STATS Module 8

• Estimation


  The assignment of value(s) to a population parameter based on a value of the corresponding sample statistic

  
• Point estimate


  The value of a sample statistic that is used to estimate a population parameter

  
• Interval estimate


  An interval constructed around the point estimate, where it is stated that this interval contains the corresponding population parameter with a certain confidence level

  
• Estimating population parameters


  • Mean fuel consumption for a car model
  • Average time taken by new employees to learn a job
  • Proportion of adults in favor of raising taxes on rich people
• Estimator


  The sample statistic used to estimate a population parameter
• Estimation procedure


  1. Select a sample

  2. Collect required information from sample

  3. Calculate value of sample statistic

  4. Assign value(s) to corresponding population parameter

  
• Margin of error


  The number added to and subtracted from the point estimate to construct the confidence interval
• Confidence level


  The probability that the confidence interval contains the true population parameter
• Confidence coefficient


  1 - α, where α is the significance level
• A 95% confidence level means the total area under the standard normal curve between two points (at the same distance) on different sides of μ is 95%
• The width of a confidence interval can be controlled by the confidence level (z value) and the sample size (n)
• Population standard deviation


  The standard deviation of the entire population
• Estimating the mean debt of this year's college graduates

  1. Select a sample

  2. Calculate the margin of error

  3. Determine the required sample size
• Margin of error


  The maximum size of the difference between the sample mean and the population mean
• The value of z for a 99% confidence level is 2.58
• The value of the population standard deviation is $11,800
• Calculating the required sample size

  1. Use the formula: n = (z^2 * σ^2) / E^2

  2. Plug in the values: n = (2.58^2 * 11,800^2) / 800^2

  3. Solve for n: n = 1449

  
• Confidence level


  The probability that the true population mean is within the calculated confidence interval
• Estimating the mean amount of detergent in 64-ounce jugs

  1. Select a sample

  2. Calculate the margin of error

  3. Determine the required sample size
• The standard deviation of the amounts of detergent in all such jugs is 0.20 ounce
• The desired margin of error is 0.04 ounce
• The value of z for a 90% confidence level is 1.65
• Calculating the required sample size

  1. Use the formula: n = (z^2 * σ^2) / E^2

  2. Plug in the values: n = (1.65^2 * 0.2^2) / 0.4^2

  3. Solve for n: n = 68

  
• t distribution


  A bell-shaped distribution with a lower height and greater spread than the standard normal distribution
• As the sample size becomes larger, the t distribution approaches the standard normal distribution

  
• Degrees of freedom (df)


  The parameter of the t distribution, equal to n-1

  
• The mean of the t distribution is 0 and its standard deviation is sqrt(df/(df-2))

  
• The number of degrees of freedom is the number of observations that can be chosen freely
• Calculating a confidence interval for the population mean using the t distribution

  1. Calculate the margin of error: E = t * (s/sqrt(n))

  2. The confidence interval is: x +/- E
• The distribution of premiums paid for family health insurance coverage by all workers in New York City who have employer-provided health insurance coverage is approximately normally distributed
• Calculating a 95% confidence interval for the average premium paid for family health insurance coverage

  1. Calculate the standard error: s/sqrt(n) = 800/sqrt(25) = 160

  2. Find the t-value for 24 degrees of freedom and 95% confidence: 2.064

  3. The confidence interval is: 6600 +/- 2.064*160 = $6269.76 to $6930.24
• Calculating a 99% confidence interval for the mean annual expenditure on books

  1. Calculate the standard error: s/sqrt(n) = 300/sqrt(64) = 37.50

  2. Find the t-value for 63 degrees of freedom and 99% confidence: 2.656

  3. The confidence interval is: 1450 +/- 2.656*37.50 = $1350.40 to $1549.60

  
• Estimator of the standard deviation of the sample proportion


  sqrt(p_hat * (1 - p_hat) / n)

  
• Calculating a confidence interval for the population proportion


  Use the formula: p_hat +/- z * sqrt(p_hat * (1 - p_hat) / n)

  
• 75% of the people in the sample said having basic needs met is very or extremely important in their vision of the American dream
• Calculating the point estimate of the population proportion

  The point estimate is p_hat = 0.75

  
• Calculating a 99% confidence interval for the population proportion

  Use the formula: 0.75 +/- 2.58 * sqrt(0.75 * 0.25 / 1821)

  The confidence interval is 0.724 to 0.776

  
• The margin of error of the 99% confidence interval estimate is +/- 0.026 or +/- 2.6%
• Determining the sample size for estimating a population proportion


  Use the formula: n = (z^2 * p_hat * (1 - p_hat)) / E^2

  If p_hat and q_hat are unknown, use p_hat = 0.5 and q_hat = 0.5 for the most conservative estimate

  
• Determining the sample size for estimating a population proportion when p_hat and q_hat are known

  Take a preliminary sample, calculate p_hat and q_hat, then use those values in the formula