Question 1

The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a mean of 60,000 miles and a standard deviation of 4,000 miles. Suppose we select a sample of 40 tires and use a simulator to determine the tread life. What is the standard error of the mean?
A) 632.46
B) 40
C) 4000
D) Cannot be determineD.

Accepted Answer

The standard error of the mean is calculated using the formula SE = σ/√n, where σ is the standard deviation and n is the sample size. Therefore, SE = 4000/√40 = 632.46. This means that if we were to repeat the sampling process multiple times, the mean tread life of the tires would vary within ±632.46 of the population mean of 60,000 miles.

Question 2

A __________________ assumes that each item or individual in the population has an equal chance of being included.

Accepted Answer

The answer of A __________________ assumes that each item or...

Question 3

Sampling is required when crash testing cars because the products are __________.

Accepted Answer

The answer of Sampling is required when crash testing cars...

Question 4

The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find that the mean of this sample is between 98 and 103?
A) 0.3264
B) 0.9428
C) 0.4702
D) 0.7471

Accepted Answer

The standard error of the mean for a sample of 50 is 15/sqrt(50). Using the standard normal distribution, the probability that the sample mean is between 98 and 103 is approximately 0.7471.

Question 5

The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find the mean of this sample is between 95 and 105?
A) 0.0182
B) 0.9818
C) 1.00
D) Very unlikely

Accepted Answer

Since the sample size is greater than 30, we can use the Central Limit Theorem, which states that the sample means are normally distributed with a mean of the population and a standard deviation of (population standard deviation / square root of sample size).
Thus, the mean of the distribution of sample means is 100 and the standard deviation is 15/sqrt(50) = 2.12.
We can standardize this distribution by using the z-score formula: z = (x - mean) / standard deviation.
Therefore, 
z for x = 95: z = (95-100)/2.12 = -2.36
z for x = 105: z = (105-100)/2.12 = 2.36
Using the z-table, we can find the area under the normal distribution curve between these two z-scores:
Area = 0.4908 - 0.0090 = 0.4818
Therefore, the probability that we could select a sample of 50 adults and find the mean of this sample is between 95 and 105 is 0.4818 or 48.18%. Option B is correct.

Question 6

The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a mean of 60,000 miles and a standard deviation of 4,000 miles. Suppose we select a sample of 40 tires and use a simulator to determine the tread life. What is the likelihood of finding that the sample mean is between 59,050 and 60,950?
A) 0.4332
B) 0.8664
C) 1.00
D) Very likely

Accepted Answer

We can use the central limit theorem here to approximate the distribution of sample means as normal with a mean of 60,000 and a standard deviation of 4,000/sqrt(40) = 632.46. Then, we can find the z-scores for the lower and upper bounds:

z1 = (59050 - 60000)/632.46 = -1.50
z2 = (60950 - 60000)/632.46 = 1.50

Using a standard normal table or calculator, we can find the probability of the sample mean being between these bounds:

P(-1.50 < z < 1.50) = 0.8664

Therefore, the likelihood of finding that the sample mean is between 59,050 and 60,950 is 0.8664, or choice B.

Question 7

The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a mean of 60,000 miles and a standard deviation of 4,000 miles. Suppose you bought a set of four tires, what is the likelihood the mean tire life of these four tires is between 57,000 and 63,000 miles?
A) 0.4332
B) 0.8664
C) 1.00
D) Very likely

Accepted Answer

The mean tire life of four tires is calculated as the average of individual tire life. Since the individual tire life follows a normal distribution, the mean tire life also follows a normal distribution with mean 60,000 miles and standard deviation 4,000 miles/sqrt(4) = 2,000 miles. 
Therefore, we need to find the probability that a normal random variable with mean 60,000 and standard deviation 2,000 falls between 57,000 and 63,000. 
Using a standard normal distribution table or calculator, we can calculate this probability as P(-1.5 < Z < 1.5) = 0.8664, where Z is a standard normal random variable. 
Hence, the likelihood that the mean tire life of these four tires is between 57,000 and 63,000 miles is 0.8664 or 86.64%. Therefore, the best choice is B.

Question 8

Auditors may select every 20^th file starting with the 5^th file in the top drawer. Then, file numbers 25, 45, 65, 85, . . . are selected. This type of sampling is called ___________.

Accepted Answer

The answer of Auditors may select every 20^th file starting...

Question 9

Sampling from a population may be preferred, because studying all the items in a population may be too ____________.

Accepted Answer

The answer of Sampling from a population may be preferred,...

Question 10

If a population is normally distributed, then the sampling distribution of the sample means will be __________________________.

Accepted Answer

The answer of If a population is normally distributed, then...

Question 11

The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find that the mean of this sample exceeds 104?
A) 0.0294
B) 0.9412
C) Approximately one
D) 0.9706

Accepted Answer

The probability of selecting a sample of 50 adults and finding that the mean of this sample exceeds 104 is 0.0294. This is because the mean of a sample of 50 adults is normally distributed with a mean of 100 and a standard deviation of 15/sqrt(50) = 2.1213. The probability of selecting a sample with a mean greater than 104 is the area under the normal curve to the right of 104, which is 0.0294.

Question 12

The tread life of tires mounted on light duty trucks follows the normal probability distribution with a mean of 60,000 miles and a standard deviation of 4,000 miles. Suppose you bought a set of four tires, what is the likelihood the mean tire life of these four tires is more than 66,000 miles?
A) 0.0013
B) 0.9987
C) 0.4987
D) 0.9544

Accepted Answer

We need to use the Central Limit Theorem to approximate the distribution of the sample mean.

