Question 1

SCENARIO 8-3
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
-Referring to Scenario 8-3,the mean of the sample is_____ ,while the standard deviation of the sample is_____ .

Accepted Answer

The answer of SCENARIO 8-3
To become an actuary,it is necessary...

Question 2

SCENARIO 8-2
A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine.The desired mean length of the insulation is 12 feet.It is known that the standard deviation in the cutting length is 0.15 feet.A sample of 70 cut sheets yields a mean length of 12.14 feet.This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.
-Referring to Scenario 8-2,the confidence interval indicates that the machine is not working properly.

Accepted Answer

The confidence interval does not include the desired mean length of 12 feet, suggesting the machine may not be cutting to the desired specification.

Question 3

SCENARIO 8-3
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
-Referring to Scenario 8-3,if we use the same sample information to obtain a 95% confidence interval,the resulting interval would be narrower than the one obtained here with 90% confidence.

Accepted Answer

Increasing the confidence level from 90% to 95% would result in a wider confidence interval because a higher confidence level means we want to be more certain that the true mean falls within our interval, which requires expanding the range of the interval.

Question 4

SCENARIO 8-4
The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
-Referring to Scenario 8-4,a 99% confidence interval for the mean voltage of the power packs is from_____ to_____ .

Accepted Answer

The answer of SCENARIO 8-4
The actual voltages of power packs...

Question 5

SCENARIO 8-4
The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
-Referring to Scenario 8-4,a confidence interval estimate of the population mean would only be valid if the distribution of voltages is normal.

Accepted Answer

A confidence interval estimate is based on the assumption that the sample comes from a normally distributed population. Therefore, if the sample distribution of voltages is normal, then a confidence interval estimate of the population mean would be valid.

Question 6

SCENARIO 8-3
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
-Referring to Scenario 8-3,the confidence interval will be based on _____degrees of freedom.

Accepted Answer

The answer of SCENARIO 8-3
To become an actuary,it is necessary...

Question 7

SCENARIO 8-3
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
-Referring to Scenario 8-3,for the confidence interval to be valid,it is necessary that test scores of students in the special study program on the actuarial exam be normally distributed.

Accepted Answer

In order to use statistical tests and confidence intervals for a sample mean, it is necessary to assume that the population from which the sample is drawn follows a normal distribution. Therefore, for the confidence interval calculated in this scenario to be valid, it is necessary to assume that the test scores of actuarial students in the special study program on the actuarial exam are normally distributed.

Question 8

SCENARIO 8-4
The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
-Referring to Scenario 8-4,a 95% confidence interval for the mean voltage of the power pack is wider than a 99% confidence interval.

Accepted Answer

This statement is false. A 99% confidence interval will be wider than a 95% confidence interval as it requires a higher level of confidence, thus allowing for a wider range of possible values.

Question 9

SCENARIO 8-4
The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
-Referring to Scenario 8-4,the critical value for a 99% confidence interval for this sample is
.

Accepted Answer

The answer of SCENARIO 8-4
The actual voltages of power packs...

Question 10

SCENARIO 8-2
A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine.The desired mean length of the insulation is 12 feet.It is known that the standard deviation in the cutting length is 0.15 feet.A sample of 70 cut sheets yields a mean length of 12.14 feet.This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.
-Referring to Scenario 8-2,the confidence interval is valid only if the lengths cut are normally distributed.

Accepted Answer

The confidence interval is valid due to the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normal if the sample size is large enough (usually n > 30), regardless of the population distribution.

Question 11

SCENARIO 8-4
The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
-Referring to Scenario 8-4,a 99% confidence interval will contain 99% of the voltages for all such power packs.

Accepted Answer

A 99% confidence interval estimates the range in which the true mean voltage of all such power packs lies with 99% confidence, not the individual voltages of 99% of all power packs.

Question 12

SCENARIO 8-3
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
-Referring to Scenario 8-3,the critical value used in constructing a 90% confidence interval is
.

Accepted Answer

The answer of SCENARIO 8-3
To become an actuary,it is necessary...

Question 13

SCENARIO 8-3
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
-Referring to Scenario 8-3,a 90% confidence interval for the mean score of actuarial students in the special program is from _____to_____ .

Accepted Answer

The answer of SCENARIO 8-3
To become an actuary,it is necessary...

Question 14

SCENARIO 8-4
The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
-Referring to Scenario 8-4,it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the entire population.

Accepted Answer

Since the sample standard deviation is not given in the scenario, we cannot directly calculate the confidence interval. However, assuming a normal distribution and using a t-distribution with degrees of freedom equal to 6 (sample size - 1), we can calculate a range of possible values for the mean voltage with 99% confidence. If this range does not include the population mean for 12 volts, then the statement is true.

Question 15

SCENARIO 8-2
A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine.The desired mean length of the insulation is 12 feet.It is known that the standard deviation in the cutting length is 0.15 feet.A sample of 70 cut sheets yields a mean length of 12.14 feet.This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.
-Referring to Scenario 8-2,suppose the engineer had decided to estimate the mean length to within 0.03 with 99% confidence.Then the sample size would be .

Accepted Answer

The answer of SCENARIO 8-2
A quality control engineer is interested...

Question 16

SCENARIO 8-4
The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
-Referring to Scenario 8-4,a confidence interval for this sample would be based on the t distribution with _____degrees of freedom.

Accepted Answer

The answer of SCENARIO 8-4
The actual voltages of power packs...

Question 17

SCENARIO 8-4
The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
-Referring to Scenario 8-4,a 90% confidence interval calculated from the same data would be narrower than a 99% confidence interval.

Accepted Answer

A higher confidence interval means a wider range of values is being considered, so a 99% confidence interval would be wider than a 90% confidence interval.

Question 18

SCENARIO 8-3
To become an actuary,it is necessary to pass a series of 10 exams,including the most important one,an exam in probability and statistics.An insurance company wants to estimate the mean score on this exam for actuarial students who have enrolled in a special study program.They take a sample of 8 actuarial students in this program and determine that their scores are: 2,5,8,8,7,6,5,and 7.This sample will be used to calculate a 90% confidence interval for the mean score for actuarial students in the special study program.
-Referring to Scenario 8-3,it is possible that the confidence interval obtained will not contain the mean score for all actuarial students in the special study program.

Accepted Answer

A confidence interval is an estimate of the unknown population parameter and there is always a chance that the interval obtained from a sample will not contain the true population parameter. This is known as sampling error. Therefore, it is possible that the confidence interval obtained in this scenario will not contain the mean score for all actuarial students in the special study program.

Question 19

SCENARIO 8-2
A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine.The desired mean length of the insulation is 12 feet.It is known that the standard deviation in the cutting length is 0.15 feet.A sample of 70 cut sheets yields a mean length of 12.14 feet.This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.
-Referring to Scenario 8-2,the confidence interval goes from_____ to_____ .

Accepted Answer

The answer of SCENARIO 8-2
A quality control engineer is interested...

Question 20

SCENARIO 8-4
The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77.
-Referring to Scenario 8-4,it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the sample.

Accepted Answer

The sample mean voltage can be calculated as (11.77+11.90+11.64+11.84+12.13+11.99+11.77)/7 = 11.89 volts. Since the sample mean is one of the values used to calculate the confidence interval, it is guaranteed to be within the interval. Therefore, the 99% confidence interval will always contain the sample mean voltage.

Quiz 8: Confidence Interval Estimation

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a 99% confidence interval for the mean voltage of the power packs is from_ to_ .

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a confidence interval estimate of the population mean would only be valid if the distribution of voltages is normal.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a 95% confidence interval for the mean voltage of the power pack is wider than a 99% confidence interval.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,the critical value for a 99% confidence interval for this sample is .

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a 99% confidence interval will contain 99% of the voltages for all such power packs.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the entire population.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a confidence interval for this sample would be based on the t distribution with _____degrees of freedom.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a 90% confidence interval calculated from the same data would be narrower than a 99% confidence interval.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the sample.

Quiz 8: Confidence Interval Estimation

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a 99% confidence interval for the mean voltage of the power packs is from_____ to_____ .

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a confidence interval estimate of the population mean would only be valid if the distribution of voltages is normal.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a 95% confidence interval for the mean voltage of the power pack is wider than a 99% confidence interval.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,the critical value for a 99% confidence interval for this sample is .

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a 99% confidence interval will contain 99% of the voltages for all such power packs.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the entire population.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a confidence interval for this sample would be based on the t distribution with _____degrees of freedom.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a 90% confidence interval calculated from the same data would be narrower than a 99% confidence interval.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,it is possible that the 99% confidence interval calculated from the data will not contain the mean voltage for the sample.

SCENARIO 8-4 The actual voltages of power packs labeled as 12 volts are as follows: 11.77,11.90,11.64,11.84,12.13,11.99,and 11.77. -Referring to Scenario 8-4,a 99% confidence interval for the mean voltage of the power packs is from_ to_ .