s1 w5 Confidence intervals

• 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

  A sample in which 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
• Sampling bias

  • Results from the sample are not representative of the population
  • Undercoverage - having a sampling frame that lacks representation from parts of the population

  
• Nonresponse bias

  • Sampled subjects cannot be reached or refuse to participate or fail to answer some questions
• Response bias

  • Subject gives an incorrect response or the way the interviewer asks the questions (or wording of a question in print) is confusing or misleading
• Convenience sample

  A type of survey sample that is easy to obtain relatively cheaply, but unlikely to be representative of the population
• Volunteer sample

  Most common type of convenience sample, where subjects volunteer for the sample, but volunteers do not tend to be representative of the entire population
• A simple random sample of 100 people is better than a volunteer sample of thousands of people
• Steps for conducting a sample survey

  1. Identify the population of all subjects of interest

  2. Construct a sampling frame

  3. Use a random sampling design to select n subjects

  4. Be cautious about sampling bias, response bias, and nonresponse bias

  
• 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, and use the subjects in those clusters as the sample
  • Preferable if a reliable sampling frame is not available or the cost of selecting a simple random sample is excessive
  • Usually requires a larger sample size for the same level of precision
  • Selecting a small number of clusters might result in a more homogeneous sample than the population
• Stratified random sampling

  • Divide the population into separate groups (strata) based on some attribute, then select a simple random sample from each stratum
  • Advantage is you can include enough subjects from each group you want to evaluate
  • Disadvantage is you must have a sampling frame and know the stratum of each subject
• 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
• Central Limit Theorem

  Tells us that for a population with any distribution, the distribution of the sample means approaches a normal distribution as the sample size increases
• If the original population is normally distributed, then for any sample size n, the sample means will be normally distributed
• 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
• For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution
• As the sample size increases, the sampling distribution of sample means approaches a normal distribution
• Suppose an elevator has a maximum capacity of 16 passengers with a total weight of 2500 lb. Assuming a worst case scenario in which the passengers are all male, what are the chances the elevator is overloaded? Assume male weights follow a normal distribution with a mean of 182.9 lb and a standard deviation of 40.8 lb.
• 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
• Steps to find the margin of error and construct a confidence interval

  1. Verify assumptions

  2. Find the critical value zα/2

  3. Evaluate the margin of error

  4. Find the confidence interval limits

  5. Round the confidence interval limits
• Based on the 95% confidence interval, we can conclude that more than 75% of adults know what Twitter is