chapter 8: confidence intervals and levels

The golf ball manufacturer would also like to investigate the variability of the distance travelled by the golf balls by estimating the interquartile range. What is the point estimate?

Use the sample IQR as the point estimate for the true IQR

The math department wants to know what proportion of its students own a graphing calculator, so they take a random sample of 100 students and find that 28 own a graphing calculator. What is the point estimator?

Use the sample proportion p-hat as the point estimator for the true proportion p (the sample proportion is 0.28)

interpreting confidence interval

We are 95% confident that the interval from ____ to ____ captures the actual value of the (context)

point estimator

a statistic that provides a estimate of a population parameter

You have measured the systolic blood pressure of an SRS of 25 company employees. A 95% confidence interval for the mean systolic blood pressure for the employees of this company is (122, 138). Which of the following statements gives a valid interpretation of this interval? a. If the procedure were repeated many times, 95% of the resulting confidence intervals would contain the population mean systolic blood pressure. b. 95% of the population of employees have a systolic blood pressure between 122 and 138. c. The probability that the population mean blood pressure is between 122 and 138 is 0.95. d. 95% of the sample of employees have a systolic blood pressure between 122 and 138. e. If the procedure were repeated many times, 95% of the sample means would be between 122 and 138.

a. If the procedure were repeated many times, 95% of the resulting confidence intervals would contain the population mean systolic blood pressure

the point estimator can be...

potential mean, standard deviation, IQR, median, etc.

the t distribution have more _____ in the tails and less in the center that standard normal

probability

Normal conditions (CI): proportions

proportions: np and n(1-p) must both be greater than/equal to 10

t distribution normal condition

sample size less than 15: use if data is Normal/close to (don't if skewed/there are outliers) sample size at least 15: use t except if skewed/outliers large samples: t can be used if n is at least 30

as degrees of freedom increase, the t distribution approaches...

standard normal curve

when standard deviation is unknown, use...

t distribution

confidence level

tells us how likely it is that the method we're using will produce and interval that captures the population parameter if we use it many times

the confidence level does not tell us...

the chance that a particular confidence interval captures the population parameter

if the sample proportion is unknown, we use what p-hat value?

0.50

what are the z-scores for 99%, 95%, and 90% confidence intervals?

99%: 2.575 95%: 1.96 90%: 1.645

confidence interval conditions

Random (SRS or randomized experiment), Normal, Independent (10% rule)

A Census Bureau report on the income of Americans says that with 90% confidence the median income of all U.S. households in a recent year was $57,005 with a margin of error of ± $742. This means that... a. The Census Bureau got the results $57,005 ± $742 using a method that will cover the true median income 90% of the time when used repeatedly. b. 90% of the households in the sample interviewed by the Census Bureau had incomes in the range $57,005 ± $742. c. 90% of all possible samples of this same size would result in a sample median that falls within $742 of $57,005. d. 90% of all households had incomes in the range $57,005 ± $742. e. We can be sure that the median income for all households in the country lies in the range $57,005 ± $742.

a. The Census Bureau got the results $57,005 ± $742 using a method that will cover the true median income 90% of the time when used repeatedly.

Scientists collect data on the blood cholesterol levels (milligrams per deciliter of blood) of a random sample of 24 laboratory rats. A 95% confidence interval for the mean blood cholesterol level μ is 80.2 to 89.8. Which of the following would cause the most worry about the validity of this interval? a. There is a clear outlier in the data. b. None of these would be a problem because the t procedures are robust. c. The population distribution is not exactly Normal. d. A stemplot of the data shows a mild right-skew. e. You do not know the population standard deviation σ.

a. There is a clear outlier in the data.

A quality control inspector will measure the salt content (in milligrams) in a random sample of bags of potato chips from an hour of production. WHich of the following would result in the smallest margin of error in estimating the mean salt content μ ? a. 95% confidence; n = 50 b. 90% confidence; n = 50 c. 90% confidence; n = 25 d. 95% confidence; n = 25 e. n = 100 at any confidence level

b. 90% confidence; n=50

In preparing to construct a one-sample t interval for a population mean, suppose we are not sure if the population distribution is Normal. In which of the following circumstances would we not be safe constructing the interval based on an SRS of size 24 from the population? a. A histogram of the data show slight skewness. b. A stemplot of the data has a large outlier. c. The sample standard deviation is too large. d. A stemplot of the data is roughly bell-shaped. e. The t procedures are robust, so it is always safe.

b. A stemplot of the data has a large outlier

A telephone poll of an SRS of 1234 adults found that 62% are generally satisfied with their lives. The announced margin of error for the poll was 3%. Does the margin of error account for the fact that some adults do not have telephones? a. Yes. The margin of error includes all sources of error in the poll. b. No. The margin of error only includes sampling variability. c. No. The margin of error includes nonresponse but not undercoverage. d. Yes. Taking an SRS eliminates any possible bias in estimating the population proportion. e. Yes. The margin of error includes undercoverage but not nonresponse.

b. No. The margin of error only includes sampling variability

A polling organization announces that the proportion of American voters who favor congressional term limits is 64%, with a 95% confidence and margin of error of 3%. If the opinion poll had announced the margin of error for 80% confidence rather than 95% confidence, the margin of error would be... a. 3%, because the sample sample is used. b. less than 3%, because we require less confidence. c. less than 3%, because the sample is smaller. d. greater than 3%, because we require less confidence. e. greater than 3%, because the sample size is smaller.

