Chapter 8: Confidence Interval Estimation

Confidence level

Confidence the interval will contain the unknown population parameter and will be a percentage LESS THAN 100% 1-a (a/alpha=error) •Suppose confidence level = 95% •Also written (1 - a) = 0.95, (so a = 0.05)

We understand that a statistic is just an _______ and doesn't likely reflect the true ________.

estimate, population parameter

Ideally, you should have a _____ confidence level with a _____ interval width.

high, small

Holding the sample size constant, as you decrease your confidence level, the interval width ____

increases

Confidence interval formula

point estimate +/- margin of error Point estimate +/- (critical value)(standard error)

The percentage of left handers is somewhere between 8% and 12%. Or the percentage of left handers is somewhere between 2% and 22%. Which is more useful?

somewhere between 8% and 12%.

critical value

table value based on the sampling distribution of the point estimate and the desired confidence level.

Standard error

the standard deviation of the point estimate

statistical tie

when there is an overlapping of confidence intervals after accounting for your margin of error.

Suppose a 95% confidence interval for µ has been constructed. If it is decided to take a larger sample and to decrease the confidence level of the interval, then the resulting interval width would _______________. (Assume that the sample statistics gathered would not change very much for the new sample).

would be narrower than the current interval width

confidence level is chosen by

you, the statistician

Holding the confidence level constant, as you increase the sample size, interval width _____

decreases

Confidence interval in practice...

- You do NOT know the true population parameter - You do NOT know if the interval actually contains the true population parameter - You DO know that is 95% formed in this manner WILL contain the true population parameter. .... Therefore, based on the ONE sample, you CAN be 95% confident your interval will contain the true population parameter. (This is a 95% confidence interval)

Point estimate

- is a single value (or point) used to approximate a population parameter - sample statistic estimating the population parameter of interest.

A major department store chain is interested in estimating the mean amount of its credit card customers spent on their first visit to the chain's new store in the mall. 15 credit card accounts were randomly sampled and analyzed and a 95% confidence interval was constructed: $50.50 +/- $11.08. Identify the margin of error.

11.08

The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following point estimate: 830 books a day. If a 95% confidence interval is constructed and the margin of error is 30, what is the resulting confidence interval?

800,860

A relative frequency interpretation

95% of all the confidence intervals that can be constructed will contain the unknown true parameter •A specific interval either will contain or not contain the true parameter -No probability involved in a specific interval

Which is more useful: a wider or narrower interval? Why?

A narrower interval is more useful because we cannot make very many conclusions if the estimate is too wide. Consider the following example: The percentage of left handers is somewhere between 8% and 12%. Or the percentage of left handers is somewhere between 2% and 22%. Which is more useful?

Can be constructed around 1 sample statistic to give some leeway for an estimation.

Confidence Interval

To increase our level of confidence from 95% to 99%, we should expect the width of the confidence interval to increase/decrease.

INCREASE

Suppose a 95% confidence interval for µ turns out to be (1,000, 2100). Give a definition of what it means to be "95%" confident as an inference.

In repeated sampling, 95% of the intervals constructed would contain the population mean.

Suppose a 95% confidence interval for µ turns out to be (1000, 2100). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Which of the following will result in a reduced interval width?

Increase the sample size

A narrow confidence interval is more/less useful than a wide one.

MORE

A confidence interval was used to estimate the proportion of statistics students who are females. A random sample of 72 statistics students generated the following 90% confidence interval (.438, .642). Based on the interval above, is the proportion of females equal to .6?

Maybe... .60 is a believable value of the population proportion based in the information given.

Holding the SAMPLE SIZE constant, if we decrease our confidence, this will result in a more narrow/wide interval.

NARROW **If we increase our confidence, this will result in wider interval

Holding the CONFIDENCE LEVEL constant, if we increase the sample size, this will result in a more narrow/wide interval.

NARROW If we decrease the sample size, this will result in a wider interval.

confidence interval (Interval estimate)

Provides additional information about the variability of an estimate. - Gives a range of what part the possible parameter could be. - Provides more information about a population characteristic than a point estimate. - Calculated based on the level of confidence (decided upon by the statistician; most common is 95%) and the standard error. - Provides a range of values - Takes into consideration variation in sample statistics from sample to sample. - Based on observations from 1 sample - Gives information about closeness to unknown population parameters - Stated in terms of level of confidence. (can never be 100% confident)

T/F The t distribution is used to develop a confidence interval estimate of the population proportion when the population standard deviation is unknown

TRUE

Interpret the following confidence interval: We are 90% confident that the percentage of COVID positive patients who are asymptomatic is [5%, 15%].

We are 95% confident that the true percentage of COVID positive patients are asymptomatic is between 5% and 15%. If we were to take all possible samples and calculate a confidence interval for each, then 95% of the confidence intervals would contain the true percentage of asymptomatic COVID positive patients.

