Question 1

What effect does increasing the sample size have on the sampling distribution of  ?
A) The mean of the distribution gets smaller.
B) Sample size has no effect on the sampling distribution of  .
C) The variability of the sampling distribution gets smaller.
D) The mean of the distribution gets larger.

Accepted Answer

Increasing the sample size leads to a smaller standard error and hence the variability of the sampling distribution gets smaller. The mean of the distribution does not change with sample size.

Question 2

A department store claims that its average customer satisfaction rating is 28.5.We believe that this rating is too high, so we randomly sample 40 customers and find a sample average of 25.2.If the standard deviation of customer satisfaction for this store is 13.1, what is the probability of observing a sample at least this far below the claimed mean, assuming the claim is true?
A) 0.9441
B) 0.4013
C) 0.0559
D) 0.2500

Accepted Answer

We can use a one-sample t-test since we have a sample size of 40 and do not know the population standard deviation. The formula for the t-test statistic is:

t = (sample mean - hypothesized mean) / (standard error of the mean)

The standard error of the mean can be calculated as:
standard error of the mean = standard deviation / sqrt(sample size)
standard error of the mean = 13.1 / sqrt(40)
standard error of the mean = 2.07

Plugging in the values:
t = (25.2 - 28.5) / 2.07
t = -1.60

The degrees of freedom for this test is 39 (sample size - 1). Using a t-distribution table or calculator with 39 degrees of freedom, we find that the probability of observing a t-value of -1.60 or lower is approximately 0.0559.

Therefore, the answer is C.

Question 3

In the natural world, the number of measurable variables that follow a normal distribution is quite remarkable.Many of these measurable variables are actually the sum of other independent variables.The Central Limit Theorem helps explain this phenomenon because:
A) the Central Limit Theorem states that all measurable variables follow a normal distribution.
B) the Central Limit Theorem states that for a sufficiently large number of variables, the sum of these variables will be approximately normally distributed.
C) the Central Limit Theorem states that for large sample sizes, all measurable variables will follow a normal distribution.
D) the Central Limit Theorem states that because nature is normative, variables found in nature will be at least approximately normal.

Accepted Answer

The Central Limit Theorem states that for a sufficiently large number of independent variables, their sum will be approximately normally distributed. This helps explain why many measurable variables in the natural world follow a normal distribution, as they are often the sum of other independent variables.

Question 4

A simple random sample (SRS) of size n is best defined as:
A) a sample selected in such a way that every possible sample of size n will be selected.
B) a sample selected in such a way that every possible sample of size n has the same chance of being selected.
C) a sample selected in such a way that every possible sample of size n has a randomly assigned chance of being selected.
D) a sample selected in such a way that every possible sample of size n has some chance of being selected.

Accepted Answer

A simple random sample (SRS) of size n is a sample selected in such a way that every possible sample of size n has the same chance of being selected. This means that each member of the population has an equal chance of being selected and each combination of members has an equal probability of being chosen. Therefore, option B is the correct definition of a SRS.

Question 5

Suppose you are interested in determining the average word length in Lincoln's Gettysburg Address.You decide to estimate this average word length by using a sample of 10 randomly selected words.Which of the following methods should result in a simple random sample of 10 words from the Address?
A) Have a student select 10 "representative" words from a copy of the Address.
B) Obtain a copy of the Address on a page of paper.Close your eyes and point the tip of your pencil on the page five times, each time selecting the word that your pencil lands on or nearest.
C) Obtain a list of the 268 words in the Address.Number the words.Use a random number generator to determine which 10 words to select.
D) Select every 27th word in the Address.

Accepted Answer

Using a random number generator to select 10 words from the list of 268 words in the Address would result in a simple random sample, as each word would have an equal chance of being selected. Option A is not random and relies on subjective judgment. Option B is also not random as the words chosen could be dependent on the location of the pencil on the page. Option D is systematic sampling and could potentially introduce bias if there is a pattern in the word lengths.

Question 6

If the monthly rents of all one-bedroom apartments in a small town are known to be normally distributed with a mean of $175 and a standard deviation of $35, which of the following would be the least likely individual rent that you might expect to find?
A) $135
B) $140
C) $210
D) $225

Accepted Answer

Using the formula for z-score, we find that the z-score for $135 is -1, for $140 it is -0.857, for $210 it is 1.17, and for $225 it is 1.5. Therefore, the least likely rent would be the one with the highest absolute z-score, which is $225.

Question 7

Let X be the fuel efficiency, in miles per gallon, of a particular vehicle. Suppose X ~ N(μ = 18.5, σ = 2.2).Find P(18.2 ≤  ≤ 19) if n = 5.
A) 0.6950
B) 0.9839
C) 0.1467
D) 0.3129

Accepted Answer

The answer of Let X be the fuel efficiency, in...

