Statistics for College Students-part 2

8/6/2019 Statistics for College Students-part 2

1/43

OVERVIEW


2/43

STATISTICS

Modules 5-8

Prepared by:

Mrs. Cristina H. Price


3/43

Module 5 The Normal Curve

The normal curve or the normal frequency distribution is a hypothetical distribution of scoresthat is widely used in statistical analysis. Since many psychological and physicalmeasurements are normally distributed, the concept of the normal curve can be used withmany scores. The characteristics of the normal curve make it useful in education and in thephysical and social sciences.

Characteristics of the Normal Curve

Some of the important characteristics of the normal curve are:

1. The normal curve is a symmetrical distribution of scores with an equal number of scoresabove and below the midpoint of the abscissa (horizontal axis of the curve).

2. The tails of the curve are asymptotic to the horizontal axis.

3. Since the distribution of scores is symmetrical the mean, median, and mode are all at thesame point on the abscissa. In other words, the mean = the median = the mode.

4. If we divide the distribution up into standard deviation units, a known proportion of scoreslies within each portion of the curve. The total area of the curve is equal to 1.

5. Tables exist so that we can find the proportion of scores above and below any part of thecurve, expressed in standard deviation units. Scores expressed in standard deviation unitsare referred to as Z-scores.


4/43

Standard score

It is the distance of an observed value (x) from the

mean in terms of the standard deviation. It tells

how many standard deviations the observed value

lies above or below the mean of its distribution.

s

xxz

=

=

xzOR

Where: x = observed value or raw score

= sample mean

s = sample standard deviation

= population mean = population standard deviation

x


5/43

Using MS Excels Statistical

Functions

Conversion of raw score to standard score

=standardize(x, , s)

Finding the area/probability value given the value of z=normsdist(z)

Finding the value of z given the probability value

=normsinv(p-value)

x


6/43

Sample problem

The average daily income of 2000 workers isP362.00 with a standard deviation ofP15.00. Assuming that the daily incomesare normally distributed,

a) what percent of the workers earn at leastP380.00 per day?

b) what percent of the workers earn below

P350.00 per day?c) determine the number of workers who earn

from P350.00 to P375.00 per day.


7/43

Exercises

1. In a departmental examination in statistics, the mean grade was74 and the standard deviation was 10. If the grades areapproximately normally distributed and 40 students got gradesbetween 70 and 80, how many students took the examination?

2. The experience of a certain hospital showed that the distributionof length of stay of its patients is normal with a mean of 11.5 days

and a standard deviation of 2 days.a) What percent of the patients stayed 9 days or less?

b) If a new method in nursing care is to be administered to themiddle 95% of the group, how long should a patient stay to beincluded in the study?

3. A study finds that the time spent on advertisement per hour on a

certain TV station is approximately distributed with mean equal to12.8 minutes and standard deviation equal to 2.2 minutes. Duringa randomly selected hour, what is the probability that between 14and 16 minutes were devoted to advertisements?


8/43

Hypothesis statement that is formulated which

cannot be accepted to be true unless otherwise

proven

Assumption statement that is formulated andaccepted to be true without the necessity of a

proof. It serves as the springboard of the study

Types of hypothesis null and alternative

Module 6 - Hypothesis Testing


9/43


10/43

Other key concepts

Types of test one-tailed and two-tailed

Level of significance alpha (0.01, 0.05, 0.1)

Observed value the obtained computed

value based on the data gathered

Critical value the value obtained from the

table; the value that divides the distribution of

the test into the rejection and the acceptanceregion


11/43

Critical Values of z

Test Type Level of significance

0.05 0.01One-tailed 1.645 2.33

Two-tailed 1.96 2.575


12/43

Steps to performing hypothesis testing

1. Write the original claim and identify whether it is the nullhypothesis or the alternative hypothesis.

2. