Question 1

Which of the following is the best description of a robust estimator?
A) It is one that is unbiased and symmetrical about 0.
B) It is one that is consistent and also mound-shaped.
C) It is one that is efficient and less spread out.
D) It is one that is not sensitive to a moderate departure from the assumption of normality in the population.

Accepted Answer

A robust estimator is one that is not sensitive to a moderate departure from the assumption of normality in the population. It can still provide accurate estimates even if the underlying distribution is not exactly normal. The other choices (A, B, and C) are not necessarily characteristics of a robust estimator.

Question 2

On which of the following does the shape of the Student's t distribution primarily depend?
A) the population variance
B) the number of its degrees of freedom
C) the population mean
D) the range of the data values

Accepted Answer

The shape of the Student's t distribution primarily depends on the number of its degrees of freedom. The distribution approaches the normal distribution as the degrees of freedom increase. The population variance and mean are used in calculating the t-statistic, but do not determine the shape of the distribution. The range of the data values is not directly relevant to the t distribution.

Question 3

To estimate with 95% confidence the average number of kilometres that students living off-campus commute to classes every day, a random sample of 20 students was taken and produced a mean equal to 5.2 km and a standard deviation of 3.05 km. Under these circumstances, what is the appropriate critical value?
A) 1.7291
B) 1.96
C) 2.086
D) 2.093

Accepted Answer

D

Question 4

A random sample is selected from a normally distributed population. The following sample statistics are obtained: n = 20,  = 30, and s = 10. Based on this information, and using a 95% confidence level, which of the following is a valid calculation from the sample statistics?
A) The critical value is 1.7921.
B) The critical value is 1.96.
C) The standard deviation of the sampling distribution ofis 0.50.
D) The margin of error is approximately4.68.

Accepted Answer

The answer of A random sample is selected from a...

Question 5

A random sample of size 15 taken from a normally distributed population revealed a sample mean of 75 and a sample variance of 25. What would the upper limit of a 95% confidence interval for the population mean equal?
A) 77.769
B) 77.273
C) 72.727
D) 72.231

Accepted Answer

The upper limit of a 95% confidence interval for the population mean is calculated using the formula: mean + (t-value * standard error). The t-value for 14 degrees of freedom at 95% confidence is approximately 2.145. The standard error is the square root of the variance divided by the sample size, which is sqrt(25/15). Thus, the upper limit is 75 + (2.145 * sqrt(25/15)) = 77.769.

Question 6

Under which of the following circumstances is it impossible to construct a confidence interval for the population mean?
A) a non-normal population with a large sample and an unknown population variance
B) a normal population with a large sample and a known population variance
C) a non-normal population with a small sample and an unknown population variance
D) a normal population with a small sample and an unknown population variance

Accepted Answer

When the population is non-normal and the sample size is small, the Central Limit Theorem does not apply, making it impossible to construct a reliable confidence interval without additional assumptions.

Question 7

Suppose that a one-tailed t-test is being applied to find out if the population mean is less than 100. The level of significance is 0.05 and 25 observations were sampled. Which of the following best describes the rejection region? A) t > 1.708 B) t < -1.711 C) t > 1.318 D) t < -1.316

Accepted Answer

Since the test is one-tailed with the alternative hypothesis being that the population mean is less than 100, we need to look for the critical t-value in the left tail of the t-distribution. With a level of significance of 0.05 and 25 degrees of freedom, the critical t-value is -1.711. Therefore, any calculated t-value less than -1.711 will fall into the rejection region, which corresponds to rejecting the null hypothesis and concluding that the population mean is significantly less than 100.

Question 8

What do the values in a t distribution table measure?
A) the area under a specified curve, defined by a given number of degrees of freedom, that lies to the left of a specified value of t
B) the area under a specified curve, defined by a given number of degrees of freedom, that lies to the right of a specified value of t
C) the heights of a specified curve, defined by a given number of degrees of freedom, at various specified values of t
D) the specified values of t for different heights of a specified curve, defined by a given number of degrees of freedom

Accepted Answer

The answer of What do the values in a t...

Question 9

Which of the following best describes the Student's t distribution?
A) It is a sampling distribution for a random variable, t, derived from a normally distributed population.
B) It is single-peaked distribution above the random variable's mean, median, and mode of value 1.
C) It is always skewed, either to the right or to the left.