Question 8

The ______________ of a statistic is the probability distribution of the statistic.
A) limiting distribution
B) exponential distribution
C) normal distribution
D) sampling distribution

Accepted Answer

The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. Therefore, D is the best choice as it directly addresses the definition of the term. The other options are probability distributions that may be relevant to certain statistics but do not directly answer the definition of the term.

Question 9

Homeowners who recently bought their houses from a local developer allege that 30% of the houses this developer constructs have some major defect that will require substantial repairs.To test this allegation, we randomly sample 20 houses constructed by the developer and find that two of the homes do, indeed, have some major defect.If the allegation is correct, what is the probability of observing at most two defective homes out of a random sample of 20?
A) 0.0256
B) 0.1000
C) 0.0300
D) 0.9744

Accepted Answer

The probability of observing at most two defective homes can be calculated using the binomial probability formula with n=20, p=0.3, and x=0,1,2. The sum of these probabilities is approximately 0.0256.

Question 10

An industrial fan is designed to move 2500 ft3of air per minute, on average.Under normal operating conditions, the standard deviation of air volume moved by this fan is 300 ft3/min.If the fan is operating as designed, what is the probability that the average of 30 randomly selected measurements of fan capacity will be less than 2000 ft3/min?
A) 0.0475
B) 0.0000
C) 0.9525
D) 0.913

Accepted Answer

We can use the central limit theorem to approximate the distribution of sample means. The mean of the sample means will be equal to the population mean (2500 ft3/min), and the standard deviation of the sample means will be equal to the standard deviation of the population (300 ft3/min) divided by the square root of the sample size (sqrt(30)).

z = (x̄ - μ) / (σ / sqrt(n))
z = (2000 - 2500) / (300 / sqrt(30))
z = -10.392

Using a standard normal distribution table, we find that the probability of z being less than -10.392 is essentially 0. Therefore, the probability that the average of 30 randomly selected measurements of fan capacity will be less than 2000 ft3/min is 0.

Question 11

Consider the following discrete population distribution: ​  ​
We randomly draw samples of size 5 from this population using the simple random sampling technique.If we know that μx = 5.4, σx= 2.069, what are the mean and the standard deviation of the sampling distribution of sample means? Assume the Central Limit Theorem applies.
A) μ  = 5.4, σ  = 2.069
B) μ  = 1.08, σ  = 0.4138
C) μ  = 5.4, σ  = 0.9253
D) μ  = 5.4, σ  = 0.4138

Accepted Answer

According to the Central Limit Theorem, the sampling distribution of sample means is normally distributed with a mean equal to the population mean (μx) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (σx/sqrt(n)). Therefore, the mean of the sampling distribution of sample means is 5.4 and the standard deviation is 2.069/sqrt(5) = 0.9253. Option C is the only choice that matches both of these values.

Question 12

A psychologist randomly samples 28 clinically depressed patients.She subjects each patient to a standardized battery of tests.The average battery test score for the sample is 128.In this example, the number 128 is a:
A) parameter.
B) sampling distribution.
C) statistic.
D) random variable.

Accepted Answer

The number 128 represents the average battery test score for the sample of 28 clinically depressed patients, which is a characteristic or attribute of the sample only. Therefore, 128 is a statistic. A parameter would be a characteristic or attribute of the entire population of clinically depressed patients, which is unknown in this case. A sampling distribution is a distribution of statistics from multiple samples, which is not relevant to this example. A random variable is a variable whose possible values are numerical outcomes of a random phenomenon, which is also not relevant to this example.

Question 13

A _________ is any quantity computed from values in a sample.
A) sampling distribution
B) parameter
C) statistic
D) parametric statistic

Accepted Answer

A parameter is a quantity computed from values in a population, while a statistic is a quantity computed from values in a sample. Therefore, a quantity computed from values in a sample is a statistic.

Question 14

Concerning the sampling distribution of  , what is the significance of the conditions np ≥ 5 and n(1 - p) ≥ 5?
A) It is impossible to make any inference on p using  unless both these conditions are met.
B) The sampling distribution of  will be approximately normal if both these conditions are met.
C) It is possible to use  to estimate p only if one of these conditions is met.
D) The value of  will be very close to p if and only if the sample size is large enough for both these conditions to be met.

Accepted Answer

The conditions np ≥ 5 and n(1-p) ≥ 5 ensure that the sampling distribution of the sample proportion will be approximately normal. This is important for making inferences about the population proportion p. If these conditions are not met, the distribution may be skewed or have other non-normal characteristics that could affect the accuracy of our statistical tests and confidence intervals. Therefore, it is important to check these conditions before using 11eb86f3_5f6c_25f8_883a_37e1fa58bc71_TB8195_11 to make any inference about p.

