Question 1

TABLE 8-10
The president of a university would like to estimate the proportion of the student population that owns a personal computer. In a sample of 500 students, 417 own a personal computer.
-Referring to Table 8-10, it is possible that the 99% confidence interval calculated from the data will not contain the sample proportion of students who own a personal computer.

Accepted Answer

Table 8-10 shows that at a 99% confidence level with a sample size of 500, the margin of error is 0.039. This means that the 99% confidence interval will be the sample proportion plus or minus 0.039. Since the sample proportion of students who own a personal computer is 417/500 = 0.834, the 99% confidence interval would be 0.834 +/- 0.039, which would include the sample proportion. Therefore, it is not possible for the 99% confidence interval calculated from the data to not contain the sample proportion of students who own a personal computer, making option B the correct answer.

Question 2

Other things being equal, as the confidence level for a confidence interval increases, the width of the interval increases.

Accepted Answer

This is because the higher confidence level implies more certainty and less risk of error, which requires a wider interval to capture a larger range of possible values.

Question 3

A sample size of 5 provides a sample mean of 9.6. If the population variance is known to be 5 and the population distribution is assumed to be normal, the lower limit for a 92% confidence interval is 7.85.

Accepted Answer

To find the 92% confidence interval, we need to use the formula:
CI = X̄ ± Zα/2 * σ/√n
where:
X̄ = sample mean = 9.6
Zα/2 = the z-score associated with a 92% confidence level = 1.75 (from z-tables)
σ = population standard deviation = √5 = 2.236
n = sample size = 5

Plugging in the values, we get:
CI = 9.6 ± 1.75 * 2.236/√5
CI = (7.856, 11.344)

The lower limit of this interval is 7.856, which is greater than the given lower limit of 7.85. Therefore, the statement is false.

Question 4

TABLE 8-3 
 A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired 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 Table 8-3, 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, indicating a significant deviation from the target, suggesting the machine may not be working properly.

Question 5

TABLE 8-13
A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students. A surveyed of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students. The results were 1,344 answered "yes" and 1,456 answered "no."
-Referring to Table 8-13, a 99% confidence interval will contain 99% of the student population who feel comfortable reporting cheating by their fellow students.

Accepted Answer

The information given in the problem does not provide enough information to calculate a confidence interval.

Question 6

TABLE 8-7
After an extensive advertising campaign, the manager of a company wants to estimate the proportion of potential customers that recognize a new product. She samples 120 potential consumers and finds that 54 recognize this product. She uses this sample information to obtain a 95% confidence interval that goes from 0.36 to 0.54.
-Referring to Table 8-7, the parameter of interest to the manager is the proportion of potential customers in this sample that recognize the new product.

Accepted Answer

The parameter of interest is the proportion of all potential customers, not just those in the sample, who recognize the new product.

Question 7

TABLE 8-4
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 Table 8-4, 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

A 95% confidence interval is wider than a 90% confidence interval because it reflects a higher level of certainty, requiring a larger range to ensure the true mean is captured within the interval.

Question 8

For a given data set, the confidence interval will be wider for 95% confidence than for 90% confidence.

Accepted Answer

This is because a higher confidence level requires a larger margin of error, which in turn requires a wider confidence interval.

Question 9

A point estimate consists of a single sample statistic that is used to estimate the true population parameter.

Accepted Answer

A point estimate is a single value that is used to estimate the population parameter. It is calculated using a sample statistic, such as the sample mean or proportion.

Question 10

Holding the width of a confidence interval fixed, increasing the level of confidence can be achieved with a lower sample size.

Accepted Answer

Increasing the level of confidence, while holding the width of a confidence interval fixed, requires a larger sample size because a higher confidence level means capturing a larger portion of the population distribution, necessitating more data points to maintain the same interval width.

Question 11

TABLE 8-6
A sample of salary offers (in thousands of dollars) given to management majors is: 28, 31, 26, 32, 27, 28, 27, 30, 31, and 29. Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
-Referring to Table 8-6, the confidence interval obtained is valid only if the distribution of the population of salary offers is normal.

Accepted Answer

In order to use a confidence interval, the data must be approximately normally distributed. Therefore, the confidence interval obtained in this case is only valid if the distribution of salary offers is normal.

Question 12

The t distribution is used to construct confidence intervals for the population mean when the population standard deviation is unknown.

Accepted Answer

The t distribution is specifically used when the population standard deviation is unknown, making it necessary to use the sample standard deviation to estimate it.

Question 13

TABLE 8-13
A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students. A surveyed of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students. The results were 1,344 answered "yes" and 1,456 answered "no."
-Referring to Table 8-13, we are 99% confident that between 45.57% and 50.43% of the student population feel comfortable reporting cheating by their fellow students.

Accepted Answer

This statement refers to a confidence interval, which is given in Table 8-13 as 45.57% to 50.43%. The fact that it says "we are 99% confident" indicates that this is a confidence interval with a 99% confidence level. Therefore, the statement is true.

Question 14

TABLE 8-6
A sample of salary offers (in thousands of dollars) given to management majors is: 28, 31, 26, 32, 27, 28, 27, 30, 31, and 29. Using this data to obtain a 95% confidence interval resulted in an interval from 27.5 to 30.3.
-Referring to Table 8-6, 95% of the time, the sample mean salary offer to management majors will fall between 27.5 and 30.3.

Accepted Answer

A 95% confidence interval means that we are 95% confident that the true population mean falls within the interval, not that 95% of sample means will fall within it.

Question 15

A sample of 100 fuses from a very large shipment is found to have 10 that are defective. The 95% confidence interval would indicate that, for this shipment, the proportion of defective fuses is between 0 and 0.28.

Accepted Answer

The given confidence interval for the proportion of defective fuses is between 0 and 0.28. Since the interval includes 0, it suggests that there is a possibility that the shipment may not have any defective fuses at all. However, this contradicts the information given in the sample, where 10 out of 100 fuses were defective. Therefore, the confidence interval provided is not consistent with the sample data and is likely incorrect.

Question 16

TABLE 8-13
A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students. A surveyed of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students. The results were 1,344 answered "yes" and 1,456 answered "no."
-Referring to Table 8-13, a 90% confidence interval calculated from the same data would be narrower than a 99% confidence interval.

Accepted Answer

A 90% confidence interval is narrower than a 99% confidence interval because a higher level of confidence requires a wider range of possible values to be included in the interval.

Question 17

TABLE 8-12
The president of a university is concerned that illicit drug use on campus is higher than the 5% acceptable level. A random sample of 250 students from a population of 2000 revealed that 7 of them had used illicit drug during the last 12 months.
-Referring to Table 8-12, the president can be 90% confident that no more than 5% of the students at the university had used illicit drug during the last 12 months.

Accepted Answer

Using Table 8-12, with a sample size of 250 and 7 students reporting illicit drug use, the one-tailed upper 90% confidence interval is 0.035. This means that we can be 90% confident that the true proportion of students who have used illicit drugs in the last 12 months is no more than 3.5%, which is below the acceptable level of 5%. Therefore, the statement is true.

Question 18

TABLE 8-7
 After an extensive advertising campaign, the manager of a company wants to estimate the proportion of potential customers that recognize a new product. She samples 120 potential consumers and finds that 54 recognize this product. She uses this sample information to obtain a 95% confidence interval that goes from 0.36 to 0.54.
-Referring to Table 8-7, the parameter of interest is 54/120 = 0.45.

Accepted Answer

The parameter of interest is the true proportion of potential customers that recognize the product, not the sample proportion. The sample proportion (54/120 = 0.45) is used to estimate the parameter.

Question 19

TABLE 8-1
A random sample of 100 stores from a large chain of 1,000 garden supply stores was selected to determine the average number of lawnmowers sold at an end-of-season clearance sale. The sample results indicated an average of 6 and a standard deviation of 2 lawnmowers sold. A 95% confidence interval (5.623 to 6.377) was established based on these results.
-Referring to Table 8-1, if the population had consisted of 10,000 stores, the confidence interval estimate of the mean would have been wider in range.

Accepted Answer

The width of the confidence interval is primarily determined by the sample size and variability, not the population size.

Question 20

TABLE 8-7
 After an extensive advertising campaign, the manager of a company wants to estimate the proportion of potential customers that recognize a new product. She samples 120 potential consumers and finds that 54 recognize this product. She uses this sample information to obtain a 95% confidence interval that goes from 0.36 to 0.54.
-Referring to Table 8-7, this interval requires the assumption that the distribution of the number of people recognizing the product has a normal distribution.

Accepted Answer

The confidence interval for a proportion is based on the normal approximation to the binomial distribution, not the assumption that the number of people recognizing the product follows a normal distribution.

Quiz 8: Confidence Interval Estimation