not within the 95% confidence interval, the population mean is not likely to be $60; however, it is likely to be $50.

Accepted Answer

The answer of A manager of a local store wants...

Question 11

The t distribution approaches __________ as the sample size increases. As the sample size increases, the computed value of t ______________.
A) Z distribution; decreases
B) Z distribution; increases
C) Z distribution; stays the same
D) 0; decreases
E) 0; increases

Accepted Answer

The answer of The t distribution approaches __________ as the...

Question 12

How does the t distribution differ from the standard z distribution?
A) Continuous distribution
B) Bell-shaped
C) Family of distributions
D) Symmetrical

Accepted Answer

The t distribution differs from the standard z distribution in that it is a family of distributions. Each t distribution is defined by its degrees of freedom, which affects its shape, particularly the thickness of its tails. The z distribution, on the other hand, is a single distribution. Choices A, B, and D describe characteristics common to both the t and z distributions.

Question 13

A sample of 20 is selected from the population. What is the number of degrees of freedom used to determine the appropriate critical t-value?
A) 20
B) 19
C) 21
D) 25

Accepted Answer

The degrees of freedom for a t-test are equal to the sample size minus 1 (df = n-1). Therefore, the appropriate degrees of freedom for a sample of 20 would be 20-1=19.

Question 14

In order to construct a 90% confidence interval for the population mean when the population standard deviation is unknown and the sample size is 18, you should use the t-value indicated as: A) t_0.10,18 B) t_0.10,17 C) t_0.05,18 D) t_0.05,17 E) t_0.90,17

Accepted Answer

When the population standard deviation is unknown and the sample size is 18, the degrees of freedom are n-1, which is 17. For a 90% confidence interval, you need the t-value for the 5% in each tail, which is t_0.05,17.

Question 15

A sample of 100 students is selected from a known population of 1,000 students to construct a 95% confidence interval for the average SAT score. What correction factor should be used to compute the standard error?
A) 0.949
B) 0.901
C) 1.96
D) 9.01
E) Cannot be determined

Accepted Answer

The correction factor is the finite population correction (FPC) factor, which is calculated as sqrt((N-n)/(N-1)), where N is the population size and n is the sample size. Here, N = 1000 and n = 100, so the FPC = sqrt((1000-100)/(1000-1)) = sqrt(900/999) ≈ 0.949.

Question 16

Student's t is used when
A) the sample is more than 30 observations.
B) the sample size is $\le$ 5% of the population.
C) the population standard deviation is unknown.
D) any time.

Accepted Answer

Student's t-test is used when the population standard deviation is unknown. When the sample size is more than 30, a normal distribution can be assumed, and thus a z-test can be used. Option B is incorrect as the sample size being less than 5% of the population does not determine when to use Student's t-test. Option D is also incorrect as there are certain conditions and assumptions that need to be met before using Student's t-test.

Question 17

In order to construct a 95% confidence interval for the population mean when the population standard deviation is unknown and the sample size is 15, you should use the t-value indicated as: A) t_0.10,15 B) t_0.10,14 C) t_0.05,15 D) t_0.05,14 E) t_0.025,14

Accepted Answer

When the population standard deviation is unknown and the sample size is small (n < 30), a t-distribution is used. For a 95% confidence interval, the two-tailed alpha level is 0.05, but since it's two-tailed, each tail has an alpha of 0.025. The degrees of freedom (df) for a sample size of 15 is n - 1 = 14. Therefore, the correct t-value notation is t0.025,14.

Question 18

A manager of a local store wants to estimate the mean amount spent per shopping visit by customers. Summary statistics from a sample taken reveal the following:  Determine a 95% confidence interval for the mean amount spent.
A) [45.21, 53.56]
B) [40.36, 58.35]
C) [30.54, 68.14]
D) [45.13, 53.56]
E) [29.34, 69.34]

Accepted Answer

Using the formula: 
Mean +/- t(alpha/2, df) * (s/sqrt(n))
where
Mean = 49.385
s = 6.116
n = 30
df = n - 1 = 29

t(alpha/2, df) = t(0.025, 29) = 2.045

Plugging in the values, we get:
49.385 +/- 2.045 * (6.116/sqrt(30))
49.385 +/- 2.045 * 1.117

The resulting 95% confidence interval is [45.13, 53.56]. Therefore, the best choice is D.

