Question 1

Which statement(s)is(are)correct about the t-distribution?
A)The mean is zero.
B)Its shape is symmetric.
C)Its dispersion is based on degrees of freedom.
D)All apply.

Accepted Answer

A) The mean of the t-distribution is not always zero, it depends on the degrees of freedom.
B) The shape of the t-distribution is approximately symmetric, but it can be slightly skewed depending on the sample size and underlying population distribution.
C) The dispersion of the t-distribution is determined by the degrees of freedom, as it becomes more dispersed with fewer degrees of freedom.

Question 2

When a confidence interval for a population mean is constructed from sample data,________.
A)we can conclude that the population mean is in the interval
B)we can conclude that the population mean is not in the interval
C)we can conclude,for an infinite number of samples and corresponding confidence intervals,that the population mean is in a stated percentage of the intervals
D)we cannot make any inferences

Accepted Answer

Option C is correct because when constructing confidence intervals for a population mean from sample data, the interpretation is that if we were to take an infinite number of samples and construct confidence intervals in the same way, a stated percentage of these intervals would contain the population mean. This is the essence of what a confidence interval represents.

Question 3

A sample mean is the best point estimate of ________.
A)the population standard deviation
B)the population median
C)the population mean
D)the sample standard deviation

Accepted Answer

A point estimate is an estimate of a population parameter based on a sample statistic. The sample mean is the best point estimate of the population mean, since it provides the most likely estimate of the true population mean based on the available data.

Question 4

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, meaning it has equal probabilities of positive and negative deviations from the mean. This is a characteristic of a normal distribution.

Question 5

Which of the following is not necessary to determine how large a sample to select from a population?
A)The level of confidence in estimating the population parameter
B)The size of the population
C)The maximum allowable error in estimating the population parameter
D)An estimate of the population variation

Accepted Answer

The size of the population is often not necessary to determine the sample size for many statistical sampling methods, especially when the population is large. The level of confidence, maximum allowable error, and an estimate of the population variation are more critical factors in determining the appropriate sample size.

Question 6

Suppose 1,600 of 2,000 registered voters sampled said they planned to vote for the Republican candidate for president. Using the 0.95 degree of confidence,what is the interval estimate for the population proportion (to the nearest 10th of a percent)?
A)78.2% to 81.8%
B)69.2% to 86.4%
C)76.5% to 83.5%
D)77.7% to 82.3%

Accepted Answer

The sample proportion (p) is 1,600/2,000 = 0.8 or 80%. The formula for the confidence interval is $p \pm Z \sqrt{\frac{p(1-p)}{n}}$, where Z is the Z-score for the confidence level (for 95% confidence, Z ≈ 1.96), p is the sample proportion, and n is the sample size. Plugging in the numbers: $0.8 \pm 1.96 \sqrt{\frac{0.8(0.2)}{2000}}$, we get the interval 0.782 to 0.818, or 78.2% to 81.8%.

Question 7

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

Accepted Answer

The t-distribution is a continuous probability distribution, as it can take on any value between negative infinity and positive infinity.

Question 8

Of the following characteristics,the t-distribution and z-distribution are the same in all but one. Which one is it?
A)Continuous
B)Symmetrical
C)Bell-shaped
D)Mean = 0,and standard deviation = 1

Accepted Answer

The t-distribution has a mean of 0, but its standard deviation is not always 1. It varies based on the sample size and degrees of freedom used to calculate it. The z-distribution always has a mean of 0 and a standard deviation of 1. Therefore, D is the only characteristic that is not the same for both distributions.

Question 9

What is the interpretation of a 96% confidence level?
A)There's a 96% chance that the given interval includes the true value of the population parameter.
B)Approximately 96 out of 100 such intervals would include the true value of the population parameter.
C)There's a 4% chance that the given interval does not include the true value of the population parameter.
D)The interval contains 96% of all sample means.

Accepted Answer

A 96% confidence level means that if we were to repeat the sampling process many times, we would expect approximately 96 out of 100 intervals to contain the true value of the population parameter. It does not mean that there is a probability or chance associated with the given interval, nor does it refer to the percentage of sample means contained in the interval.

Question 10

A confidence interval for a population mean ________.
A)estimates the population range
B)estimates a likely interval for a population mean
C)estimates likelihood or probability
D)estimates the population standard deviation

Accepted Answer

A confidence interval for a population mean estimates a likely interval for the population mean with a certain level of confidence, typically 95% or 99%. It does not estimate the population range, likelihood or probability, or the population standard deviation.

Question 11

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

Accepted Answer

When using Student's t to compute an interval estimate, we assume that the samples are collected from normally distributed populations. The assumptions of normality and independence of sample observations are required for the validity of the t-test. Therefore, option D is the best choice. Option A is incorrect as uniform distribution is a different concept. Option B is partially correct as we do estimate the population mean based on the sample mean, but it is not the defining feature of using the t-distribution. Option C is incorrect as we use the t-distribution, not the z-distribution, because the population standard deviation is not known and must be estimated from the sample data.

Question 12

A sample standard deviation is the best point estimate of the ________.
A)population range
B)population skewness
C)population mode
D)population standard deviation

Accepted Answer

A sample standard deviation is the best point estimate of the population standard deviation. It provides an estimate of the amount of variation in the population based on the data from the sample. It is not a point estimate for population range, skewness, or mode.

