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 used
• A confidence interval is a range of values used to estimate the true value of a population parameter
• A 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 times
• How 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 significant
• The 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 values
  • degrees 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 distribution
• A bootstrap sample is another random sample of n values obtained with replacement from the original sample
• An effective use of the bootstrap method typically requires the use of software to generate 1000 or more bootstrap samples

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