b. less than 3%, because we require less confidence

A researcher plans to use a random sample of n = 500 families to estimate the mean monthly family income for large population. A 99% confidence interval based on the sample would be...than a 90$ confidence interval. a. wider, but it cannot be determined whether the risk of being incorrect would be larger or smaller. b. wider and would involve a smaller risk of being incorrect. c. narrower and would involve a larger risk of being incorrect. d. wider and would involve a larger risk of being incorrect. e. narrower and would involve a smaller risk of being incorrect.

b. wider and would involve a smaller risk of being incorrect.

the spread of t distributions is a...

bit greater than standard Normal distribution

which of the following has the highest probability? a. randomly selecting a value greater than 3 from a standard normal distribution b. randomly selecting a value greater than 3 from a t-distribution with 4 degrees of freedom. c. randomly selecting a value greater than 3 from a t-distribution with 20 degrees of freedom d. randomly selecting a value less than 3 from a standard normal distribution e. randomly selecting a value less than 3 from a t-distribution with 20 degrees of freedom.

d. randomly selecting a value less than 3 from a standard normal distribution

Normal conditions (CI): means

means: the sampling distribution is exactly normal if the population distribution is normal. when the population distribution isn't normal, then the Central Limit Theroem tells us the sampling distribution will be approx. Normal if n is at least 30

the report of a sample survey of 1041 adults says, "With 95% confidence, between 9% and 15% of all Americans expect to spend more money on gifts this year than last year." What does the phrase "95% confidence" mean? a. 95% of all Americans will spend between 9% and 15% more than what they spend last year. b. 9% to 15% of all Americans will spend 95% of what they spent last year. c. there is a 95% chance that the percent who expect to spend more is between 9% and 15%. d. the method used to get the interval from 9% to 15%, when used over and over, produces the intervals which include the true population percentage about 95% of the time. e. we can be 95% confident that the method used to get the interval always gives the right answer.

d. the method used to get the interval from 9% to 15%, when used over and over, produces the intervals which include the true population percentage about 95% of the time.

A Gallup Poll found that only 28% of American adults expect ot inherit money or valuable possessions from a relative. The poll's margin of error was ±3 percentage points at a 95% confidence level. This means that... a. if Gallup takes another poll on this issue, the results of the second poll will lie between 25% and 31%. b. the percent of all adults who expect an inheritance is between 25% and 31%. c. there's a 95% chance that the percent of all adults who expect an inheritance is between 25% and 31 % d. the poll used a method that gets an answer within 3% of the truth about the population 95% of the time. e. Gallup can be 95% confident that between 25% and 31% of the sample expect an inheritance.

d. the poll used a method that gets an answer within 3% of the truth about the population 95% of the time.

t distribution

does not have normal distribution--it's symmetric w/single peak at 0, but has more area in the tails (remember degrees of freedom is n-1)

The Gallop Poll interviews 1600 people. Of these, 18% say that they jog regularly. The new report adds: "The poll had a margin of error of plus or minus three percentage points at a 95% confidence level." You can safely conclude that... a. the percent of the population who jog is certain to be between 15% and 21%. b. 95% of the population jog between 15% and 21% of the time. c. we can be 95% confident that the sample proportion is captured by the confidence interval. d. if Gallup took many samples, 95% of them would find that 18% of the people in the sample jog. e. 95% of all Gallup Poll samples like this one give answers within ± 3% of the true population value.

e. 95% of all Gallup Poll samples like this one give answers within a 3% of the true population value

In a poll, I. Some people refused to answer questions. II. People without telephones could not be in the sample. III. Some people never answered the phone in several calls. Which of theses sources is included in the ±2% margin of error announced for the poll? a. I, II, and III b. III only c. I only d. II only e. None of these

e. None of these

I collect an SRS of size n from a population and compute a 95% confidence interval for the population proportion. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data? a. Nothing can guarantee absolutely that you will get a larger interval. One can only say that the chance of obtaining a larger interval is 0.05. b. Use the same confidence level, but compute the interval n times. Approximately 5% of these intervals will be larger. c. Increase the sample size. d. Use a smaller confidence level. e. Use a larger confidence level.

e. Use a larger confidence level

A study was done by calling land-line telephones, and those conducting the survey are concerned about the possibility of undercoverage, since some people don't own a phone or only own a cell phone. Which of the following is the best way for them to correct for this source of bias? a. use a lower confidence level, such as 80% b. use a higher confidence level, such as 99% c. take a larger sample d. use a t-interval instead of a z-interval e. throw this sample out and start over again with a better sampling method.

e. throw this sample out and start over again with a better sampling method.

One reason for using a t distribution instead of the standard Normal curve to find critical values when calculating a level C confidence interval for a population mean is that... a. z can be used only for large samples. b. the standard Normal table doesn't include confidence levels at the bottom. c. z requires that you can regard your data as an SRS from the population. d. a z critical value will lead to a wider interval than a t critical value. e. z requires that you know the population standard deviation σ

e. z requires that you know the population standard deviation σ

when is an inference procedure considered robust?

if probability calculations involved remain fairly accurate when condition for using procedures is violated

interpreting confidence level

in 95% of all possible samples of the same size, the resulting confidence interval would capture the true (context)

greater confidence requires...

larger critical value (z-score)

the standard deviation depends on..

the sample size n

point estimate

the value of that statistic from a sample; "best guess" at the value of an unknown parameter

how does the width of a confidence interval change if you take a larger sample?

the width would become narrower because standard deviation is smaller; therefore, the interval is smaller also

when does the margin of error decrease?

when confidence level decreases and sample size increases (graph becomes narrower)

are t procedures robust?

yes, especially population is non-normal (in large sample sizes). but not robust against outliers