Deck 19: The Diversity of Samples From the Same Population

Question 1

For Questions , use the following narrative
Narrative: Politics
Suppose a population contains 60% Republicans and 40% Democrats.
{Politics narrative} Suppose you take a random sample of 10 people from this population.Does the rule for sample proportions apply in this situation? Explain your answer.

Answer 1

No, the sample size is not large enough.the expected number of democrats is only 4, which is less than the 5 needed for the rule to hold.

Question 2

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} What is the chance that the proportion of cell phone owners in a sample of size 2,500 from this population will be more than two standard deviations from the expected mean?

Answer 2

5% or .05

Question 3

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} What range of proportions of cell phone owners is reasonable to expect from this population (assuming your sample size is 2,500)? Justify your answer

Answer 3

95% of the sample proportions should lie between .184 and .216

Question 4

For Questions , use the following narrative
Narrative: Bananas
Suppose a researcher asks the question: What is the average weight of bananas selected for purchase by customers in grocery stores? Assume the distribution of weights is the same across all stores.
{Bananas narrative} Suppose the researcher takes a sample of 100 shoppers and finds the average weight of their banana purchases to be 2.1 pounds.Can he just go ahead and report that the average banana purchase for all grocery store customers is 2.1 pounds? Why or why not?

Question 5

For Questions , use the following narrative
Narrative: Politics
Suppose a population contains 60% Republicans and 40% Democrats.
{Politics narrative} Suppose numerous random samples of size 1,000 are taken from this population.How will the shape, mean, and standard deviation of the frequency curve for the proportions of Democrats in the samples differ from the shape, mean, and standard deviation of the frequency curve for the proportions of Republicans in the samples?

Question 6

Which of the following statements is false?

A)Sample results will always be very close to their respective population values.
B)Sample results vary from one sample to the next.
C)The key to interpreting statistical results is to understand what kind of dissimilarity we should expect to see in various samples from the same population.
D)None of the above statements are false.

Question 7

For Questions , use the following narrative
Narrative: Bananas
Suppose a researcher asks the question: What is the average weight of bananas selected for purchase by customers in grocery stores? Assume the distribution of weights is the same across all stores.
{Bananas narrative} How can the researcher go about answering this question? Give a general description of the process.

Question 8

For Questions , use the following narrative
Narrative: Politics
Suppose a population contains 60% Republicans and 40% Democrats.
{Politics narrative} Is it possible to get a random sample that does not represent the population well, in terms of Democrats and Republicans? Explain your answer.

Question 9

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} The chance of having at least 20% cell phone owners in a random sample from this population is 50%, no matter what size the sample is (as long as it is large enough for the Rule for Sample Proportions to apply).Why?

Question 10

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} The frequency curve made from proportions of cell phone owners from the various samples of size 2,500 from this population will have what approximate mean and standard deviation?

Question 11

For Questions , use the following narrative
Narrative: Politics
Suppose a population contains 60% Republicans and 40% Democrats.
{Politics narrative} Suppose you take a random sample of 10 people from this population.Are you certain that you would get 6 Republicans and 4 Democrats in your sample? Explain your answer.

Question 12

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} Suppose you took a random sample of size 2,500 from this population and found that 17.6% of them owned a cell phone.Is this considered to be a reasonable value given the size of this sample? Use the standardized score in your answer.

Question 13

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} Suppose your sample size was only 250.What range of proportions of cell phone owners is reasonable to expect from this population? Justify your answer

Question 14

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} The frequency curve made from proportions of cell phone owners from the various samples of size 2,500 from this population will have what approximate shape?

Question 15

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} Suppose you took a random sample of size 2,500 from this population and found that 21.6% of the people in this sample owned a cell phone.Is this considered to be a reasonable value given the size of this sample? Use the standardized score in your answer.

Question 16

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} How would the frequency curve made from proportions of cell phone owners from the various samples of size 2,500 compare to the frequency curve made from proportions of cell phone owners from the various samples of size 250?

Question 17

In practice you don't know the population value, and you take a sample in order to estimate what the population value is.Once you take a specific sample, is it possible to determine whether or not that sample is an accurate reflection of the population? Explain your answer.

