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 µ) 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
13.
For which of the following situations would the histogram of the sample mean (sampling distribution) NOT be normally distributed (follow a normal curve)?
A) A random sample of size 20 is drawn from a skewed population.
B) A random sample of size 50 is drawn from a skewed population.
C) A random sample of size 20 is drawn from a normally distributed population.
D) A random sample of size 50 is drawn from a normally distributed population.
Histograms of sample statistic values (i.e. sampling distributions) will be centered at the population value (µ or p) and will have an average spread around the population value that is measured by the standard deviation of the sampling distribution.
The Standard Deviation of our sampling distribution
= formula for a sampling distribution for sample means (where σ is the population standard deviation and n is the sample size)
= formula for a sampling distribution for sample proportions (where p is the population proportion and n is the sample size)
14.
We are given that the average amount spent on textbooks by all incoming freshmen was $400 (μ) with a standard deviation of $25 (σ). The sampling distribution of the sample mean for randomly selected samples of 100 freshmen has a standard deviation of $2.50. Which of the following is the correct interpretation of the population standard deviation compared to the sampling distribution’s standard deviation?
A) The population standard deviation of $25 gives the variation in amount spent amongst all freshmen. The sampling distribution’s standard deviation of $2.50 gives the variation in the average amount spent amongst all samples of 100 freshmen.
B) The population standard deviation of $25 gives the variation in the average amount spent amongst all samples of 100 freshmen. The sampling distribution’s standard deviation of $2.50 gives the variation in amount spent amongst all freshmen.
15.
Forty percent (40% or 0.40, as a proportion) of all cars owned by residents of Lordstown, OH and the surrounding area are the Chevrolet brand. If a random sample of 81 car owners is selected in that area, then the sampling distribution of the sample proportion of those who own a Chevrolet brand car will be centered around 0.40 (40%) with a standard deviation of:
A) .09 or 9%
B) .054 or 5.4%
C) .946 or 94.6%
D) .044 or 4.4%
16.
A kitchen cabinet business has primarily homeowners as customers, but also has a few customers that are developers who build multi-unit condominiums. Overall, the average sale is $15,000, with a median sale of $10,000 and a standard deviation of $4,800. The standard deviation of the averages for samples of 100 customers would be which of the following? In other words, what is the standard deviation of the sampling distribution of the sample means from samples of 100 customers?
A) $4,800
B) $480
C) $48
D) $4.80
If the sampling distribution is normally distributed, we can use the Standard Normal Table (Table 8.1) to compute the probabilities of certain intervals of sample statistic values. We calculate the Z score using a formula given below that uses the standard deviation of the sampling distribution in the denominator and that uses the observed sample statistic and population parameter in the numerator.
17.
Twenty percent (20%) of all students who live on campus at Penn State’s University Park campus have a car that they have registered to park in one of the University’s parking lots. You randomly select 100 students who live on campus and find that 24 of the 100 have a car registered to park in one of the University’s parking lots. What is the standard score (i.e. Z score) of your sample proportion?
A) +.24
B) + 1.00
C) -1.00
D) + .20
18.
Twenty percent (20%) of all students who live on campus at Penn State’s University Park campus have a car that they have registered to park in one of the University’s parking lots. You randomly select 100 students who live on campus and find that 18 of the 100 have a car registered to park in one of the University’s parking lots. A sample proportion of 0.18 has a standard score (Z score) of -0.50. What is the probability of having a sample proportion less than 0.18? Use Table 8.1.
A) 69%
B) 31%
C) 50%
D) 95%
19.
The average monthly cell phone bill for all customers of a particular cell phone carrier in Pennsylvania is $85 with a standard deviation of $10. What is the standard score (Z score) for a random sample of 25 customers with an average amount spent monthly of $90?
A) +.50
B) – .50
C) +2.5
D) -2.5
20.
The average monthly cell phone bill for all customers of a particular cell phone carrier in Pennsylvania is $85 with a standard deviation of $10. For a sample size of 25, a sample mean of $87 has a Z score of +1.00, and a sample mean of $83 has a Z score of -1.00. We want to know the probability of obtaining a sample of size 25 that has a sample mean between $83 and $87. To find this we must find the probability of obtaining a Z score between -1 and +1. What is this probability?
A) 90%
B) 95%
C) 68%
D) none of the above
