A population has mean mu = 50 and standard deviation sigma = 12. Random samples of size n = 36 are taken. What is the standard error of the sample mean?
2
✓
12
0.33
6
Correct Answer
2
The standard error of the mean is sigma divided by the square root of n, so 12 / sqrt(36) = 12 / 6 = 2. The answer 12 is the population standard deviation, which describes individual values, not sample means. The answer 0.33 comes from dividing sigma by n instead of by the square root of n. The answer 6 is just the square root of the sample size.
Category: Standard Error
Question 2
A researcher repeatedly draws samples of 49 students from a population whose commute times have mean 50 minutes and standard deviation 14 minutes, recording each sample's average. Around what value are the recorded averages centered?
2
50
✓
2450
14
Correct Answer
50
The mean of the sampling distribution of x-bar equals the population mean, so the sample averages center on 50 minutes no matter what the sample size is. The value 2 is the standard error, 14 / sqrt(49), a measure of spread rather than center. The value 2450 is n times mu, the mean of the total of 49 commute times, not of their average. The value 14 is the population standard deviation.
Category: Sampling Distribution of Means
Question 3
Values are drawn from a population with mu = 8 and sigma = 3. For random samples of n = 25 values, what is the mean of the sum of the 25 values?
8
40
200
✓
15
Correct Answer
200
By the central limit theorem for sums, the mean of a sum of n values is n times mu, so 25 x 8 = 200. The value 8 is the mean of a single value or of the sample mean, not of the total. The value 40 multiplies mu by the square root of n instead of by n. The value 15 is the standard deviation of the sum, sqrt(25) x 3.
Category: CLT for Sums
Question 4
A delivery van carries 49 randomly chosen boxes whose weights have mean 20 pounds and standard deviation 4 pounds. What is the standard deviation of the total weight of the 49 boxes?
196
0.57
4
28
✓
Correct Answer
28
The standard deviation of a sum of n values is the square root of n times sigma, so sqrt(49) x 4 = 7 x 4 = 28 pounds. The value 196 multiplies sigma by n, which overstates the spread because high and low weights partly cancel when added. The value 0.57 is sigma divided by the square root of n, the standard error of the mean weight. The value 4 is the standard deviation of a single box.
Category: CLT for Sums
Question 5
A population has mu = 100 and sigma = 20. A random sample of n = 16 has mean x-bar = 105. What is the z-score of this sample mean?
1
✓
0.25
4
0.05
Correct Answer
1
The standard error is 20 / sqrt(16) = 5, so z = (105 - 100) / 5 = 1. The value 0.25 divides by sigma instead of by the standard error, which treats the mean like a single value. The value 4 divides by sigma / n = 1.25 instead of sigma / sqrt(n). The value 0.05 divides the difference by mu instead of by a standard deviation.
Category: Z-Scores and Percentiles
Question 6
If the sample size is multiplied by 4 while the population standard deviation stays the same, what happens to the standard error of the mean?
It is cut to one fourth
It is cut in half
✓
It doubles
It stays the same
Correct Answer
It is cut in half
The standard error is sigma / sqrt(n), and multiplying n by 4 multiplies sqrt(n) by 2, so the standard error becomes half as large. It is cut to one fourth would be true only if the standard error were sigma / n. It doubles reverses the direction, since larger samples give less variable means. It stays the same ignores that n appears in the denominator.
Category: Standard Error
Question 7
A population is strongly skewed to the right. Random samples of a large size n are taken and the sample mean of each is recorded. What is the approximate shape of the distribution of these sample means?
Strongly right-skewed
Uniform
Approximately normal
✓
Strongly left-skewed
Correct Answer
Approximately normal
The central limit theorem says that for large enough samples the distribution of sample means is approximately normal, whatever the shape of the population. Strongly right-skewed describes the population, not the means, and the skew fades as n grows. Uniform is not the shape the theorem predicts. Strongly left-skewed reverses the skew and is also not what the theorem predicts.
Category: Sampling Distribution of Means
Question 8
In the central limit theorem for sample means, what does the letter n represent?
The number of samples drawn
The size of the whole population
The number of possible sample means
The number of values averaged in each sample
✓
Correct Answer
The number of values averaged in each sample
In the central limit theorem, n is the sample size, the number of values averaged together to form each sample mean. The number of samples drawn is how many times the experiment is repeated, which does not enter the standard error formula. The size of the whole population is usually unknown and is not n. The number of possible sample means is not part of the theorem.
Category: Sampling Distribution of Means
Question 9
Exam scores have a known mean and standard deviation. A question asks for the probability that ONE randomly chosen student scores above 90. Which distribution should be used?
The distribution of the original variable X
✓
The sampling distribution of the sample mean
The distribution of the sum of the scores
The t-distribution with n - 1 degrees of freedom
Correct Answer
The distribution of the original variable X
A probability about a single individual uses the distribution of the original variable X, with the population mean and standard deviation. The sampling distribution of the sample mean applies to averages of several values, not to one value. The distribution of the sum of the scores applies to a total of several values. The t-distribution with n - 1 degrees of freedom is used for inference about a mean when sigma is unknown, not for one value.
Category: Means Versus Individuals
Question 10
A shipping company wants the probability that the combined weight of 40 randomly chosen packages exceeds the limit of a truck. Which tool fits this question best?
The central limit theorem for sample means
The central limit theorem for sums
✓
The distribution of a single package weight
The binomial distribution
Correct Answer
The central limit theorem for sums
A question about a combined total of many values calls for the central limit theorem for sums, with mean n times mu and standard deviation sqrt(n) times sigma. The central limit theorem for sample means fits questions about an average, not a total. The distribution of a single package weight describes one package, not 40 of them. The binomial distribution counts successes in trials and does not describe a total weight.
