1037Y

• Confidence intervals


  Statistical method for estimating a population parameter from a sample
• Topics covered


  • Review and Preview
  • Sampling
  • The Central Limit Theorem
  • Estimating a Population Proportion
  • Estimating a Population Mean
  • Estimating a Population Standard Deviation or Variance
• Sampling frame


  List of subjects in the population from which the sample is taken
• Simple random sample


  • Each possible sample of that size has the same chance of being selected
• Selecting a simple random sample


  1. Number the subjects in the sampling frame

  2. Generate a set of those numbers randomly

  3. Sample the subjects whose numbers were generated
• Bias


  When results from the sample are not representative of the population

  
• Types of bias


  • Undercoverage
  • Sampling bias
  • Nonresponse bias
  • Response bias
• Convenience sample


  A type of survey sample that is easy to obtain relatively cheaply
• Volunteer sample


  Most common type of convenience sample where subjects volunteer for the sample
• A simple random sample of 100 people is better than a volunteer sample of thousands of people
• Steps in sampling


  1. Identify the population

  2. Construct a sampling frame

  3. Use a random sampling design to select n subjects

  4. Be cautious about sampling bias and other biases
• Random sampling methods


  • Simple random sampling
  • Cluster random sampling
  • Stratified random sampling
• Cluster random sampling


  • Divide the population into a large number of clusters, select a simple random sample of the clusters, use the subjects in those clusters as the sample
• Stratified random sampling


  • Divide the population into separate groups (strata), select a simple random sample from each stratum
• The Central Limit Theorem states that for a population with any distribution, the distribution of the sample means approaches a normal distribution as the sample size increases
• Mean of the sample means

  Equal to the population mean μ
• Standard deviation of the sample means


  Equal to σ/√n, where σ is the population standard deviation and n is the sample size
• For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution
• If the original population is normally distributed, then for any sample size n, the sample means will be normally distributed
• As the sample size increases, the sampling distribution of sample means approaches a normal distribution
• As we proceed from n = 1 to n = 50


  The distribution of sample means is approaching the shape of a normal distribution
• Elevator capacity


  Maximum capacity of 16 passengers with a total weight of 2500 lb
• Male weights


  Follow a normal distribution with a mean of 182.9 lb and a standard deviation of 40.8 lb
• If the elevator is filled to capacity with all males, there is a very good chance the safe weight capacity of 2500 lb will be exceeded
• Finite population correction factor

  When sampling without replacement and the sample size n is greater than 5% of the finite population of size N, adjust the standard deviation of sample means by multiplying it by the finite population correction factor
• Topics covered


  • Sampling
  • The Central Limit Theorem
  • Estimating a Population Proportion
  • Estimating a Population Mean
  • Estimating a Population Standard Deviation or Variance
• Point estimate


  A single value (or point) used to approximate a population parameter
• Sample proportion


  The best point estimate of the population proportion
• Confidence interval


  A range (or an interval) of values used to estimate the true value of a population parameter
• Confidence level


  The probability 1-α (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times
• The correct interpretation of a confidence interval is that we are X% confident that the interval contains the true value of the population parameter
• Critical value


  The number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur
• The z score separating the right-tail region is commonly denoted by zα/2 and is referred to as a critical value
• Critical Values


  • 90% confidence level, zα/2 = 1.645
  • 95% confidence level, zα/2 = 1.96
  • 99% confidence level, zα/2 = 2.575
• Margin of error


  The maximum likely difference (with probability 1-α) between the observed proportion and the true value of the population proportion
• The assumptions required for using the margin of error formula are: 1) simple random sample, 2) binomial distribution conditions satisfied, 3) at least 5 successes and 5 failures
• Steps to find the margin of error and confidence interval

  1. Verify assumptions

  2. Find critical value zα/2

  3. Evaluate margin of error E

  4. Find confidence interval limits p̑-E < p < p̑+E

  5. Round confidence interval limits to 3 significant digits
• When analyzing polls, the key is to ensure the required assumptions are satisfied
• Finding the margin of error

  1. Use the formula

  2. 2020/2021

  3. SIS 1037Y

  4. 59
• Margin of error (E)

  The amount the sample percentage is likely to differ from the true population percentage