Chapter 9: Estimating the Value of a Parameter

9.1 Estimating a Population Proportion

This section discusses methods for estimating a population proportion, including point estimation, constructing confidence intervals, and determining required sample sizes.

Obtain a Point Estimate for the Population Proportion

• Point Estimate: A single value used to estimate a population parameter. For a population proportion, the point estimate is the sample proportion, denoted as .

• Formula: , where is the number of individuals with the specified characteristic and is the sample size.

• Example: In a survey of 2018 adults, 1184 said they frequently worry about their financial situation. The point estimate is .

Construct and Interpret a Confidence Interval for the Population Proportion

• Confidence Interval: An interval of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter.

• Formula for a (1-α)100% Confidence Interval: Lower bound: Upper bound: where is the critical value from the standard normal distribution.

• Margin of Error:

• Interpretation: A 95% confidence interval means that if many samples were taken, about 95% of the intervals would contain the true population proportion.

• Critical Values Table:

• Example: Construct a 95% confidence interval for the proportion of adults who frequently worry about their financial situation, given and .

Determine the Sample Size Necessary for Estimating a Population Proportion within a Specified Margin of Error

• Formula:

• If no prior estimate is available, use for maximum variability.

• Example: To estimate a population proportion with 95% confidence and a margin of error of 0.03, use and :

9.2 Estimating a Population Mean

This section covers point estimation and confidence intervals for a population mean, including the use of the Student's t-distribution when the population standard deviation is unknown.

Obtain a Point Estimate for the Population Mean

• Point Estimate: The sample mean is used as the point estimate for the population mean .

• Example: The mean amount spent on Valentine's Day gifts by a sample of students is calculated from the data: .

State Properties of Student's t-Distribution

• The t-distribution is used when the population standard deviation is unknown and the sample size is small.

• Properties:

  1. Different for different degrees of freedom (df = n - 1).

  2. Centered at zero and symmetric about zero.

  3. Has more area in the tails than the standard normal distribution.

  4. As the sample size increases, the t-distribution approaches the standard normal distribution.

Construct and Interpret a Confidence Interval for a Population Mean

• Formula for a (1-α)100% Confidence Interval: Lower bound: Upper bound: where is the critical value from the t-distribution with n-1 degrees of freedom, is the sample standard deviation, and is the sample size.

• Example: For a sample mean of , , , and 95% confidence, find the interval using the appropriate value.

The Normality Condition

• For small samples, the population must be approximately normal to use the t-distribution.

• For large samples (n > 30), the Central Limit Theorem allows use of the t-distribution even if the population is not normal.

Determine the Sample Size Needed to Estimate a Population Mean within a Specified Margin of Error

• Formula:

• If is unknown, use the sample standard deviation as an estimate.

• Example: To estimate a mean with , , and 95% confidence ():

(round up to 62)

9.3 Putting It Together: Which Method Do I Use?

This section provides guidance on selecting the appropriate confidence interval method based on the type of variable (proportion or mean) and the information available (population standard deviation known or unknown).

• For proportions, use the z-interval for population proportions.

• For means, use the t-interval if the population standard deviation is unknown; use the z-interval if it is known.

• Verify model requirements (e.g., normality, random sampling) before constructing intervals.

9.4 Estimating a Population Standard Deviation (Text Only)

This section introduces the chi-square distribution for constructing confidence intervals for population variance and standard deviation.

Find Critical Values for the Chi-Square Distribution

• The chi-square distribution is used for inference about population variance and standard deviation.

• Properties:

  1. Different for different degrees of freedom (df = n - 1).

  2. Values are always non-negative.

  3. As df increases, the distribution becomes more symmetric.

Construct and Interpret Confidence Intervals for the Population Variance and Standard Deviation

• Formula for a (1-α)100% Confidence Interval for Variance: Lower bound: Upper bound: where and are critical values from the chi-square distribution.

• For standard deviation, take the square root of the interval endpoints.

9.5 Estimating with Bootstrapping (Text Only)

This section introduces the bootstrap method for estimating parameters and constructing confidence intervals without relying on parametric assumptions.

Estimate a Parameter Using the Bootstrap Method

• Bootstrapping: A resampling technique that involves repeatedly sampling with replacement from the observed data to estimate the sampling distribution of a statistic.

• Bootstrap Algorithm:

  1. Select B independent bootstrap samples of size n with replacement from the original data.

  2. Compute the statistic of interest for each bootstrap sample.

  3. Use the distribution of the bootstrap statistics to estimate confidence intervals (e.g., the 2.5th and 97.5th percentiles for a 95% interval).

• Example: Estimate a 95% confidence interval for the mean or proportion using 1000 bootstrap samples.

Some Final Thoughts on Bootstrapping

• Bootstrapping is especially useful when the theoretical distribution of the statistic is unknown or the sample size is small.

• The standard error of bootstrap distributions tends to be smaller by a factor of compared to the original sample.

• It is recommended to use at least 10,000 bootstrap samples for more accurate results.