Applied Business Research and Analysis (assessment 2)

Cindy Yu

qr__dap__lecture.pdf

Home >Business & Finance homework help >Management homework help >Applied Business Research and Analysis (assessment 2)

Salford Business School

Applied Business Research and Analysis

Quantitative Research

Data Analysis and Presentation

Lecture

Professor David F. Percy

Contents

1. Summaries

2. Probability

3. Inference

4. Modelling

5. Forecasting

Bibliography

1. Rees D.G. (2000) Essential Statistics, Chapman & Hall

2. Weiss N.A. (2012) Introductory Statistics, Addison-Wesley

3. Freund J.E. & Perles B.M. (2006) Modern Elementary Statistics, Prentice Hall

Syllabus

1. Summaries

C o s t

Sample size

� � 24 Measures of location

mode (most likely value): £51.00

median (middle ordered value): £51.00

mean (average value): £50.50

1. Summaries Gas heating costs (£) per unit area in 24 UK factories

51 52 50 47 55 57 43 59

52 51 40 48 53 47 54 63

51 45 55 53 48 56 36 46

Measures of spread

range (difference between maximum and minimum): £27.00

interquartile range (difference between upper and lower quartiles): £7.75

standard deviation (root mean square distance about mean): £6.03

1. Summaries

Summary Statistics

Consider the current ratios (assets÷liabilities) of 8 market traders:

1.43, 1.02, 2.07, 2.35, 0.81, 1.73, 2.99, 1.26

Sample median:

median � �. ��.�� 1.58

Sample mean:

�̅ � �� ∑ �� . ��⋯��.�� 1.71

Sample range:

maximum � minimum � 2.99 � 0.81 � 2.18

Sample standard deviation:

s � ��#� ∑ �� ̅ � � ��

�. �#�.�� $�⋯� �.��#�.�� $ � � 0.73

1. Summaries

Accuracy of Data

• Express different observations of a measurement, such as monthly sales figures, to

the same degree of precision.

• Most observations are rounded up or down in the last decimal place: heights of

1.341 metres and 1.417 metres become 1.34m and 1.42m respectively.

• Retain as much accuracy as possible in intermediate calculations: do not round the

sample mean when calculating a sample standard deviation.

• Avoid displaying results of calculations to more accuracy than is needed: present a

sample mean as 5.2 rather than 5.166666667.

• Enter data carefully into computer spread sheets and perform simple numerical and

graphical checks that the data are reasonable.

• Use a special symbol for missing data such as * rather than a space, zero or minus

number to ensure that these are excluded from your analysis.

• Avoid guessing missing data values without proper justification.

1. Summaries

Event % collection of outcomes Probability & % chance that % occurs Complement %′ & % ( & %′ � 1

impossible evens certain

0 ½ 1

2. Probability

Union % ∪ * % or * or both Intersection % ∩ * % and * Conditioning *|% * given %

% *

2. Probability

% = “motorist makes insurance claim this year” * = “motorist makes insurance claim next year”

& % � & * � -. and & *|% � -$ then

& %′ � 1 � & % � � & % ∩ * � & % / & * % � ��

& % ∪ * � & % ( & * � & % ∩ * � ��

2. Probability

95% probability interval 479,521 has limits 3 4 1.966

2. Probability

3. Inference

7~9 3,6�

�̅ � 1� : ��

��

;� � 1� � 1 : �� ̅ �

�

��

95% confidence interval for 3 has limits

�̅ 4 < / ;�

0.1000 0.0500 0.0250 0.0100 0.0050 0.0025 0.0010 0.0005

1 3.078 6.314 12.71 31.82 63.66 127.3 318.3 636.6

2 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31.60

3 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12.92

4 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610

5 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869

6 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.959

7 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408

8 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.041

9 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781

10 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587

11 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.437

12 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.318

13 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.221

14 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140

15 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.073

16 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.015

17 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.965

18 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.922

19 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883

20 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.850

21 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.819

22 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.792

23 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.768

24 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.745

25 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.725

26 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.707

27 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.690

28 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.674

29 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.659

30 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.646

40 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.551

60 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.460

120 1.289 1.658 1.980 2.358 2.617 2.860 3.160 3.373

∞∞∞∞ 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291

4 3 2 1 0 1 2 3 4

0.1

0.2

0.3

−

critical value

<=.=�> � � 1

3. Inference

7~? @

@A � 1� :��

��

95% confidence interval for @ has limits

@A 4 1.96 / @ A 1 � @A

�

3. Inference

0.13

3. Inference

null hypothesis for:

