Market and Social Research Part 9

1

Quantitative Analysis

Learning Outcomes

•Successful students will have a knowledge and understanding of:-

•Hypothesis testing

The ideaChi-squared rest

•Regression

•Multivariate analysis

Study Skills and Learning Resources - http://www.griot.org

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Basics

Should all be familiar with this material

Specialist statisticians take 2 semesters

Aim is to understand statements like

“I used a chi-squared test to investigate this relationship”


Different types of data:

Nominal data “names” items Ordinal data “orders” items Interval and ratio data indicate

“distances” between items


Study Skills and Learning Resources - http://www.griot.org 3

4


Car owners 44 22.0% 87 34.8%

Total 200 100.0% 250 100%

Percentages

18-29 30-49

Percentage for 18-49 is:-

%1.29100x450

131100x

250200

8744

It is not 22.05+34.8%/2 – the 2 groups are different sizesStudy Skills and Learning

Resources - http://www.griot.org

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Location (`Central Tendency')

The mode = the commonest value in the data (highest peak in a histogram)

The median = the half-way point when the data are sorted; Not distorted by atypically high/low values.

The mean = the average. Usually denoted as x


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Proctor example

Raw data: 9 7 6 4 3 8 7 6 4 2 8 6 6 3 2 7 6 5 3 1 7 6 5 3 1

Sorted data: 11 22 3333 44 55 666666 7777 88 9

Mode 6 is most common, so the mode is [6].

Median twenty-five values, so the 13th is half-way; the 13th value in the sorted list is 6, so the median is [6]

Mean total = 125 and divide this total by the number of values, 25, to get a mean of [5]


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The best of these three is the median, but

nobody knows what it is;

mathematical properties are such it can not be used in calculation

If the mean and median are

similar then use the mean, (best known)

very different, then draw a histogram to find out why, but use the median.


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Dispersion (Spread)

The range is Biggest -Smallest.

The variance - the difference between a sample value and the sample mean squared .

The standard deviation is the square root of the variance;

on the University approved calculators (FX-82, FX-83 or FX-85);

the University approved calculator can work out the mean - you should learn how to get it to do this.

1

)( 22

n

xxs

n

i i

1x n


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Proctor's example

Raw data 9 7 6 4 3 8 7 6 4 2 8 6 6 3 2 7 6 5 3 1 7 6 5 3 1

Range biggest is 9, smallest is 1 , range is 9-1=8.

Variance add up the squared deviations to get 124, and divide by the number of values minus one, 24, gives 5.2

Standard Deviation square-root the variance to get a standard deviation of 2.273 .


10


9 5 4 167 5 2 46 5 1 14 5 -1 13 5 -2 48 5 3 97 5 2 46 5 1 14 5 -1 12 5 -3 98 5 3 96 5 1 16 5 1 13 5 -2 42 5 -3 97 5 2 46 5 1 15 5 0 03 5 -2 41 5 -4 167 5 2 46 5 1 15 5 0 03 5 -2 41 5 -4 16

124

xxi ix x 2)( xxi


Bivariate analysis takes two variables at a time and inspects the pattern between them

Consider the game of “noughts and crosses”

Bivariate Analysis


This is a simple grid -a player must consider the columns and rows just like bivariate analysis

Now think of the game in three dimensions…it helps with the next step..

Bivariate Analysis


Multivariate analysis

This is trivariate analysis

add another level and we can picture a more complex shape

add yet another level and the human mind is almost incapable of playing the game


A correlation is a single number that describes the degree of relationship (association)between two variables.

Strong: . |r|≥0.8. Moderate: 0.5<|r|<0.8. Weak: . |r|≤0.8.

Correlation



Hypothesis Tests

We see a pattern in our data.

How do we know whether this pattern is a peculiarity of our sample (Just chance),

Is this pattern 'real','reliable','generalisable' (is a characteristic of the target population)?

We use hypothesis tests



The idea

Form two hypothesis (theories)

the null hypothesis or H0, - the pattern we see is a random fluctuation;

the alternative hypothesis or H1, is that the pattern is real.



Calculate a test statistic from the data.

Is your test statistic typical of the sort of value the test statistic ought to have if H0 was true?

No - conclude H0 is not true, and so, decide H1 is true.

Yes - conclude H0 could easily be true, and so, no evidence to claim that the pattern is real.



Drink Male Female Total

Coke 66 52 118

Pepsi 67 48 115

7up 35 38 73

Dr Pepper 34 21 55

Sprite 32 41 73

Lilt 33 31 64

Blackjack 18 25 43

Tango 34 24 56

Total 319 280 599

Chi-squared Test

Favourite drinks for 599 people

The male-to-female ratio varies across the drinks:

more males than females for Coke,

fewer males than females for 7 Up.Is this pattern real?



