chapter 17: statistical analysis. contents the statistics approach statistical tests – types of...

Chapter 17: Statistical Analysis

CONTENTS

• The statistics approach• Statistical tests

– Types of data and appropriate tests– Chi-square– Comparing two means: the t-test– A number of means: one-way analysis of variance– A table of means: factorial analysis of variance– Correlation– Linear regression– Multiple regression– Factor and cluster analysis

The statistics approach

• Probabilistic statements• The normal distribution• Probabilistic statement formats• Significance• The null hypothesis• Dependent and independent variables.

A. J. Veal & S. Darcy (2014) Research Methods for Sport Studies and Sport Management: A practical guide. London: Routledge

Probabilistic statements

• descriptive: e.g. : 10% of adults play tennis• comparative: e.g. : 10% play tennis, but 12% play golf• relational: e.g. 15% of people with high incomes play tennis

but only 7% of people with low incomes do so: there is a positive relationship between tennis-playing and income.

• However: when based on a samples, the above must be made using a probabilistic format


Probabilistic statements contd

• We can be 95% confident that the proportion of adults that plays tennis is between 9% and 11%

• The proportion of golf players is significantly higher than the proportion of tennis players (at the 95% level of probability)

• There is a positive relationship between level of income and level of tennis playing (at the 95% level)

• (See discussion of Confidence intervals: Chapt 13).


Probabilistic statement formats

• 95% probability– sometimes expressed as 5% – sometimes as 0.05

• 99% probability is also used – also expressed 1% or 0.01


Normal distribution (Fig. 17.1):

Popn Value

-4 -3 -2 -1 +1 +2 +3 +4

Samples

Sample valuesSample values

a. Drawing repeated samples (theory)


Normal distribution contd

Popn Value

-4 -3 -2 -1 +1 +2 +3 +4

Samples

Sample valuesSample values

b. Normal distribution/curve


Normal curve (Fig. 13.1)

NUMBER OF SAMPLES

-4 -3 -2 -1 +1 +2 +3 +4

Popn Value

2.5%2.5%

95%

Standard errors Standard errors


Significance

• Statistically significant: unlikely to have happened by chance (highly probable)

• Level of significance is affected by sample size (not by population size)

• Probability of finding happening by chance related to normal curve and similar theoretical distributions.

• But NB: small differences or weak relationships may not be socially or managerially significant – even when they are statistically significant


Null hypothesis

• H0 – Null hypothesis: there is no significant difference or relationship

• H1 – Alternative hypothesis: there is a significant difference or relationship

• eg.– H0 tennis and golf participation levels are the same;

– H1 tennis and golf participation levels are significantly

different.


Dependent and independent variables

Independent variable 2


Dependent variable



Statistical testsTask Format of data No. of

var’blesTypes of variable

Test

Relationship between 2 variables

Crosstabulation of frequencies

2 Nominal Chi-square

Difference between 2 means - paired

Means: for a whole sample

2 Two scale/ordinal

t-test - paired

Difference between 2 means – indep-endent samples

Means: for 2 sub-groups

2 1. scale/ordinal (means)2. nominal (2 grps only)

t-test – independent samples


Statistical tests contd

Task Format of data

No. of var’bles

Types of variable Test


Means - for 3+sub-groups

2 1. scale/ ordinal (means)2. nominal (3+ groups)

One-way analysis of variance

Relationship between 3 or more variables

Means: crosstabulated

3+ 1. scale/ordinal (means)2. Two or more nominal

Factorial analysis of variance


Statistical tests contd

Task Format of data

No. of var’bles

Types of variable

Test


Individual measures

2 Two scale/ ordinal

Correlation

Linear relationship between 2 variables

Individual measures

2 Two scale/ ordinal

Linear regression

Linear relationship between 3+ variables

Individual measures

3+ Three or more scale/ ordinal

Multiple regression

Relationships between large nos of variables

Individual measures

Many Large nos of scale/ ordinal

Factor analysisCluster analysis


Data

• Extended version of Campus Sporting Life survey with– additional variables – additional cases

• See Appendix 17.2• SPSS used, as in Chapt. 16


Chi-square (X2)

• Testing the relationship between two variables presented in a frequency crosstabulation.

