Deck 32: Variability in Samples

Question 1

A report says that 80% of the children are overweight. Which of the following is more likely to happen? Explain your answer.

A) Five children are weighed, and three are overweight.
B) One hundred children are weighed, and 60 are overweight.

Answer 1

Choice A is more likely since the larger sample in B should give an estimate closer to 80%.

Question 2

A radio report says, "Only 40% of the children are physically fit." Which sample size would be the more informative one to test the statement?

A) 10 children
B) 50 children

Answer 2

50 children

Question 3

A radio report says, "Only 40% of the children are physically fit." Which scenario would be more likely to happen (or equally likely)? Explain your answer.

A) Ten children are tested, and seven are physically fit.
B) Fifty children are tested, and 35 are physically fit.

Answer 3

Choice A would be more likely to happen because the larger sample in B should give a proportion closer to the 40%.

Question 4

Suppose that population A has 10 million items. If a sample of size 2000 is used, then a sample of size 4000 would give twice as much confidence in the result. Explain your decision.

Question 5

In testing a life-or-death drug, a company increased its sample size from 25 to 100 before asking for Food and Drug Administration approval. Why would they do this?

Question 6

Which answer BEST explains of the meaning of the following statement?
"Based on the results in our survey, a confidence interval from 30% to 38% has a 90% confidence level."

A) Ninety percent of the samples surveyed were in the 30% to 38% range.
B) Thirty percent to 38% of the population parameter has a probability of 90%.
C) Thirty percent to 38% of the samples surveyed has a confidence level of 90%.
D) There is a 90% chance that the population percent is in the 30% to 38% range.

Question 7

The Federal Trade Commission studied 300 ads for diets and diet aids and found that 55% made at least one misleading claim. Give a confidence interval for the percentage of diet or diet aid ads that contain misleading claims. Show your work.

Question 8

You hire 100 people to survey city residents about the question, "Do you think the county park should be closed in order to resolve the budget crisis?" Each person interviews 25 people. They report back, and here is a tabulation of the 100 samples.
$$\begin{array} { | l | l | } \hline \begin{array} { c } \text { Number of } 100 \text { samples giving } \\\text { indicated sample percents }\end{array} & \begin{array} { l } \text { Number of } 100 \text { samples } \\\text { giving indicated sample } \\\text { percents }\end{array} \\\hline \text { Percent of sample who are for } \\\text { the closing } & \begin{array} { l } \text { Number of samples of } \\\text { size } 25\end{array} \\\hline 12 \% & 2 \text { samples } \\\hline 16 \% & 3 \text { samples } \\\hline 20 \% & 6 \text { samples } \\\hline 24 \% & 7 \text { samples } \\\hline 28 \% & 13 \text { samples } \\\hline 32 \% & 19 \text { samples } \\\hline 36 \% & 18 \text { samples } \\\hline 40 \% & 14 \text { samples } \\\hline 44 \% & 13 \text { samples } \\\hline 48 \% & 3 \text { samples } \\\hline 52 \% & 2 \text { samples } \\\hline\end{array}$$

A) From the data above, give a good estimate of the percent of the whole population who are for closing the park. Explain your decision.
B) Give a 90% confidence interval for the percentage of people in the population who favor the county park closing. Use the table, rather than the $\frac { 1 } { \sqrt { n } }$ idea. With a 90% confidence interval:
C) Why is the $\frac { 1 } { \sqrt { n } }$ idea not good here?
D) If there were an added column for 100 samples, each sample of size 100, how would the column look compared to the column for samples of size 25?

Question 9

You hire 100 people to survey the city's residents about the question, "Do you approve of the job the mayor is doing?" Each person interviews 25 people. They report back, and here is a tabulation of the 100 samples.
Number of samples (of 100 samples of 25 each) giving indicated sample percents
A) From the data above, give a good estimate of the percent of the whole population who are for the mayor. Explain your decision. $$\begin{array} { | l | l | } \hline \begin{array} { r } \text { Percent of sample who are for } \\\text { the mayor }\end{array} & \text { Number of samples } \\\hline 48 \% & 1 \text { sample } \\\hline 44 \% & 4 \text { samples } \\\hline 40 \% & 5 \text { samples } \\\hline 36 \% & 4 \text { samples } \\\hline 32 \% & 14 \text { samples } \\\hline 28 \% & 21 \text { samples } \\\hline 24 \% & 19 \text { samples } \\\hline 20 \% & 16 \text { samples } \\\hline 16 \% & 11 \text { samples } \\\hline 12 \% & 2 \text { samples } \\\hline 8 \% & 3 \text { samples } \\\hline\end{array}$$