The sample mean follows a normal distribution with a mean of 60,000 and a standard deviation of 2,000 (since we are dividing the standard deviation by the square root of the sample size, which is 4 for 4 tires).

Using the standard normal distribution, we can find the z-score for a mean tire life of 66,000 miles:

z = (66000 - 60000)/2000 = 3

Now we need to find the probability of getting a z-score of 3 or more. This can be found from a normal distribution table or calculator:

P(z > 3) = 0.0013

Therefore, the likelihood of the mean tire life of these four tires being more than 66,000 miles is 0.0013.

Question 13

A total of _______ different samples of size 3 can be collected from a population of 10 items.

Accepted Answer

The answer of A total of _______ different samples of...

Question 14

For a given population with a normal probability distribution, the sampling distribution of  is a normal probability distribution for:
A) small sample sizes only.
B) any sample size.
C) large sample sizes only.
D) sample sizes greater than 30 only.

Accepted Answer

The central limit theorem states that the sampling distribution of means will be approximately normally distributed regardless of the size of the sample as long as the sample is taken from a population that has a normal distribution or a sample size larger than 30. Therefore, the sampling distribution of means will be a normal probability distribution for any sample size if the population has a normal probability distribution.

Question 15

The following data was collected from a simple random sample of a population of college freshmen. It represents their ages on the first day of the fall semester. 17 21 19 18 18 19
The point estimates of the population mean and standard deviation are:
A) 18 and 1.75.
B) 19 and 1.78.
C) 17.67 and 1.63.
D) 18.67 and 1.36.

Accepted Answer

The answer of The following data was collected from a...

Question 16

Sampling from a population may be preferred because, contacting or listing the entire population would be too ___________ consuming.

Accepted Answer

The answer of Sampling from a population may be preferred...

Question 17

To collect a sample, a population is first divided into subgroups and then a sample is selected from each subgroup. This is called _________ sampling.

Accepted Answer

The answer of To collect a sample, a population is...

Question 18

The Office of Student Services at a large western state university maintains information on the study habits of its full-time students. Their studies indicate that the mean amount of time undergraduate students study per week is 20 hours. The hours studied follows the normal distribution with a standard deviation of six hours. Suppose we select a random sample of 144 current students. What is the probability that the mean of this sample is between 19.25 hours and 21.0 hours?
A) 0.9544
B) 0.0160
C) 0.0986
D) 0.9104

Accepted Answer

The sampling distribution of the sample mean will also be normally distributed with a mean of 20 hours and a standard deviation of 6/sqrt(144) = 0.5 hours. Using the standard normal distribution, the probability that the sample mean is between 19.25 and 21.0 hours is approximately 0.9104.

Question 19

How many different samples of size 4 can be selected from a population of size 8?
A) 8
B) 32
C) 70
D) 1680

Accepted Answer

The number of different samples of size 4 that can be selected from a population of size 8 is calculated using the combination formula, which is C(n, k) = n! / [k!(n - k)!], where n is the population size and k is the sample size. Here, n = 8 and k = 4, so C(8, 4) = 8! / [4!(8 - 4)!] = 70.

Question 20

The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find the mean of this sample is less than 95?
A) 0.0091
B) 0.9818
C) 0.4909
D) 0.9544

Accepted Answer

To solve this problem we use the central limit theorem. The sample mean follows a normal distribution with a mean of 100 and a standard deviation of 15/√50=2.12. We need to find the probability that the sample mean is less than 95.

Using the z-score formula: 
z = (sample mean - population mean) / (standard deviation / sqrt(sample size)) 
z = (95 - 100) / (15 / sqrt(50)) 
z = -2.36

Looking up the z-score in a standard normal distribution table, we find the probability to be 0.0091. Therefore, the answer is A.

Quiz 8: Sampling Methods and the Central Limit Theorem

The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a mean of 60,000 miles and a standard deviation of 4,000 miles. Suppose we select a sample of 40 tires and use a simulator to determine the tread life. What is the standard error of the mean?

A __________________ assumes that each item or individual in the population has an equal chance of being included.

Sampling is required when crash testing cars because the products are __________.

The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find that the mean of this sample is between 98 and 103?

The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find the mean of this sample is between 95 and 105?

Auditors may select every 20th file starting with the 5th file in the top drawer. Then, file numbers 25, 45, 65, 85, . . . are selected. This type of sampling is called ___________.

Sampling from a population may be preferred, because studying all the items in a population may be too ____________.

If a population is normally distributed, then the sampling distribution of the sample means will be __________________________.

The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find that the mean of this sample exceeds 104?

The tread life of tires mounted on light duty trucks follows the normal probability distribution with a mean of 60,000 miles and a standard deviation of 4,000 miles. Suppose you bought a set of four tires, what is the likelihood the mean tire life of these four tires is more than 66,000 miles?

A total of _______ different samples of size 3 can be collected from a population of 10 items.

For a given population with a normal probability distribution, the sampling distribution of is a normal probability distribution for:

The following data was collected from a simple random sample of a population of college freshmen. It represents their ages on the first day of the fall semester. 17 21 19 18 18 19 The point estimates of the population mean and standard deviation are:

Sampling from a population may be preferred because, contacting or listing the entire population would be too ___________ consuming.

To collect a sample, a population is first divided into subgroups and then a sample is selected from each subgroup. This is called _________ sampling.

How many different samples of size 4 can be selected from a population of size 8?

The Intelligence Quotient (IQ) test scores for adults are normally distributed with a mean of 100 and a standard deviation of 15. What is the probability we could select a sample of 50 adults and find the mean of this sample is less than 95?

Auditors may select every 20^th file starting with the 5^th file in the top drawer. Then, file numbers 25, 45, 65, 85, . . . are selected. This type of sampling is called ___________.