Normal distribution: a probability distribution 99% of scores are within 3sd of mean
Who cares… The most useful distribution in inferential statistics. We can translate any normal variable, X, into the standardized value, Z to make assumptions about the whole population. Use when comparing means or proportions. Example: Suppose you were the city police and you wanted to know how many photo radar tickets you could expect to collect next year so that you can develop your budget... $
Last year the mean number of tickets for all locations was 9000 with a standard deviation of 500 tickets. What is the probability that you will give out between 7500 tickets (your lowball guess) and 9625 (your highball guess)? Calculate Z score …what type of scale must you have to calculate Z scores? …what reasons can you think of for wanting to calculate a Z score for your research?
Z tests, another application You have been asked to conduct a survey on customer satisfaction at the food court. Customers indicate their perceptions on a 5 point scale where 1=very unfriendly and 5=very friendly. Assume this is an interval scale and that previous studies have shown that a normal distribution of scores is expected.
Z tests, assumptions about mean You think: perhaps customers think that the service is neither friendly nor unfriendly Ho: mean is equal to 3.0 H1: mean is not equal to 3.0 Establish significance/confidence level=0.05/95% confidence therefore Z= +/- 1.96 You do a study with a sample of 225 interviews and the mean is 3.78. The standard deviation is 1.5. Do we accept or reject the null hypothesis?
LOWER LIMIT UPPER LIMIT A Sampling Distribution
Critical values of Critical value - upper limit
Critical value - lower limit Critical values of
3.0 3.196 2.804 3.78 sample mean, therefore reject Ho and say that the sample results are significant at.05 level of significance Range of acceptability
Type I and Type II Errors Accept nullReject null Null is true Null is false Correct- no error Type I error Type II errorCorrect- no error
If sample is small… Small usually means less than 30 Do a t test instead
Is this statistically significant? Chi-square test: a hypothesis test that allows for investigation of statistical significance in the analysis of a frequency distribution (or cross tab) Categorical data such as sex, education or dichotomous answers may be statistically analyzed Tests the “goodness of fit” of the sample with expected population results
Chi-square example Through observation research we have identified that of the sample of 100 people who got photo radar tickets, 60 were female and 40 were male. We expected that the proportions should be equal (.5 probability for each sex). Our null hypothesis is that the population data will be consistent with our sample data at 0.05 level of significance. If the calculated chi square is above the critical chi square for this level (3.84) we reject the null hypothesis. This is the case. The observed values are not comparable to expected values
Estimation of population parameters: Confidence The population mean and standard deviation are unknown; we do know the sample mean and standard deviation…. We take a sample of a number of students with children and ask them to identify how much they would be willing to pay per hour for on campus childcare. Our sample size is 30. The student population with children is estimated to be 300.
The sample mean is $2.60. This is called a point estimate. How close is this sample mean to the population mean? How confident are we? Confidence interval: the percentage indicating the long run probability that the results will be correct. Usually 95%
Relationship between variables Correlation and regression analysis
Types of questions Is employee productivity associated with pay incentives? Is salary level correlated with type of degree or designation? Is willingness to pay student fees levies for daycare correlated with whether one has a child? Are students grades influenced by length of term?
Measures of association A general term that refers to a number of bivariate statistical techniques used to measure the strength of a relationship between two variables Correlation coefficient (r): most popular. Is a measure of the covariation or association between two variables. It ranges from +1 to -1
Measures of association Coefficient of determination (r 2 ) The proportion of the total variance of a variable that is accounted for by knowing the value of another variable. Often shown as a correlation matrix. We have calculated r=-.65 when investigating whether the number of years of university is correlated with unemployment. If r 2 =.38, we know that about 40% of the variance in unemployment can be explained by variance in years of university
Regression analysis Bivariate linear regression: a measure of linear association that investigates a straight line relationship. Assuming that there is an association between students’ performance and length of term, can we predict a students GPA given the distribution of their courses along semesters Uses interval data
Regression analysis Multiple regression analysis: an analysis of association that simultaneously investigates the effect of two or more variables on a single, interval-scaled dependent variable
Summary Chi-square allows you to test whether an observed sample distribution fits some given distribution. Are the groups in your cross tab independent? Z and t tests are used to determine if the means or proportions of two samples are significantly different. Simple correlation measures the relationship of one variable to another. Correlation coefficient (r) indicates the strength of the association and direction of the association. The coefficient of determination measures the amount of the total variance in the DV that is accounted for by knowing the value of the independent variable. The results are often shown in a correlation matrix. Bivariate regression investigates a straight-line relationship between one IV and one DV. This can be done by plotting a scatter diagram or least squares method. This is used to forecast values of the DV given values of the IV. The goodness of fit may be evaluated by calculating the correlation of determination. Multiple regression analysis allows for simultaneous investigation of two or more IV on the DV
Type of ScaleNumerical Operation Descriptive Statistics NominalCountingFrequency; cross tab Percentage; mode OrdinalRank ordering(plus…)Median Range; Percentile IntervalArithmetic operations on intervals bet numbers (plus…) Mean; Standard deviation; variance RatioArithmetic operations on actual quantities (plus…) Geometric mean; Co-efficent of variation
Selecting appropriate univariate statistical method ScaleBusiness Problem Statistical question to be asked Possible test of statistical significance Nominal Scale Identify sex of key executives Is the number of female executives equal to the number of males executives? Chi-square test
ScaleBusiness Problem Statistical question to be asked Possible test of statistical significance Nominal Scale Indicate percentage of key executives who are male Is the proportion of male executives the same as the hypothesized proportion? Z test
ScaleBusiness Problem Statistical question to be asked Possible test of statistical significance Ordinal scaleCompare actual and expected evaluations Does the distribution of scores for a scale with categories of poor,good, excellent differ from an expected distribution? Chi-square test
ScaleBusiness Problem Statistical question to be asked Possible test of statistical significance Interval or Ratio scale Compare actual and hypothetical values of average salary Is the sample mean significantly different from the hypothesized population mean? Z-test (sample is large) T-test (sample is small)
Determining Sample Size What data do you need to consider –Variance or heterogeneity of population –The degree of acceptable error (confidence interval –Confidence level –Generally, we need to make judgments on all these variables
Determining Sample Size Variance or heterogeneity of population –Previous studies? Industry expectations? Pilot study? –Sequential sampling –Rule of thumb: the value of standard deviation is expected to be 1/6 of the range.
Determining Sample Size Formula N= (ZS/E) 2 Z= standardization value indicating confidence level S= sample standard deviation E= acceptable magnitude of error Its not the size that matters….