Question 13

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 and a standard deviation that depends on the sample size, specifically the degrees of freedom (n-1). This is because the t-distribution is used for calculating confidence intervals and hypothesis tests for small sample sizes when the population standard deviation is unknown. The sample size affects the degree of uncertainty, thus varying the standard deviation of the t-distribution.

Question 14

A random sample of 20 items is selected from a population. When computing a confidence interval for the population mean,what number of degrees of freedom should be used to determine the appropriate t-value?
A)20
B)19
C)21
D)25

Accepted Answer

When computing a confidence interval for the population mean with a sample size of n=20, we should use n-1 degrees of freedom, which is 19. This is because we are estimating the population variance using the sample variance, which creates one less degree of freedom in the calculation.

Question 15

How is the t-distribution similar to the standard z-distribution?
A)Both are discrete distributions.
B)Both are skewed distributions.
C)Both are families of distributions.
D)Both are continuous distributions.

Accepted Answer

Both the t-distribution and the standard z-distribution are continuous distributions. However, they differ in their shape and properties. The t-distribution has heavier tails and a more pronounced peak compared to the standard z-distribution. Additionally, the t-distribution takes into account the sample size when calculating probabilities or confidence intervals, whereas the standard z-distribution assumes a known population standard deviation. Overall, while the two distributions have some similarities, it is important to understand their differences and appropriate use cases.

Question 16

A bank wishes to estimate the mean credit card balance owed by its customers. The population standard deviation is estimated to be $300. If a 98% confidence interval is used and an interval of $75 is desired,how many customers should be sampled?
A)44
B)212
C)629
D)87

Accepted Answer

The answer of A bank wishes to estimate the mean...

Question 17

Knowing the population standard deviation,a 95% confidence interval infers that the population mean ________.
A)is between 0 and 100%
B)is within ±1.96 standard deviations of the sample mean
C)is within ±1.96 standard errors of the sample mean
D)is too large

Accepted Answer

A 95% confidence interval for the population mean, when the population standard deviation is known, is calculated as the sample mean plus or minus 1.96 times the standard error of the mean.

Question 18

Mileage tests were conducted on a randomly selected sample of 100 newly developed automobile tires. The results showed that the mean tread life was 50,000 miles,with a standard deviation of 3,500 miles. What is the best estimate of the mean tread life in miles for the entire population of these tires?
A)50,000
B)3,500
C)500
D)35

Accepted Answer

Since the sample was randomly selected and the sample size is relatively large (n=100), it can be assumed that the sample is representative of the population. Therefore, the mean tread life of the sample (50,000 miles) can be used as the best estimate of the mean tread life for the entire population of these tires.

Question 19

Which of the following is a point estimate for the population mean (µ)?
A)σ
B)x/n
C) 
D)s

Accepted Answer

The answer of Which of the following is a point...

Question 20

A random sample of 42 college graduates revealed that they worked an average of 5.5 years on the job before being promoted. The sample standard deviation was 1.1 years. Using the 0.99 degree of confidence,what is the confidence interval for the population mean?
A)5.04 and 5.96
B)5.06 and 5.94
C)2.67 and 8.33
D)4.40 and 6.60

Accepted Answer

The confidence interval for the population mean can be calculated using the formula: $\bar{x} \pm z\frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $z$ is the z-score corresponding to the confidence level. For a 99% confidence level, the z-score is approximately 2.58. Plugging in the given values: $5.5 \pm 2.58\frac{1.1}{\sqrt{42}}$, we get the interval $5.04$ to $5.96$.

Quiz 9: Estimation and Confidence Intervals

Which statement(s) is(are) correct about the t-distribution?

When a confidence interval for a population mean is constructed from sample data,________.

A sample mean is the best point estimate of ________.

The distribution of Student's t is ________.

Which of the following is not necessary to determine how large a sample to select from a population?

Suppose 1,600 of 2,000 registered voters sampled said they planned to vote for the Republican candidate for president. Using the 0.95 degree of confidence,what is the interval estimate for the population proportion (to the nearest 10th of a percent) ?

What kind of distribution is the t-distribution?

Of the following characteristics,the t-distribution and z-distribution are the same in all but one. Which one is it?

What is the interpretation of a 96% confidence level?

A confidence interval for a population mean ________.

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

A sample standard deviation is the best point estimate of the ________.

The distribution of Student's t has ________.

A random sample of 20 items is selected from a population. When computing a confidence interval for the population mean,what number of degrees of freedom should be used to determine the appropriate t-value?

How is the t-distribution similar to the standard z-distribution?

A bank wishes to estimate the mean credit card balance owed by its customers. The population standard deviation is estimated to be $300. If a 98% confidence interval is used and an interval of $75 is desired,how many customers should be sampled?

Knowing the population standard deviation,a 95% confidence interval infers that the population mean ________.

Which of the following is a point estimate for the population mean (µ) ?

A random sample of 42 college graduates revealed that they worked an average of 5.5 years on the job before being promoted. The sample standard deviation was 1.1 years. Using the 0.99 degree of confidence,what is the confidence interval for the population mean?