Does this pattern apply to all people in the target population, or just the 599 in the sample?

Hypotheses

The null hypothesis (H0) - that brand preference is the same for both genders

The alternative hypothesis (H1 ) - that brand preference is different for the two genders



What would we expect to see if H0 was true?

If brand preference is the same for the two sexes, then the male-to-female ratio ought to be the same for all the drinks.

There are 319 males and 280 females in the sample

(if H0 was true)

the common male-to-female ratio ought to beMales = 319 : Females = 280

as percentages

Males = 53% :Females = 47% .



Drink Male Female Total

Coke 62.84 55.16 118

Pepsi 61.24 53.76 115

7up 38.88 34.12 73

Dr pepper 29.29 25.71 55

Sprite 38.88 34.12 73

Lilt 34.08 29.92 64

Blackjack 22.90 20.10 43

Tango 30.89 27.11 58

Total 319 280 599

118 people who had Coke as their favourite brand.

Expected male : female

Male =53% of 118 = 62.84.

Female =47% of 118 = 55.16

The expected counts



General formula for the expected values

Row Total x Column Total Grand Total

Coke and males:

118 x 319 = 62.84 599



Calculate the chi-squared (test) statistic

For the Ith cell of a table we have:-

Oi – the observed countEi – the expected count

The chi-squared statistic is:-

ii

ii

E

EOX

22 )(

Larger as the difference between the observed and expected increases –Larger values less typical of random patternsDifferences between observed and expected less important when expected count is large.



Drink Male Female Total Drink Male Female Total

Coke 66 52 118 Coke 62.84 55.16 118

Pepsi 67 46 115 Pepsi 61.24 53.76 115

7up 35 38 73 7up 38.88 34.12 73

Dr Pepper 34 21 55 Dr pepper 29.29 25.71 55

Sprite 32 41 73 Sprite 38.88 34.12 73

Lilt 33 31 64 Lilt 34.08 29.92 64

Blackjack 18 25 43 Blackjack 22.90 20.10 43

Tango 34 24 56 Tango 30.89 27.11 58

Total 319 280 599 Total 319 280 599

OBSERVED EXPECTED



533.911.27

)11.2724(...

16.55

)16.5552(

84.62

)84.6266( 2222

X



Is the test statistic typical of H0?

Degrees of freedom=(n of rows-1)x(n of columns-1)

lf Ho is true then should look like a typical observation from the probability distribution with 7 (i.e., (8- I ) x (2- I )) degrees of freedom.

Two approaches

1. Use critical value from statistical tables = 14.07 at 5%.

2. Find p-value using computer (Excel) = 22 % .

2X2X



2X

The p-value

there is a 22 % chance of Getting an > 9.533 when H0 is true.

Patterns in the drinks data are similar to the random fluctuations expected if brand and gender are not related.

We CANNOT claim the patterns in our sample are anything other than random sampling fluctuations.

2X


Simple Linear regressionIncome vs Shoe Size

Income (£) Shoe size18,368 929,432 9.525,624 8.530,325 823,345 1020,439 7.5


Income vs Shoe Size

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

16,000 20,000 24,000 28,000 32,000

Income (£)

Sh

oe

siz

e



SUMMARY OUTPUTRegression Statistics

Multiple R 0.051R Square 0.003Adjusted R Square -0.247Standard Error 5352.495Observations 6.000ANOVA

df SS MS F Significance FRegression 1.000 301628.929 301628.929 0.011 0.923Residual 4.000 114596817.905 28649204.476


Income vs Shoe Size

“R Square” = 0.003 or 0.3%

0.3% of the variation in the shoe sizes is being explained by income.

“Sig. F”/“P-value” = 0.923

For a linear regression with just one explanatory variable (here: income), the “Sig. F” value from the “ANOVA” and the “P-value” associated with the explanatory variable will be the same.

there is a probability (p) of 0.923 of being wrong if we say that income really does have a relationship with shoe size.


Income vs Shoe Size

y = 0.00000998x + 8.5042

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

16,000 20,000 24,000 28,000 32,000

Income (£)

Sh

oe

siz

e



Multivariate analysis

Deals with many variables and investigates their Inter relationships, simultaneously

•Multiple regression

•Cluster analysis

•Factor (principal components) analysis

•Multidimensional scalingStudy Skills and Learning

Resources - http://www.griot.org 33


•Hypothesis testing

The ideaChi-squared restAnalysis of variance

•Regression


Market and Social Research Part 9

Education

Transcript of Market and Social Research Part 9