Chapter 7 - Estimating Parameters and Determining Sample Sizes

**7-1 Estimating a Population Proportion**

**Point estimate:**the sample proportion is the best point estimate of the population proportion p**Confidence interval:**we can use a sample proportion to construct a confidence interval estimate of the true value of a population proportion, and we should know how to construct and interpret such confidence intervals**Sample size:**we should know how to find the sample size necessary to estimate a population proportion**Unbiased estimator:**we use p hat as the point estimate of p because it is unbiased and it is the most consistent of the estimators that could be usedA

**confidence interval**is a range of values used to estimate the true value of a population parameterA

**confidence level**is the probability 1-alpha that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of timesHow to interpret a CI: "We are __% confident that the interval from ___ to ___ actually does contain the true value of the population proportion p"

A

**critical value**is the number on the borderline separating sample statistics that are significantly high or low from those that are not significantThe difference between the sample proportion p hat and the population proportion p is an error

The maximum likely amount of that error is the margin of error, denoted by E

p hat = (upper CI limit + lower CI limit) / 2

E = (upper CI limit - lower CI limit) / 2

The coverage probability of a CI is the actual proportion of such confidence intervals that contain the true population proportion

**7-2 Estimating a Population Mean**

The sample mean x bar is the best point estimate of the population mean mu

Requirement of "normality of n>30" means that the distribution should be somewhat symmetric / sample size must be greater than 30

A student t distribution is commonly referred to as a t distribution

**Degrees of freedom**for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data valuesdegrees of freedom = n -1

The overlapping of confidence intervals should not be used for making formal / final conclusions about the equality of means

When dealing with unknown sigma when finding sample sizes, sigma is about range/4 is a rule of thumb

**7-3 Estimating a Population Standard Deviation or Variance**

When constructing a confidence interval estimate of a population standard deviation, we construct the confidence interval using the X squared distribution

The sample statistic X^2 (chi-squared) has a sampling distribution called the

**chi-square distribution**

**7-4 Bootstrapping: Using Technology for Estimates**

Important requirements such that the sample is a simple random sample:

CI for proportion: there are at least 5 successes and at least 5 failures

CI for mean: the population is normally distributed or n > 30

CI for sigma or sigma squared: the population must have normally distributed values, even if the sample is large

**Nonparametric or distribution-free method**means the method does not require the sample to be collected from a normal or any other particular distributionA

**bootstrap**sample is another random sample of n values obtained with replacement from the original sampleAn effective use of the bootstrap method typically requires the use of software to generate 1000 or more bootstrap samples

**7-1 Estimating a Population Proportion**

**Point estimate:**the sample proportion is the best point estimate of the population proportion p**Confidence interval:**we can use a sample proportion to construct a confidence interval estimate of the true value of a population proportion, and we should know how to construct and interpret such confidence intervals**Sample size:**we should know how to find the sample size necessary to estimate a population proportion**Unbiased estimator:**we use p hat as the point estimate of p because it is unbiased and it is the most consistent of the estimators that could be usedA

**confidence interval**is a range of values used to estimate the true value of a population parameterA

**confidence level**is the probability 1-alpha that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of timesHow to interpret a CI: "We are __% confident that the interval from ___ to ___ actually does contain the true value of the population proportion p"

A

**critical value**is the number on the borderline separating sample statistics that are significantly high or low from those that are not significantThe difference between the sample proportion p hat and the population proportion p is an error

The maximum likely amount of that error is the margin of error, denoted by E

p hat = (upper CI limit + lower CI limit) / 2

E = (upper CI limit - lower CI limit) / 2

The coverage probability of a CI is the actual proportion of such confidence intervals that contain the true population proportion

**7-2 Estimating a Population Mean**

The sample mean x bar is the best point estimate of the population mean mu

Requirement of "normality of n>30" means that the distribution should be somewhat symmetric / sample size must be greater than 30

A student t distribution is commonly referred to as a t distribution

**Degrees of freedom**for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data valuesdegrees of freedom = n -1

The overlapping of confidence intervals should not be used for making formal / final conclusions about the equality of means

When dealing with unknown sigma when finding sample sizes, sigma is about range/4 is a rule of thumb

**7-3 Estimating a Population Standard Deviation or Variance**

When constructing a confidence interval estimate of a population standard deviation, we construct the confidence interval using the X squared distribution

The sample statistic X^2 (chi-squared) has a sampling distribution called the

**chi-square distribution**

**7-4 Bootstrapping: Using Technology for Estimates**

Important requirements such that the sample is a simple random sample:

CI for proportion: there are at least 5 successes and at least 5 failures

CI for mean: the population is normally distributed or n > 30

CI for sigma or sigma squared: the population must have normally distributed values, even if the sample is large

**Nonparametric or distribution-free method**means the method does not require the sample to be collected from a normal or any other particular distributionA

**bootstrap**sample is another random sample of n values obtained with replacement from the original sampleAn effective use of the bootstrap method typically requires the use of software to generate 1000 or more bootstrap samples