Stat Chapter 16

confidence interval

Uses sample data to estimate an unknown population parameter with an indication of how accurate the estimate is and of how confident we are that the result is correct. It has two parts: an interval calculated from the data and a confidence level. It often has the form: estimate ± margin of error.

inference

Using information from a sample to describe a population is called ______________.

confidence interval

Using the critical values, we can calculate a confidence interval for any confidence level: Draw an SRS of size n from a Normal population having unknown mean μ and known standard deviation σ. A level C ______________ _________ for μ is:

confidence level. The confidence level is the success rate for the confidence interval estimation method.

What describes how often confidence intervals will capture the true parameter value in repeated sampling?

state

What is the practical question that required estimating a parameter?... thinking about what we want to do

correct

A high confidence level indicates that our methods commonly yield __________ answers.

1.645

Confidence level C: 90% Whats critical value z*?

The margin of error for the population mean will contain constant, standard deviation, and sample size n.

The margin of error for the population mean will contain constant, standard deviation, and sample size n. Copy.

narrower

As the sample size increases, the confidence interval becomes ____________.

conclude

Return to the practical question to describe your results in this setting... communicating our results in context.

different from Because the sample will consist of only a few observations from the population, the sample mean will likely be different from the population mean.

The weight of four-ounce bags of cashews packaged in a cashew plant follows Normal distribution with a standard deviation of 0.03 ounce. Suppose a random sample of 9 bags is to be taken from production and the mean weight of the sample is computed. The mean of this sample will likely be _____________ _______ the population mean.

sample mean

If we want to estimate the population mean, it is logical to use the __________ _______ as our basis. But, the sample mean might not be exactly equal to the population mean, so we'll need some "room for error."

critical values

In general, for a level C confidence interval, we need to find the value z* such that we have area C between -z* and z*. These are the __________ ________ of the standard Normal distribution.

four-step process

The broader method of calculating confidence intervals, involves the ______-______ _________ "S.P.S.C.": 1. State 2. Plan 3. Solve 4. Conclude

success rate

The confidence level is the __________ _______ of the method that produces the interval. We don't know whether the 95% confidence interval from a particular sample is one of the 95% that capture μ or one of the unlucky 5% that miss.

the value of the unknown parameter We hope to capture the value of the unknown population parameter within the range of a confidence interval. A confidence interval gives a range of reasonable values for the unknown population parameter.

What do we hope to capture within a confidence interval?

the value of the unknown parameter. We hope to capture the value of the unknown population parameter within the range of a confidence interval. A confidence interval gives a range of reasonable values for the unknown population parameter.

What do we hope to capture within a confidence interval?

Drawing conclusions about a population mean based on a sample mean. Using information from a sample to estimate a population parameter value is statistical inference.

What is statistical inference on μ?

To provide us with a range of reasonable values for parameter This is the purpose of a confidence interval.

What is the purpose of a confidence interval?

C. Estimate of the population parameter, the confidence level, and the margin of error. To obtain a confidence interval, we have to know the estimate of the parameter from the sample, the confidence level and the margin of error. (We also need to know the population standard deviation and the sample size, which are contained in the margin of error.)

Which of the following are three components that would completely specify a confidence interval? A. An estimate of the confidence level, the sample size, and the margin of error. B. Mean of the sampling distribution of x¯, the confidence level, and the margin of error. C. Estimate of the population parameter, the confidence level, and the margin of error. D. Estimate of the population parameter, the sample size and the standard deviation of the population.

B. the population size. The population size does not affect the confidence interval for μ. The population size does not matter as long as it is large relative to the sample size.

Which one of the following is NOT part of a confidence interval for μ? A. the confidence level. B. the population size. C. the margin of error. D. the sample mean.

smaller

___________ margins of error are quite desirable and can be attained mathematically through: 1. Lower confidence levels... these result in smaller values of z*. 2. Smaller standard deviations... less error in the data, with smaller values of standard deviation. 3. Increased sample sizes... this results in dividing standard deviation by a larger number... quadrupling the sample size cuts the margin of error in half.

Nothing needs to be fixed; the interpretation is correct. The confidence level is given, the parameter is stated correctly and in context, and the interval is given. This is a correct interpretation of the confidence interval.