mean B= ∶ 3 � 8.3 proportion B= ∶ @ � 0.1

two-sided alternative hypothesis for:

mean B� ∶ 3 E 8.3 proportion B� ∶ @ E 0.1

one-sided alternative hypothesis for:

mean B� ∶ 3 F 8.3 B� ∶ 3 G 8.3 proportion B� ∶ @ F 0.1 B� ∶ @ G 0.1

3. Inference

calculate test statistic and compare with critical value

reject B= or do not reject B= at 5% level of significance p<0.05 (reject) or p>0.05 (do not reject)

3. Inference

4. Modelling

Chi-square Test for Association

Test for association between two factors with B=:"no association" and B�:"association".

For a table with r rows and c columns, compare the test statistic

N� � : obs � exp �

expQRR STRRU with the upper critical value from the N� distribution with V � 1 W � 1 degrees of freedom.

4. Modelling

0.995 0.990 0.975 0.950 0.900 0.100 0.050 0.025 0.010 0.005

1 0.000 0.000 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879

2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.60

3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.34 12.84

4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.14 13.28 14.86

5 0.412 0.554 0.831 1.145 1.610 9.236 11.07 12.83 15.09 16.75

6 0.676 0.872 1.237 1.635 2.204 10.64 12.59 14.45 16.81 18.55

7 0.989 1.239 1.690 2.167 2.833 12.02 14.07 16.01 18.48 20.28

8 1.344 1.646 2.180 2.733 3.490 13.36 15.51 17.53 20.09 21.95

9 1.735 2.088 2.700 3.325 4.168 14.68 16.92 19.02 21.67 23.59

10 2.156 2.558 3.247 3.940 4.865 15.99 18.31 20.48 23.21 25.19

0 5 10 15 20 25 30

5. Forecasting

Business applications often involve data that are collected sequentially over

time (time series) with the aim of predicting (forecasting) future values.

Trend and Seasonality

If a time series 7X is not stationary, we remove trend and seasonality by transforming the original data. Trend is removed by taking lag-one differences,

also called integrating, to generate a new time series YX where

YX � 7X � 7X#� and seasonality is removed by taking seasonal differences to generate a new

time series YX where

YX � 7X � 7X#Z and ; typically takes the values 24 (hours per day), 7 (days per week), 12 (months per year) and 365 (days per year), though other values can arise.

5. Forecasting

Autoregressive AR(1) Model

For a stationary time series 7X , an autoregressive model of order 1 is defined by

7X � [7X#� ( \X where % \X � 0, var \X � 6� and cov \X-,\X$ � 0 for <� E <�, for weight [ that is estimated from data.

10 20 30 40 50 60 70 80 90 100

4 4

4−

1001 t

5. Forecasting

Moving Average MA(1) Model

For a stationary time series 7X , a moving average model of order 1 is defined by

7X � _\X#� ( \X where % \X � 0, var \X � 6� and cov \X-,\X$ � 0 for <� E <�, for weight _ that is estimated from data.

10 20 30 40 50 60 70 80 90 100

4 4

4−

1001 t

5. Forecasting

The Forecast

The one-step-ahead forecast for theAR(1) model is:

7AX��|X � [�X

The one-step-ahead forecast for the MA(1) model is:

7AX��|X � 0

5. Forecasting

10 20 30 40 50 60 70 80 90 100

4 4

4−

1001 t

10 20 30 40 50 60 70 80 90 100

4 4

4−

1001 t