Question 18

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} What is the chance that a sample of size 2,500 from this population will contain at least 20% cell phone owners?

Question 19

For Questions , use the following narrative
Narrative: cell phone owners
Suppose numerous random samples of size 2,500 are taken from a population made up of 20% cell phone owners.
{Cell phone owners narrative} What is the chance that less than 20.8% of the people in a sample of size 2,500 from this population will own a cell phone?

Question 20

Suppose you knew that most samples were likely to provide an answer that is within 10% of the population value.What would also be true in that case?

A)The population value should be within 10% of whatever our specific sample gave us.
B)10% of the population values should be close to whatever our specific sample gave us.
C)The chance that a specific sample answer is correct (equal to the population value) is 90%.
D)All of the above statements are true.

Question 21

For Questions , use the following narrative
Narrative: Test scores
Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and that they have a bell-shaped curve.
{Test scores narrative} Suppose you take a single random sample of size 100 people from this population.What is the chance that their average test score will be above 77?

Question 22

For Questions , use the following narrative
Narrative: Test scores
Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and that they have a bell-shaped curve.
{Test scores narrative} Suppose you randomly select a single individual from this population.Where would you expect his/her test score to fall?

Question 23

If numerous large random samples or repetitions of the same size are taken from a population, the frequency curve made from means from the various samples will have what approximate shape?

A)A flat shape; each outcome should be equally likely.
B)A bell-shape.
C)A histogram.
D)Unknown; it can change every time.

Question 24

In which of the following situations does the rule for sample proportions apply?

A)A pollster takes a random sample of 1,000 Americans and asks their opinion on the President (approve/disapprove/neutral).He is interested in the percentage who approve of the President.
B)You want to know whether or not people like the new CD by your favorite artist.You ask 5 people and record the percentage who say they like it.
C)A researcher weighs the same newborn baby each week for one year, and records whether or not the child is within the normal weight range.At the end of the year, he records the percentage of times that the child was within the normal weight range.
D)All of the above.

Question 25

For Questions , use the following narrative
Narrative: Test scores
Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and that they have a bell-shaped curve.
{Test scores narrative} Suppose that the test scores are not bell-shaped, but are skewed to the right.You want to take a random sample and estimate the average test scored for this population.You want to be able to use the rule of sample means to interpret your results in this case.Under what (if any) conditions is this possible?

Question 26

Explain, in words that a non-statistics student would understand, why the standard deviation of the various sample means taken from a population is smaller than the standard deviation of the individuals in the population? (In the first situation, assume all samples are of the same size, and that size is large.)

Question 27

For Questions , use the following narrative
Narrative: Test scores
Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and that they have a bell-shaped curve.
{Test scores narrative} Suppose you randomly select a sample of size 100 from this population.Where would you expect their average test score to fall? Compare your answer to what you would expect from a single individual selected at random from this population.

Question 28

The standard deviation of the proportions from numerous random samples of size 1,000 from a population will be __________ the standard deviation of the proportions from numerous random samples of size 10,000 from the same population.

Question 29

For Questions , use the following narrative
Narrative: Test scores
Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and that they have a bell-shaped curve.
{Test scores narrative} Suppose you take a single random sample of size 100 from this population, and you get a mean test score of 79.Is this something that you would have expected? Use a probability to justify your answer.

Question 30

If numerous large random samples or repetitions of the same size are taken from a population, the frequency curve made from proportions from the various samples will have an approximate __________ shape.

Question 31

Which of the following are examples where you would be interested in estimating the population mean?

A)About how long do left-handed people live?
B)Do most people support or oppose the President's foreign policy?
C)What size was the viewing population who tuned in to the ABC News special last night?
D)All of the above.

Question 32

If numerous large random samples or repetitions of the same size are taken from a population, the proportions from the various samples will have what approximate mean?

A)The true population proportion.
B)The true population average.
C)95% because most of them will be within 2 standard deviations of the true population value.
D)None of the above.