A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. The 95% confidence interval for Π is .59 +/- .07. Interpret this interval.

We are 95% confident that the true proportion of all students receiving financial aid is between 0.52 and 0.66.

What are ways we can decrease the width of the interval

We can decrease our confidence level or increase the sample size

Suppose that we are 95% confident that the percentage of Americans who don't have health insurance is 8.5% with a margin of error 3%. What is the confidence interval?

[5.5%, 11.5%] 8.5-3=5.5 8.5+3=11.5

A random sample of 100 male giraffes' height was taken and generated the following 95% confidence interval (13.0, 19.6). Based on this interval, determine if each of the following statements are true or false. a. The true population height could be between 13.0 and 19.6 b. The true population height could be outside of this interval. c. 95% of giraffes will be between 13 and 19.6 feet. d. If we take a sample of 500 giraffes, we can increase our confidence level.

a) True - we are 95% confident the true mean height is somewhere between 13 and 19.6 b) True - 95% of the intervals produced would contain the true mean but 5% of the intervals do not contain the true mean. It's possible that the interval we have is part of this 5% c) False - The 95% should refer to the confidence level or intervals. The given interval is for the parameter. d) False - The confidence level is chosen by the statistician. By increasing the sample size, we can decrease the width of the interval

A random sample of five hundred 18 to 29 year olds was taken to find the percentage who do not have a primary care provider, and the following 95% confidence interval was generated [42%,48%]. Based on this interval, determine which of the following conclusions are valid. a. We are 95% confident that between 42% and 48% of 18 - 29 year olds do not have a primary care provider. b. If we were to take all possible samples of 500 18-29 year olds, find the percentage who do not have a primary care provider, and create a confidence interval for each sample, then 95% of the confidence intervals would contain the true population parameter. c. We have a 5% chance that the true percentage of 18 - 29 year olds is not between 42% and 48%. d. 95% of 18 - 29 year olds do not have a primary health provider. e. 95% of the samples obtained produce a statistic that is between 42% and 48%. f. We are 95% confident that the true percentage of 18-29 year olds who don't have a primary health care provider is 45%. g. We are 95% confident that the percentage of 18-29 year olds who don't have a primary health care provider is greater than 40%. h. We are 95% confident that the percentage of 18-29 year olds who don't have a primary health care provider is greater than 45%. i. We are 95% confident that the percentage of 18-29 year olds who don't have a primary health care provider is less than 50%.

a) VALID b) VALID c) VALID d) INVALID - the confidence level should refer to confidence or intervals e) INVALID - Conclusions for a confidence interval need to be for the parameter, not the samples f) INVALID - While 45% is a point estimate, our confidence level is for the interval, not the point estimate and is somewhere BETWEEN 42% and 48%. g) VALID - If we are 95% confident that the true percentage is between 42 and 48, all of these numbers in this range are greater than 40. h) INVALID - The confidence interval includes percentages that are lower than 45%. All values in the interval are possible values for the parameter, so we cannot say that it is greater than 45% i) VALID - If we are 95% confident that the true percentage is between 42 and 48, all of these numbers in this range are less than 50.

The owner of a small bank is interested in determined the amount of credit card debt his customers have. She randomly selects 200 customers and finds the 85% confidence interval is $4,717 +/- $1,222. Based on this interval, determine if each of the following statements are true or false. a. To reduce the width, she could have taken a sample of 400. b. If she increased the level of confidence to 95%, she would get a narrower interval. c. 85% of bank customers have $4,717 of credit card debt. d. If all possible samples of 200 are taken and confidence intervals are developed, 85% of them would contain the true population parameter.

a)True - To decrease the width, we can increase the sample size. b) False - Increased confidence results in a wider interval c) False - The confidence level should refer to confidence or intervals d) True - This is why we can we say we are 85% confident the true population parameter is within this interval

A 99% confidence interval estimate can be interpreted to mean that... a) If all possible samples are taken and confidence interval estimates are developed, 99% of them would include the true population mean somewhere within their interval. b) You have 99% confidence that you have selected a sample whose interval does include the population mean. c) All of the above d) None of the above

c) All of the above - If all possible samples are taken and confidence interval estimates are developed, 99% of them would include the true population mean somewhere within their interval. - You have 99% confidence that you have selected a sample whose interval does include the population mean.

Which of the following statements about confidence intervals is FALSE? a) For a given data set, the confidence interval will be wider for a 95% confidence interval than for a 90% confidence interval b) Holding the sample size fixed, increasing the level of confidence in a confidence interval will necessarily lead to a wider confidence interval. c) Holding the width of a confidence interval, increasing the level of confidence can be achieved with a lower sample size. d) All of the above statements are TRUE.

c) Holding the width of a confidence interval, increasing the level of confidence can be achieved with a lower sample size.