Accepted Answer

The Student's t distribution is a sampling distribution that is derived from a normally distributed population. It is used for hypothesis testing and constructing confidence intervals when the sample size is small (less than 30) or when the population standard deviation is unknown. The distribution is bell-shaped like the normal distribution but has heavier tails, which means it is more spread out than the normal distribution, especially in the tails.

Question 10

Suppose that a t test is being conducted at the 0.05 level of significance to test vs. . A sample of size 20 is randomly selected. What is the rejection region? A) t > -2.093 B) t < -1.729 C) t > 1.725 D) t < 2.086

Accepted Answer

The answer of Suppose that a t test is being...

Question 11

Which of the following correctly describes the term "degrees of freedom"?
A) They are equal to the difference between the point estimate located at the interval centre and either the lower or the upper limit of the interval between which the parameter is free to move.
B) They determine the shape of the t distribution, and refer to the number of independent squared deviations inthat are available for estimatingand, in this sense, can be "freely chosen."
C) They equal sample size minus 2 in the case of a single sample.

Accepted Answer

Degrees of freedom refer to the number of independent squared deviations available for estimating the population variance and standard deviation, and therefore determine the shape of the t distribution. They can be "freely chosen" in the sense that they are determined by the sample size minus 1. Option A describes the margin of error, and option C describes the degrees of freedom for a single-sample t-test, but not in general.

Question 12

To estimate with 95% confidence the average number of kilometres that students living off-campus commute to classes every day, a random sample of 20 students was taken and produced a mean equal to 5.2 km and a standard deviation of 3.05 km. In this case, what would be the approximate value of the upper limit for a 95% confidence interval estimate for the true population mean?
A) 6.63
B) 5.20
C) 3.22
D) 2.15

Accepted Answer

First, we need to calculate the margin of error:
Margin of Error = t-value x (standard deviation/square root of sample size)
Since the sample size is 20, the degrees of freedom is 19 (df=n-1).
For a 95% confidence interval, the t-value with 19 degrees of freedom is 2.093.
Margin of Error = 2.093 x (3.05/sqrt(20)) = 1.922

To find the upper limit of the confidence interval, we add the margin of error to the sample mean:
Upper limit = sample mean + margin of error
Upper limit = 5.2 + 1.922 = 7.122

Therefore, the approximate value of the upper limit for a 95% confidence interval estimate for the true population mean is 6.63 (rounded to two decimal places).

The answer is A.

Question 13

A major department store chain is interested in estimating the average amount its credit card customers spent. Fifteen credit card accounts were randomly sampled and analyzed with the following results:  = $50.50 and  = 400. Assuming the distribution of the amount spent is approximately normal, what is the shape of the sampling distribution of the sample mean that will be used to create the desired confidence interval for  ?
A) approximately normal with a mean of $50.50
B) a standard normal distribution
C) a t distribution with 15 degrees of freedom
D) a t distribution with 14 degrees of freedom

Accepted Answer

The answer of A major department store chain is interested...

Question 14

To estimate, with 95% confidence, the average number of kilometres that students living off-campus commute to classes every day, a random sample of 20 students was taken and produced a mean equal to 5.2 km and a standard deviation of 3.05 km. In this case, which of the following is the point estimate for the true population mean?
A) 6.63
B) 5.20
C) 3.22
D) 2.15

Accepted Answer

The point estimate for the true population mean is equal to the sample mean, which is 5.2 km.

Question 15

For a sample of size 20 taken from a normally distributed population with standard deviation equal to 5, which of the following would be required to construct a 90% confidence interval for the population mean?
A) z = 1.96
B) t = 1.729
C) z = 1.645
D) t = 1.328

Accepted Answer

The answer of For a sample of size 20 taken...

Question 16

A major department store chain is interested in estimating the average amount its credit card customers spent. Fifteen credit card accounts were randomly sampled and analyzed with the following results:  = $50.50 and  = 400. Which of the following is a 95% confidence interval for the average amount the credit card customers spent?
A) $50.50$9.09
B) $50.50$10.12
C) $50.50$11.00
D) $50.50$11.08

Accepted Answer

The sample mean is $50.50 and the sample standard deviation is $400. To calculate the confidence interval, we will use the t-distribution with 14 degrees of freedom (since n-1=14). Using a t-table or calculator, we find that the t-value for a 95% confidence interval with 14 degrees of freedom is approximately 2.145.