Category: CLT for Sums
Question 11
According to the law of large numbers, what tends to happen to the sample mean x-bar as the sample size increases?
It gets closer to zero
It becomes more variable
It gets closer to the population mean
✓
It moves farther from the population mean
Correct Answer
It gets closer to the population mean
The law of large numbers says that as n grows, the sample mean tends to get closer to the population mean mu. It gets closer to zero is wrong because the target is mu, which need not be zero. It becomes more variable reverses the effect, since the standard error sigma / sqrt(n) shrinks as n grows. It moves farther from the population mean is the opposite of what the law states.
Category: Law of Large Numbers
Question 12
A binomial experiment has n = 100 trials with probability of success p = 0.4. What is the mean of the normal distribution used to approximate it?
24
4.9
60
40
✓
Correct Answer
40
The normal approximation to a binomial uses mean np, so 100 x 0.4 = 40. The value 24 is npq = 100 x 0.4 x 0.6, the variance. The value 4.9 is the standard deviation, sqrt(24), rounded to one decimal. The value 60 is nq, the expected number of failures.
Category: Normal Approximation to Binomial
Question 13
A fair coin is tossed 100 times. When the number of heads is approximated by a normal curve, what standard deviation is used?
5
✓
25
50
0.05
Correct Answer
5
The standard deviation is sqrt(npq) = sqrt(100 x 0.5 x 0.5) = sqrt(25) = 5. The value 25 is npq, the variance, before taking the square root. The value 50 is np, the mean. The value 0.05 comes from sqrt(pq / n), which describes a proportion rather than a count of successes.
Category: Normal Approximation to Binomial
Question 14
Values come from a population with mu = 10 and sigma = 2. For random samples of n = 16, the total of the 16 values is close to normal. Using z = 1.645, what is the 95th percentile of the total?
160.82
173.16
✓
212.64
10.82
Correct Answer
173.16
The total has mean 16 x 10 = 160 and standard deviation sqrt(16) x 2 = 8, so the 95th percentile is 160 + 1.645 x 8 = 173.16. The value 160.82 uses the standard error of the mean, 0.5, with the mean of the total. The value 212.64 uses n times sigma, 32, as the standard deviation. The value 10.82 is the 95th percentile of the sample mean, not of the total.
Category: CLT for Sums
Question 15
A population has mu = 60 and sigma = 10. For random samples of n = 25, what value lies 2 standard errors above the mean of the sampling distribution of x-bar?
80
62
64
✓
60.8
Correct Answer
64
The standard error is 10 / sqrt(25) = 2, so 2 standard errors above 60 is 60 + 2 x 2 = 64. The value 80 uses sigma, 10, in place of the standard error. The value 62 goes up only 1 standard error. The value 60.8 uses sigma / n = 0.4 as the standard error.
Category: Z-Scores and Percentiles
Question 16
Sample means follow a nearly normal distribution. About what percentage of sample means fall within 2 standard errors of the population mean?
About 68%
About 99.7%
About 50%
About 95%
✓
Correct Answer
About 95%
For an approximately normal distribution, about 95% of values lie within 2 standard deviations of the mean, and the standard error is the standard deviation of the sample means. About 68% is the share within 1 standard error. About 99.7% is the share within 3 standard errors. About 50% is the share on either side of the mean, not within 2 standard errors.
Category: Z-Scores and Percentiles
Question 17
A population has sigma = 12. What sample size n makes the standard error of the mean equal to 3?
16
✓
4
36
144
Correct Answer
16
Setting 12 / sqrt(n) = 3 gives sqrt(n) = 4, so n = 16. The value 4 is the square root of the needed sample size, not the sample size itself. The value 36 comes from 12 x 3 instead of (12 / 3) squared. The value 144 is sigma squared, which ignores the target standard error.
Category: Standard Error
Question 18
Waiting times follow an exponential distribution with mean 4 minutes, so sigma is also 4 minutes. For random samples of n = 64 waiting times, what is the standard error of the sample mean?
4
0.5
✓
0.0625
8
Correct Answer
0.5
The standard error is sigma / sqrt(n) = 4 / sqrt(64) = 4 / 8 = 0.5 minute. The value 4 is sigma for a single waiting time. The value 0.0625 divides sigma by n instead of by the square root of n. The value 8 is the square root of the sample size.
Category: Standard Error
Question 19
Values come from a population with mu = 5 and sigma = 2. For random samples of n = 36, the sum of the 36 values is 192. What is the z-score of this sum?
6
0.17
1
✓
36
Correct Answer
1
The sum has mean 36 x 5 = 180 and standard deviation sqrt(36) x 2 = 12, so z = (192 - 180) / 12 = 1. The value 6 divides by sigma, 2, instead of the standard deviation of the sum. The value 0.17 divides by n times sigma, 72. The value 36 divides by the standard error of the mean, 2 / 6, which belongs to x-bar rather than to the sum.
Category: CLT for Sums
Question 20
Which statement describes what the standard error of the mean measures?
How much individual values typically vary from mu
How far the sample median is from the sample mean
The proportion of samples whose mean equals mu
How much sample means typically vary from mu
✓
Correct Answer
How much sample means typically vary from mu
The standard error of the mean, sigma / sqrt(n), is the standard deviation of the sampling distribution of x-bar, so it describes how much sample means typically vary from mu across repeated samples. How much individual values typically vary from mu is described by sigma itself. How far the sample median is from the sample mean relates to skewness, not standard error. The proportion of samples whose mean equals mu is not a measure of spread at all.
Category: Standard Error