Question 33

For Questions , use the following narrative
Narrative: Test scores
Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and that they have a bell-shaped curve.
{Test scores narrative} Suppose you take numerous random samples of size 100 from this population.Describe the shape and give the mean and standard deviation of the resulting frequency curve.

Question 34

In which of the following situations does the rule for sample means not apply?

A)A pollster takes a random sample of 1,000 Americans and asks them to give their opinion of the President on a scale from 1 (completely disapprove) to 100 (completely approve).He is interested in the average rating.
B)You take a random sample of 20 students' scores from the ACT exam and record the average score.Assume ACT scores are bell-shaped.
C)A sports fan takes a random sample of 20 NBA players and records their salaries.He wants to estimate the average salary for the entire NBA.
D)The rule for sample means does not apply in any of these situations.

Question 35

For Questions , use the following narrative
Narrative: Test scores
Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and that they have a bell-shaped curve.
{Test scores narrative} Find and compare the answers to the following two questions; explain why your answers are the same or different.1) One individual is selected at random from the population.What range of test scores is reasonable to expect for this person? 2) A sample of 100 individuals is selected at random from the population.What range of average test scores is reasonable to expect for this group?

Question 36

For Questions , use the following narrative
Narrative: Test scores
Suppose that test scores on a particular exam have a mean of 77 and standard deviation of 5, and that they have a bell-shaped curve.
{Test scores narrative} Suppose you take a single random sample of size 100 from this population, and you get a mean test score of 76.Is this something that you would have expected? Use a probability to justify your answer.

Question 37

If numerous large random samples or repetitions of the same size are taken from a population, the frequency curve made from proportions from the various samples will have what approximate shape?

A)A bar graph with two bars, one for the proportion having the trait of interest, and the other for the proportion not having the trait of interest.
B)A bell-shape.
C)A flat shape; each outcome should be equally likely.
D)Unknown; it can change every time.

Question 38

{Politics narrative} Suppose numerous random samples of size 1,000 are taken from this population.The proportions of Republicans from the various samples of size 1,000 will have what approximate standard deviation?

A).24 or 24%
B).00024 or .024%
C).015 or 1.5%
D)None of the above.

Question 39

{Politics narrative} Suppose numerous random samples of size 1,000 are taken from this population.The proportions of Democrats from the various samples of size 1,000 will have what approximate standard deviation?

A).24 or 24%
B).00024 or .024%
C).015 or 1.5%
D)None of the above.

Question 40

If numerous large random samples or repetitions of the same size are taken from a population, the frequency curve made from proportions from the various samples will have a mean that is __________ the true population proportion.

Question 41

A __________ provides a range of possible values for the correct value in a population, whether that correct value is a mean, a proportion, or something else.

Question 42

Suppose someone has a theory that 'something interesting is happening' in the population.What type of statistical technique will they most likely use to 'prove' or 'disprove' their theory, a confidence interval or a hypothesis test?

Question 43

Suppose you do not know anything about what the true population mean is, and you want to estimate it using a sample mean.Which statistical technique would you use, a confidence interval or a hypothesis test?

Question 44

When someone reports that their results are found to be 'statistically significant', which type of statistical technique was most likely used?

A)A confidence interval.
B)A hypothesis test.
C)A significance interval.
D)Not enough information to tell.

Question 45

Name the two basic techniques researchers use to summarize and draw conclusions about a population based on their statistical results from a sample.

Question 46

Suppose a certain population was reported in the news as being 60% Republican.You take a random sample of 100 people from this population and find that 67% of your sample is Republican.What can you conclude about the news report, based on your sample results?

A)You can't say they are wrong; your results are within the expected limits for a sample this size.
B)A sample of size 100 is never large enough to make any conclusions either way about an entire population.
C)You can say they are wrong because your sample results are too far away from what was reported in the news.
D)None of the above.

Question 47

Samples of size 2,500 will produce estimates of the population value that are __________ times more accurate than samples of size 25.(Assume the population is bell-shaped.Use standard deviation as a measure of accuracy.)

Question 48

Larger samples tend to result in __________ accurate estimates of population values than do smaller samples.

Question 49

Achieving statistical significance is equivalent to __________ the idea that chance alone can explain the observed results.