The margin of error is: 
$E = t*\frac{s}{\sqrt{n}} = 2.145*\frac{400}{\sqrt{15}} \approx 111.10$

Therefore, the 95% confidence interval is: 
$50.50 \pm 111.10$

Which simplifies to: 
$(39.40, 61.60)$

Thus, choice D is correct as it is the only option that contains 50.50 and a range of 11.08 or more.

Question 17

Researchers determined that 60 tissues is the average number of tissues used during a cold. Suppose a random sample of 100 tissue users yielded the following data on the number of tissues used during a cold: = 52 and s = 22. Suppose the alternative we wanted to test is . Which of the following best describes the rejection region for = 0.05? A) Rejectif t > 1.6604. B) Rejectif t < -1.6604. C) Rejectif t > 1.9842 or Z < -1.9842. D) Rejectif t < -1.9842.

Accepted Answer

The answer of Researchers determined that 60 tissues is the...

Question 18

A random sample of size n has been selected from a normally distributed population. In hypothesis testing for the population mean, when should the t test be used instead of the z test?
A) when n < 30 andis unknown
B) when n > 30 andis unknown
C) when n < 30 andis known
D) both a and c

Accepted Answer

The t test should be used instead of the z test when the sample size is small (n < 30) and the population standard deviation is unknown (option a), and also when the population standard deviation is known (option c). Therefore, the correct answer is both a and c, which is option D.

Question 19

When finding a confidence interval for a population mean based on a sample of size 8, which of the following assumptions is made?
A) The population standard deviation,, is known.
B) The sampling distribution of z is normal.
C) The sampled population is approximately normal.
D) There is no special assumption made.

Accepted Answer

When finding a confidence interval for a population mean based on a sample of size 8, the assumption that is made is that the sampled population is approximately normal. This assumption is necessary in order to use the t-distribution, which is used when the population standard deviation is unknown and must be estimated from the sample. However, if the population is not approximately normal, then the t-distribution may not be appropriate, and other methods may need to be used.

Question 20

As the degrees of freedom for the t distribution increase, which of the following distributions does it more and more closely approximate?
A) the chi-square distribution
B) the standard normal distribution
C) the normal distribution
D) the F distribution

Accepted Answer

As the degrees of freedom for the t distribution increase, it becomes more and more similar to the standard normal distribution. This is because the t distribution approaches a normal distribution as the degrees of freedom approach infinity. Therefore, as the degrees of freedom increase, the t distribution becomes more and more similar to the standard normal distribution.

Quiz 10: A: Inference From Small Samples

Which of the following is the best description of a robust estimator?

On which of the following does the shape of the Student's t distribution primarily depend?

A random sample is selected from a normally distributed population. The following sample statistics are obtained: n = 20, = 30, and s = 10. Based on this information, and using a 95% confidence level, which of the following is a valid calculation from the sample statistics?

A random sample of size 15 taken from a normally distributed population revealed a sample mean of 75 and a sample variance of 25. What would the upper limit of a 95% confidence interval for the population mean equal?

Under which of the following circumstances is it impossible to construct a confidence interval for the population mean?

Suppose that a one-tailed t-test is being applied to find out if the population mean is less than 100. The level of significance is 0.05 and 25 observations were sampled. Which of the following best describes the rejection region?

What do the values in a t distribution table measure?

Which of the following best describes the Student's t distribution?

Suppose that a t test is being conducted at the 0.05 level of significance to test vs. . A sample of size 20 is randomly selected. What is the rejection region?

Which of the following correctly describes the term "degrees of freedom"?

For a sample of size 20 taken from a normally distributed population with standard deviation equal to 5, which of the following would be required to construct a 90% confidence interval for the population mean?

A random sample of size n has been selected from a normally distributed population. In hypothesis testing for the population mean, when should the t test be used instead of the z test?

When finding a confidence interval for a population mean based on a sample of size 8, which of the following assumptions is made?

As the degrees of freedom for the t distribution increase, which of the following distributions does it more and more closely approximate?