A) From the data above, give a good estimate of the percent of the whole population who are for the mayor. Explain your decision.
B) Using the table above, give a confidence interval for the percentage of people in the population who are for the mayor. Your confidence interval should have a 90% confidence level.With a 90% confidence level:
C) Why is the $\frac { 1 } { \sqrt { n } }$ idea not good here?
D) If there were an added column for 100 samples, each sample of size 100, how would the column look compared to the column for samples of size 25?

Question 10

You hire 100 people to survey city residents about the question, "Do you approve of a new stadium for the football team?" Each person interviews 100 people. They report back, and here is a tabulation of the 100 samples.

A) One of the people sees the table and says, "Oh! So the percent of all the residents who approve is definitely 34%, just like in my sample!" What do you say? A) One of the people sees the table and says, "Oh! So the percent of all the residents who approve is definitely 34%, just like in my sample!" What do you say?
B) Using the table above, give a confidence interval for the percentage of people in the population who approve. Your confidence interval should have a 90% confidence level.
Confidence interval:
C) Why is the idea not good here?
D) If there were an added column for 100 samples, each sample of size 200, how would the column look compared to the column for samples of size 100?

Question 11

"With our survey showing 75% for, we are 95% confident that 71% to 79% will vote for."

A) What does the statement in quotes mean?
B) Using the quote, put the correct percents for "for" votes in the blanks to the right of the labels in the diagram below.

Question 12

"With our survey showing 65% for, we are 95% confident that 61% to 69% will vote for."

A) What does the statement in quotes mean?
B) Using the quote, put the correct percents for "for" votes in the blanks to the right of the labels in the diagram below.

Question 13

"Based on the test score average of 85% at Washington Elementary School, I would say scores for the whole district are in the 82% to 88% range, with 95% confidence."

A) Why does the person say, "Scores for the whole district will be in the 82% to 88% range," rather than give a definite percent?
B) Using the quote, put the correct percents for test score average in the blanks to the right in the diagram below.

Question 14

A) Suppose we took a random sample of 400 statewide voters and asked whether they would raise taxes to support schools more. Suppose 45% say they would. What (numerically) can be said about the percent of all voters who would raise taxes?
B) What is the confidence level associated with your answer in part A?
C) In a pie chart (circle graph) for the data in part A, how many degrees would these positive responses get? Which choice below is correct? Write enough (numbers or words) to make your thinking clear.

Question 15

A) Suppose we took a random sample of 400 statewide voters and asked whether they would raise taxes to support schools more. Suppose 55% say they would. What (numerically) can be said about the percent of all voters in the state who would raise taxes?
B) What is the confidence level associated with your answer in part A?
C) In a pie chart (circle graph) for the data in part A, how many degrees would these positive responses get? Which choice below is correct? Write enough (numbers or words) to make your thinking clear.
i. $55 ^ { \circ }$
ii. 73
iii. $198 ^ { \circ }$
iv. $220 ^ { \circ }$

Question 16

The $\frac { 1 } { \sqrt { n } }$ rule of thumb can be used if _____ and if _____.

Question 17

A larger sample gives a larger confidence interval for the percent of "yeses."

Question 18

A survey concludes, "So with a confidence level of 95%, our figures show that only 48% of the people, ± 5%, prefer Brand X."

A) What does the statement mean?
B) About how many people did they survey?

Question 19

State candidate X's headquarters has had local groups run polls in large cities. In City A, 486 of 900 voters polled said that they would vote for X. In City B, 56 of 100 voters polled said that they would vote for X. Headquarters has some more money to spend on advertising in City A and/or City B, and they think they should do it in City
A. However, they have decided to consult you. Should they spend more campaign money in City A or City B? Explain.

Question 20

A drug company wants to judge the effectiveness of a new drug by testing it on a sample of people. They would like to know the percent that the drug will help within 0.5%.

A) They ask you for advice on how many people they should test. What do you say?
B) What assumptions did you make in your work in part A?

Question 21

A) A random sample of 40 university graduates showed that 55% earned at least $45,000 a year. Give a confidence interval for the percentage of university graduates who earn at least $45,000 a year.
B) If you take a larger sample size in A, will your confidence interval be longer or shorter?