• Null/alternative hypotheses:– H0 - there is no relationship between student status and gender in the

population– H1 - there is a relationship between status and gender in the

population

• Findings (Fig. 17.5):– Value of Chi-square: 6.522– Significance: 0.011– Less that 0.05 (5%)– Conclusion: H0 rejected, H1 accepted: there is a relationship


Comparing two means: t-test

• Paired samples: whole sample: comparing means for two variables

• Independent samples: sample divided into two groups (eg. males and females) and comparing means for one variable


Comparing 2 means: t-test : Paired samples (Fig. 17.9)

• Example 1: Compare average times played sport in last 3 months (12.2) with average times visited national parks (9.8)

• Difference is 2.4• value of t is 1.245• Significance is 0.219, which is larger than 0.05• Null hypothesis is accepted: difference is not

significant


Comparing 2 means: t-test : Paired samples (Fig. 17.10)

• Compare course costs for males ($110.00 pa) and females ($136.60)

• Difference is $28.60• value of t is -1.245• significance is 0.219• Null hypothesis is accepted: difference is not

significant


One-way analysis of variance (ANOVA)(Fig. 17.11, 13)

• Means of one variable for groups defined by another variable• F-test rather than r-test• eg. Means of times played sport by student status:

– F/T student/no paid work: mean = 9.7 times in 3 months– F/T student/paid work: 9.6 times– P/T student – F/T job: 19.1 times– P/T student – Other: 12.2 times

• Value of F: 2.485, Significance 0.072, which is greater than 0.05• Null hypothesis accepted: no relationship between status and sport• But for ‘going out for a meal’: F = 6.64 and Sig. = 0.001, which is less than 0.05,

so null hypothesis rejected: there is a significant relationship


Factorial analysis of variance (Fig. 17.14, 15)

• Status not significant and gender not significant• But for status x gender: F = 3.681, Sig. = 0.019, which is <0.05, so

null hypothesis rejected: there is a significant relationship.

Mean number of visits in three monthsStatus Male FemaleF/T student/no paid work 3.1 1.5F/T student/paid work 1.6 5.4P/T student - F/T job 1.4 2.6P/T student/Other 3.5 3.2

• A table of means: two variables and means of a third• eg. Mean visits to theatre by gender by student status


Correlation (Fig. 17.16) Watched sport by income: weak positive: r = .46


Correlation (Fig. 7.16) Played sport by income: weak negative: r = -0.44


Correlation (Fig. 7.16) Sport exp. by income: strong positive: r= 0.91


Correlation (Fig. 17.18)

• Correlation coefficient (r) expresses the relationship numerically• No relationship: r =0• Exact relationship: r = 1 (positive) -1 (negative)• Correlation matrix shows correlations between a number of

variables


Correlation matrix (simplified Fig. 17.18)

Income Sport Watch sport

Visit park

Meal Sport exp.

Income 1.00Sport -.44** 1.00Watch sport .46** -.68** 1.00Visit park .02 .27 -.29* 1.00Meal .08 .45** -.29* -.04 1.00Sport exp. .91** -.37** .38 .06 .12 1.00

* = significant at the 0.05 level** = significant at the 0.01 level


Regression Fits best fit ‘regression line’ to scatterplot: Fig. 17.21


Regression: best fit may be a curve (Fig. 17.22)


Multi-variate analysis

• Multiple regression has one dependent variable and a number of independent, influencing, variables

• One development: Structural Equation Modelling explores inter-relationships between a number of variables

• Cluster and factor analysis: combine large numbers of variables into groups – eg. lifestyle or personality groups


chapter 17: statistical analysis. contents the statistics approach statistical tests – types of...

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