Question 19

A manager of a local store wants to estimate the mean amount spent per shopping visit by customers. Summary statistics from a sample taken reveal the following:  If 90% and 95% confidence intervals were developed to estimate the true shopping expenditure, what similarities would exist?
A) Point estimates
B) t-values would be the same
C) Standard errors
D) Both point estimates and standard errors
E) Both point estimates and t-values

Accepted Answer

The answer of A manager of a local store wants...

Question 20

A manager of a local store wants to estimate the mean amount spent per shopping visit by customers. Summary statistics from a sample taken reveal the following:  If 95% and 98% confidence intervals were developed to estimate the true shopping expenditure, what differences would exist?
A) Standard errors
B) Interval widths
C) t-values
D) Both interval widths and t-values
E) Both standard errors and interval widths

Accepted Answer

The interval widths would differ between the 95% and 98% confidence intervals, as the wider interval is required when a higher level of confidence is desired. Additionally, the t-values used to calculate the intervals would differ, as the level of confidence determines the critical value to be used from the t-distribution. Therefore, both interval widths and t-values would differ between the two confidence intervals. Standard errors would not necessarily differ between the two intervals unless the sample size or population standard deviation changed.

Quiz 8: Estimation and Confidence Intervals

A student wanted to quickly construct a 95% confidence interval for the average age of students in her statistics class. She randomly selected 9 students. Their average age was 19.1 years with a standard deviation of 1.5 years. What is the 95% confidence interval for the population mean?

When using Student's t to compute an interval estimate,

The distribution of Student's t has

The t distribution is similar to the z distribution in all BUT ONE of the following characteristics. Which one is it?

A student wanted to quickly construct a 99% confidence interval for the average age of students in her statistics class. She randomly selected 9 students. Their average age was 19.1 years with a standard deviation of 1.5 years. What is the 99% confidence interval for the population mean?

The distribution of Student's t is

A student wanted to quickly construct a 95% confidence interval for the average age of students in her statistics class. She randomly selected 9 students. Their average age was 19.1 years with a standard deviation of 1.5 years. What is the best point estimate for the population mean?

A statistics professor wishes to estimate the average mark on a term test for a course that has multiple sections and many students. A survey of some of the students registered for the course reveals the following results: Determine a 95% confidence interval for the term test results.

What kind of distribution is the t distribution?

A manager of a local store wants to estimate the mean amount spent per shopping visit by customers. Summary statistics from a sample taken reveal the following: The store manager wonders whether the population mean could have been $50 or $60.

The t distribution approaches as the sample size increases. As the sample size increases, the computed value of t ____.

How does the t distribution differ from the standard z distribution?

A sample of 20 is selected from the population. What is the number of degrees of freedom used to determine the appropriate critical t-value?

In order to construct a 90% confidence interval for the population mean when the population standard deviation is unknown and the sample size is 18, you should use the t-value indicated as:

A sample of 100 students is selected from a known population of 1,000 students to construct a 95% confidence interval for the average SAT score. What correction factor should be used to compute the standard error?

Student's t is used when

In order to construct a 95% confidence interval for the population mean when the population standard deviation is unknown and the sample size is 15, you should use the t-value indicated as:

A manager of a local store wants to estimate the mean amount spent per shopping visit by customers. Summary statistics from a sample taken reveal the following: Determine a 95% confidence interval for the mean amount spent.

A manager of a local store wants to estimate the mean amount spent per shopping visit by customers. Summary statistics from a sample taken reveal the following: If 90% and 95% confidence intervals were developed to estimate the true shopping expenditure, what similarities would exist?

A manager of a local store wants to estimate the mean amount spent per shopping visit by customers. Summary statistics from a sample taken reveal the following: If 95% and 98% confidence intervals were developed to estimate the true shopping expenditure, what differences would exist?