BA Chapter 6

Question 1

• This is on the quiz:

Population Frame is the list from which sample is selected.

Question 2

• Sampling distribution of the mean
  
  • Formed by the means of all possible samples of a fixed size n from some population.
• Standard error of the mean
  
  • The standard deviation of the sampling distribution of the mean.
  • This is referred to as the standard error.
  • Formula std error: attached image (front).
  • Larger sample sizes have less sampling error and provide greater accuracy in estimating the true population mean.

Question 3

• If the sample size is ______ enough, then the sampling distribution of the mean:
  
  • is approximately _______ distributed regardless of the distribution of the _______.
  • has a mean equal to the _______ mean.
• If the population is normally distributed, then the sampling distribution is also _______ distributed for any sample size.
• The Central Limit Theorem is one of the most important practical results in ______.

Answer 3

• large
  
  • normally; population
  • population
• normally
• statistics

Question 4

Sampling distribution is a distribution of _______.

Answer 4

means.

Question 5

• _________ is the most common form of sampling, select items from a population so that every subset of a given size has an equal chance of being selected.

There are other useful methods of obtaining a random sample from a population:

• Systematic (or Periodic) sampling
  
  • Randomly select ____ subject (NOTE: it’s not a random sample without this!!!).
  • After the above step, select every ____ subject until you get your sample size.
  • k can be _______.
• Stratified sampling
  
  • Divide population into naturally occurring ______ or strata.
  • Randomly select n subjects from ______.
  • Example: see the attached image (front).
  • Sample n_i=25 students from each strata. No problem if N₁ = N₂ = N₃ = N₄ = N₅
  • But what if the size is unequal in each strata? Then you can use a proportional method based on stratum size. See the attached image (back).

Answer 5

• Simple Random Sampling
• Systematic (or Periodic)
  
  • first
  • k^th
  • any number
• Stratified
  
  • subgroups
  • every stratum

Question 6

• Cluster sampling
  
  • Use when you have many, many ______ (clusters).
  • Divide population into _______ subgroups called clusters.
  • Randomly select clusters, then perform a census (i.e., ______ ) in each selected cluster.
  • Example: see the attached image (front)
• Sampling from a continuous process
  
  • Select a _____ at random, then select the next n items produced after that time; OR
  • Select ______ at random, then select the next item produced after these times.
  • Very useful for ________.

Answer 6

• Cluster
  
  • subgroups
  • distinctly different
  • sample everyone
• Sampling from a continuous process
  
  • time
  • n times
  • Quality Control

Question 7

Subjective Sampling Methods include:

• Convenience sampling
  
  • Samples are selected based on the ease with which the data can be collected (survey all customers who happen to visit this month).
• Judgment sampling
  
  • Expert judgment is used to select the sample (survey the best customers).

Question 8

• Interval Estimate is the ______ within which we believe the true population _______ falls.
• Confidence Interval Estimates is an interval estimate that specifies the ________ that the interval contains the true population parameter.
• Quiz question: definition of confidence interval vs. definition of interval estimate
• Margin of Error is the ______ of the Confidence Interval. Margin of error depends on the level of confidence and sample size.
• Interval Estimate = Point Estimate +/- Margin of Error
• Level of Confidence (1 – α) is the probability that the Confidence Interval contains the true population parameter, usually expressed as a _______.

Answer 8

• range; parameter
• half-width
• none
• likelihood
• percentage

Question 9

• Sampling (statistical) error occurs because samples are only a subset of the total population.
  
  • Inherent in sampling process so try to minimize it.
  • Sampling error depends on the size of the sample relative to the population.
  • Tradeoff between cost of sampling and accuracy of estimates obtained by sampling.
• Nonsampling error occurs when the sample does not adequately represent the target population.
  
  • Nonsampling error usually results from a poor sample design or choosing the wrong population frame.

Question 10

• Estimators are measures used to estimate unknown population parameters.
• Point Estimate is a single number derived from a sample that is used to estimate a population parameter.
• Example for Estimator & Point Estimate: see attached image (front).
• Unbiased Estimators - the expected value of the estimator equals the population parameter. For example, m is an unbiased estimator of m.

Question 11

• (1 – α) is the probability that the Confidence Interval contains the true population parameter, usually expressed as a percentage. A Confidence Interval of 100(1 − α)% is an interval [A, B] such that the probability of falling between A and B is 1− α.
• α is the desired significance level: 0 < = α <= 1
• 1− α is called the confidence level.

Answer 11

• Confidence intervals provide a way of assessing the accuracy of a point estimate.
• Confidence intervals estimate the value of a parameter such as a MEAN or PROPORTION

Question 12

The type of CI depends on the population parameter of interest and the information that we have available.

• CI about the mean (x̄ ), population σ known
• CI about the mean (x̄ ), population σ unknown
• CI about the proportion (p̂ )
• CI about the population variance σ²
• CI for a population total (N)

Answer 12

• Confidence Interval For the Mean, σ Known:
  
  • Use z distribution as the sampling distribution
  • Formula for CI: See the attached image (front)
  • Calculate z_a/2 using the NORM.S.INV()
  • Calculate the Standard Error = σ /sqrt(n)
  • Calculate the Margin of Error = z_a/2 * Standard Error
    
    • OR use this function: CONFIDENCE.NORM(alpha, stdev, size) to directly calculate Margin of Error.
  • Finally, add (add can get upper bound) and subtract (subtract can get lower bound) the Margin of Error from the Sample Mean to obtain a 95% Confidence Interval.
  • z_α/2 is the value of the standard normal random variable for an upper tail area of α/2 (or a lower tail area of 1−α/2).
• Confidence Interval For the Mean, σ Unknown:
  
  • When the population σ is unknown, we use the t distribution instead of the z distribution.
  • Only parameter is the degrees of freedom (df).
  • The t distribution is similar in shape to the z distribution.
  • Since we don’t know population σ, we will use sample std deviation (s) to estimate σ. See attached image (back) for the formula for CI.
  • We can find t_α/2,n-1 using the Excel function T.INV(confidence level, df) but be careful here, the confidence level represents all of the area to the left of the upper tail area and since a is divided by 2, that area is 1-α/2.
    
    • OR we can use CONFIDENCE.T(alpha, stdev, size) to directly calculate Margin of Error. And here we just use α, not α/2
  • t_α/2 is the value of the t-distribution with df = n − 1 for an upper tail area of α/2.

Question 13

The type of CI depends on the population parameter of interest and the information that we have available.

• CI about the mean (x̄ ), population σ known
• CI about the mean (x̄ ), population σ unknown
• CI about the proportion (p̂ )
• CI about the population variance σ2
• CI for a population total (N)

Answer 13

• CI about the proportion (p̂ )
  
  • Formula: see attached image (front)
  • We can find z_α/2 using the Excel function =NORM.S.INV(1-α/2)
    • 1-α/2 (is all of the area to the left of z)
  • Next, we need to calculate the standard error using the formula (attached front).
    
    • Use the SQRT() function.
  • Finally, calculate the CI.

*

Question 14

• Note that increasing the sample size:
  
  • Decreases the width of CI
  • Gives a more accurate estimate of the
    true population parameter
  • Increases costs