Question 15

An exponential population has parameter λ = 0.012 (hence, μ= 83.33, σ2= 6944.44).What is the probability that a random sample of 41 from this population will average between 81 and 84?
A) 0.5199
B) 0.0160
C) 0.0000
D) 0.0913

Accepted Answer

The sample mean follows a normal distribution with mean μ = 83.33 and standard deviation σ/√n = √6944.44/√41. Using the standard normal distribution, the probability that the sample mean is between 81 and 84 is approximately 0.0913.

Question 16

If the sample size is large (n ≥ 30), then, regardless of the population distribution, which of the following summarizes the sampling distribution of random variable  ?
A) 
B) 
C) 
D)

Accepted Answer

The answer of If the sample size is large (n...

Question 17

Because 
A)  will always be the same as p.
B)  is unbiased for p.
C)  must always be ≤ p.
D)  will always be ≥ p. ​

Accepted Answer

The answer of Because 
A)  will always be the...

Question 18

A heavily right-skewed population has a mean of 28 and a standard deviation of 2.What is the probability that a random sample of 5 from this population will average less than 27?
A) 0.1314
B) 0.3085
C) 0.6915
D) We cannot answer this question because the sample size is small and the population is non-normal; the distribution of  cannot be assumed to be normal.

Accepted Answer

The answer of A heavily right-skewed population has a mean...

Question 19

It is believed that 15% of people who fly on commercial airliners are "very concerned" about the safety of the carrier they have chosen.If this statement is accurate, what is the probability that out of 150 randomly selected people who fly on commercial airliners, between 20 and 25 of them are "very concerned" about the safety of the carrier that they have chosen?
A) 0.1333
B) 0.0333
C) 0.4314
D) 0.7157

Accepted Answer

We can use the binomial probability formula to solve this problem. Let X be the number of people out of 150 who are "very concerned" about the safety of their carrier. Then X has a binomial distribution with n=150 and p=0.15. We want to find P(20<=X<=25). We can use a calculator or software to find this probability. Using a calculator, we get P(20<=X<=25) = 0.4314. Therefore, the best choice is C.

Question 20

If a population has a variance of 64 and we take a simple random sample of size 25, what will the standard deviation of the sampling distribution of  be, assuming the Central Limit Theorem holds in this case?
A) 0.32
B) 0.625
C) 1.60
D) 3.125

Accepted Answer

The standard deviation of the sampling distribution of means can be calculated as:

$\frac{\sigma}{\sqrt{n}}=\frac{8}{\sqrt{25}}=1.6$

Therefore, the answer is (C).

Quiz 8: Sampling Distributions

What effect does increasing the sample size have on the sampling distribution of ?

In the natural world, the number of measurable variables that follow a normal distribution is quite remarkable.Many of these measurable variables are actually the sum of other independent variables.The Central Limit Theorem helps explain this phenomenon because:

A simple random sample (SRS) of size n is best defined as:

Suppose you are interested in determining the average word length in Lincoln's Gettysburg Address.You decide to estimate this average word length by using a sample of 10 randomly selected words.Which of the following methods should result in a simple random sample of 10 words from the Address?

If the monthly rents of all one-bedroom apartments in a small town are known to be normally distributed with a mean of $175 and a standard deviation of $35, which of the following would be the least likely individual rent that you might expect to find?

Let X be the fuel efficiency, in miles per gallon, of a particular vehicle. Suppose X ~ N(μ = 18.5, σ = 2.2) .Find P(18.2 ≤ ≤ 19) if n = 5.

The ______________ of a statistic is the probability distribution of the statistic.

A psychologist randomly samples 28 clinically depressed patients.She subjects each patient to a standardized battery of tests.The average battery test score for the sample is 128.In this example, the number 128 is a:

A _________ is any quantity computed from values in a sample.

Concerning the sampling distribution of , what is the significance of the conditions np ≥ 5 and n(1 - p) ≥ 5?

An exponential population has parameter λ = 0.012 (hence, μ= 83.33, σ2= 6944.44) .What is the probability that a random sample of 41 from this population will average between 81 and 84?

If the sample size is large (n ≥ 30) , then, regardless of the population distribution, which of the following summarizes the sampling distribution of random variable ?

Because

A heavily right-skewed population has a mean of 28 and a standard deviation of 2.What is the probability that a random sample of 5 from this population will average less than 27?

If a population has a variance of 64 and we take a simple random sample of size 25, what will the standard deviation of the sampling distribution of be, assuming the Central Limit Theorem holds in this case?