Statistical Inference and Hypothesis Testing: Population Percentages and Averages - Prof. , Study notes of Marketing Research

Download Statistical Inference and Hypothesis Testing: Population Percentages and Averages - Prof. and more Study notes Marketing Research in PDF only on Docsity! Chapter 12  generalization—the act of estimating a population fact from a sample finding  sample finding—any analysis value that is computed with a sample’s data  population fact—the true value when a census of a population is taken and the value is determined using all members of the population; parameter o parameter estimation—the estimation of population values  confidence interval—a range into which the researcher believes the population parameter falls with an associated degree of confidence  how to estimate a population percentage (categorical data)  population percentage estimation= p ± zαsp  p= sample percentage  zαsp= limit o zα= z value for associated level of confidence  95%: z= ±1.96 (*most commonly used)  99%: z= ±2.58 o sp= standard error of the percentage  sp= √(pq/n) o q= 1-p  EX. In a McDonald’s survey, 60% of 100 respondents were found to order an Egg McMuffin for breakfast. What is the estimated population percentage for a 95% confidence interval?  STEP 1: find sp p=60%, q=40%, n=100; sp= √[(60*40)/100]= 4.9%  STEP 2: find the limit zα= 1.96; limit= 1.96*4.9%= 9.6%  STEP 3: find the population percentage estimation lower boundary= 60-9.6= 50.4% upper boundary= 60+9.6= 69.6%  how to estimate a population average (metric data)  population average estimation= x-bar± zαsx-bar  x-bar= sample average  sx-bar= standard error of the average o sx-bar= s/√n  s= standard deviation  EX. In a New York Time’s survey, 100 readers were asked how many minutes per day they spend reading the newspaper. The sample average was 45 minutes with a standard deviation of 20 minutes. What is the estimated population average for a 95% confidence interval?  STEP 1: find sx-bar s=20, n=100; sx-bar= 20/√100= 2  STEP 2: find the limit zα= 1.96; limit= 1.96*2= 3.9  STEP 3: find the population average estimation lower boundary= 45-3.9= 41.1 minutes upper boundary= 45+3.9= 48.9 minutes  interpretation of a 95% confidence interval  standard error—a measure of the variability in a population based on the variability found in the sample  dependant on variability (pq) and sample size (n)  sampling distribution—a theoretical concept that underlies confidence intervals  more variability and/or smaller sample size= wider sampling distribution  less variability and/or larger sample size= narrower sampling distribution  A 95% confidence interval means that the research can be 95% confident that the population percentage falls within the limit.  hypothesis testing—a statistical procedure used to accept (support) or reject (not support) the hypothesis based on sample information  hypothesis—a statement about the population parameter based on prior knowledge, assumptions, or intuition o intuitive hypothesis testing—when someone uses something he has observed to see if it agrees with or refutes his belief about that topic o statistical hypothesis testing  critical value—z-value of the associated confidence level (95% or 99%)  If the computed z-score falls within the acceptance region, the hypothesis is supported. If it falls within the rejection regions, the hypothesis is rejected.  null hypothesis—a formal statement that there is no (null) difference between the hypothesized π value and the p value  directional hypothesis—one that indicates the direction in which the researcher believes the population parameter falls relative to some hypothesized parameter  specifies a “greater than” or “less than” value using only one tail of the bell-shaped distribution curve  testing a hypothesis about a percentage (categorical data)  z= (p-πH)/sp  z= number of standard errors  p= sample percent  πH= hypothesized population percentage  sp= standard error of the percentage  EX. Bill believes that 30% of drivers use seat belts. A sample of 1,000 drivers is taken, 800 of which responded that they use seatbelts. Using a 95% confidence interval, is Bill’s hypothesis supported?  STEP 1: identify πH πH= 30%  STEP 2: find sp p= 80%, q= 20%, n= 1000; sp= √[(80*20)/1000]= 1.26%  STEP 3: find z z= (80-30)/1.26= 39.7  STEP 4: compare z to the critical value 39.7>1.96; The hypothesis is rejected.  testing a hypothesis about an average (metric data)  z= (x-bar—μH)/sx-bar  z= number of standard errors  x-bar= sample average  μH= hypothesized population average  sx-bar= standard error of the average  EX. Rex believes that the typical college agent will be able to earn $2,750 in the first semester. To test this hypothesis, a survey of 100 current college agents is taken. The sample average of the amount of money made in the first semester is determined to be $2,800 and the standard deviation is $350. Using a 95% confidence interval, is Rex’s hypothesis supported?