Consider this confidence interval interpretation: We are 90% confident that the true mean weight of all black bears is between 344.8 and 361.2 pounds. What needs to be done to fix this statement?

plan

Identify the parameter, choose a level of confidence, and select the type of confidence interval that fits your situation... checking to see if we can do it

margin of error

A level C confidence interval for a parameter has two parts: An interval calculated from the data, usually of the form estimate +/- _________ ____ _______.

Remember from the 68-95-99.7 rule that the 95 we used above was approximate. We do not need to go a full 2 standard deviations away from the mean to be 95% confident. In fact, we need only go 1.96 standard deviations for 95% confidence.

Remember from the 68-95-99.7 rule that the 95 we used above was approximate. We do not need to go a full 2 standard deviations away from the mean to be 95% confident. In fact, we need only go 1.96 standard deviations for 95% confidence. Copy.

margin of error The margin of error shows how precise we believe our estimate of the parameter is based on the variability of the estimate. If the margin of error is small, we have a precise interval.

The __________ ____ ________ is a measure of how precise our estimate of a parameter is.

sample

We make statistical inference about a population parameter based on information from a ______________.

confidence level

A ______________ _______ C, which gives the probability that the interval will capture the true parameter in repeated samples. That is, the confidence level is the success rate for the method (not for an individual interval).

process

Confidence is about the ___________, not about any one interval.

standard deviation

Room for error/margin of error is a function of the ____________ ____________. But, it must also reflect our knowledge of probability theory (lets say constant c), and the sample size n.

precision

Smaller margins of error reflect increased _____________ in estimating the true value of the parameter.

The confidence interval for the population mean with known standard deviation is of the form x- +/- z* standard deviation/square root n with margin of error equaling z* standard deviation/square root n.

The confidence interval for the population mean with known standard deviation is of the form x- +/- z* standard deviation/square root n with margin of error equaling z* standard deviation/square root n. Copy.

confidence level

The success rate of the method that produces the confidence interval. Gives the probability that the interval will capture the true parameter value in repeated samples.

margin of error. We add and subtract the margin of error from the sample estimate to get a confidence interval estimate of a parameter.

A confidence interval is of the form: estimate ± ____________ _____ ________.

solve

Carry out the work in two phases: 1. Check the conditions for the interval you plan to use. 2. Calculate the confidence interval. ... doing it

statistical inference

provides methods for drawing conclusions about a population from sample data.

The true parameter needs to be stated correctly and in context. Confidence intervals address an unknown population mean. This interval talks about what the scores in the sample were. We know what the values in the sample were. A better interpretation is, With 95% confidence, the mean SAT score of all graduating students in this Connecticut school district is between 505 and 520 points.

A very large school district in Connecticut wanted to estimate the average SAT (for college admissions) score of this year's graduating class. The district took a simple random sample of 100 seniors and calculated the 95% confidence interval for μ as 505 to 520 points. Consider this interpretation of the confidence interval: For the sample of 100 graduating seniors, 95% of their SAT scores were between 505 and 520 points with 95% confidence. What needs to be done to fix this statement?

1.960

Confidence level C: 95% Whats critical value z*?

2.576

Confidence level C: 99% Whats critical value z*?

68

Confidence levels of 95%, 90%, and 99% are quite common. A confidence level of ____% is not common.

The true parameter needs to be stated correctly and in context. The parameter is not stated in context. A better interpretation is, We are 90% confident that the mean weight of all black bears is between 344.8 and 361.2 pounds, assuming the interval is about black bear weights.

Consider this confidence interval interpretation: We are 90% confident that the population mean is between 344.8 and 361.2 pounds. What needs to be done to fix this statement?

confidence level The confidence level is the success rate for the confidence interval estimation method. It describes what we expect to happen if we repeat the procedure with many different samples.

A measure of reliability in our confidence interval estimate of a parameter is the _______________ ________.

critical value

Chosen so that the standard Normal curve has area C between —z* and z*.

the population mean A confidence interval for μ gives a range of reasonable values for the unknown population mean, μ. Its basic purpose is to estimate the value of the parameter μ.

The purpose of constructing a confidence interval for μ is to give a range of reasonable values for __________________.

population mean A confidence interval for μ gives a range of reasonable values for the unknown population mean, μ. Its basic purpose is to estimate the value of the parameter μ.

The purpose of constructing a confidence interval for μ is to give a range of reasonable values for the _______________ ________.