mathematics

If we were to randomly select many, many samples of a certain size from a given population and then create a histogram of the sample statistics. To understand sampling distributions, you must be able to make the distinction between the population parameter and the sample statistic in order to know which value varies (the statistic) and which value does not vary (the parameter) from sample to sample. Questions 1-4 ask you to determine what is a parameter and what is a statistic.

 

1.

Is 20% a statistic or a parameter in this statement? 20% of all the M&Ms produced are blue M&Ms.

A) Statistic

B) Parameter

2.

Is 25% a statistic or a parameter in this statement? 25% of the M&Ms in a randomly selected bag of M&Ms are blue.

A) Statistic

B) Parameter

3.

Is 513 a statistic or a parameter in this statement? 513 is the average score on the SATM for all who have taken the SATM this past year.

A) Statistic

B) Parameter

4.

Is 450 a statistic or a parameter in this statement? 450 is the average score on the SATM for a random selection of 100 students from a particular high school who have taken the SATM in the past year.

A) Statistic

B) Parameter

 

Based on the Law of Large Numbers, we know that any sample value (statistic: sample proportion or sample mean) will tend to lie closer and closer to the true population value (parameter: p or ยต) as the sample size becomes larger and larger. Therefore, a larger sample size will yield a histogram of sample statistic values that are closer to the population parameter than will a smaller sample size. With this in mind answer the following questions.

5.

The average monthly cell phone bill (the parameter) for a particular carrier in Pittsburgh, PA is $90. Which of the following is most likely?

A) The average monthly bill (a statistic) for 50 randomly selected customers (of this carrier) is over $115.

B) The average monthly bill (a statistic) for 250 randomly selected customers (of this carrier) is over $115.

C) Both A and B are equally likely.

6.

The average monthly cell phone bill (the parameter) for a particular carrier in Pittsburgh, PA is $90. Which of the following is most likely?

A) The average monthly bill (a statistic) for 80 randomly selected customers (of this carrier) is between $80 and $100.

B) The average monthly bill (a statistic) for 350 randomly selected customers (of this carrier) is between $80 and $100.

C) Both A and B are equally likely.

7.

The average monthly cell phone bill (the parameter) for a particular carrier in Pittsburgh, PA is $90. Which of the following is most likely?

A) The average monthly bill (a statistic) for 100 randomly selected customers (of this carrier) is under $66.

B) The average monthly bill (a statistic) for 32 randomly selected customers (of this carrier) is under $66.

C) Both A and B are equally likely.

8.

Sixty percent (60% = the parameter) of all undergraduate applications to a certain University are from the state that the University is located in. At the admissions deadline for undergraduate students, the admissions officer sees that 600 applicants have applied to the College of Science and 200 have applied to the College of Arts. To which College is it more likely that between 50% and 70% (statistic = sample proportion) of the applicants are from the state that the University is located in?

A) It is more likely among the 600 students applying to the College of Science.

B) It is more likely among the 200 students applying to the College of Arts.

C) It is just as likely among either group of applicants.

9.

Sixty percent (60% = the parameter) of all undergraduate applications to a certain University are from the state that the University is located in. Of the first 50 undergraduate applicants for the upcoming Fall semester, not a single one is from the state that the University is located in. What is the probability that the next (i.e. the 51st) applicant will be from the state the University is located in?

A) Higher than 0.60 (60%)

B) Lower than 0.60 (60%)

C) Still 0.60 (60%)

D) Not 0.60 (60%), but could be higher or lower

 

Our histogram of sample statistic values will be normally distributed (follow a normal curve) if the underlying population values are normally distributed. The sampling distribution will also be normally distributed — even if the underlying population values are not normally distributed — as long as certain other criteria are met. In the questions that follow, you will be asked to determine if the histogram of the given sample statistic (i.e. the sampling distribution) follows a normal curve. Using what you know about these criteria answer the following questions.

10.

A kitchen cabinet business has primarily homeowners as customers, but the business also has a few customers who are developers that build multi-unit condominiums. Overall, the average sale is $20,000, with a median sale of $10,000. Which of the following would be closest to a normal distribution?

A) the histogram of the next 50 individual sales

B) the histogram of the last 50 individual sales

C) the histogram of the average sales (sampling distribution) for random samples of size 5

D) the histogram of the average sales (sampling distribution) for random samples of size 40

11.

It is known that 5% (p = 0.05) of a particular brand of fuses is defective. Which of the following would be closest to a normal distribution (normal curve)?

A) the histogram of sample proportions of defective fuses (sampling distribution) for samples of size 1 (n)

B) the histogram of sample proportions of defective fuses (sampling distribution) for samples of size 5 (n)

C) the histogram of sample proportions of defective fuses (sampling distribution) for samples of size 20 (n)

D) the histogram of sample proportions of defective fuses (sampling distribution) for samples of size 200 (n)

12.

For which of the following situations would the rule for sample proportions apply (i.e. we could say that the histogram for the sample proportions would approximate a normal curve)? (More than one choice may apply.)

A) a random sample of 20 (n) is taken from a population in which the proportion with the trait is 0.20 (p)

B) a random sample of 50 (n) is taken from a population in which the proportion with the trait is 0.20 (p)

C) a random sample of 25 (n) is taken from a population in which the proportion with the trait is 0.50 (p)

D) none of the above choices indicate a situation where the sampling distribution of the sample proportion is normally distributed

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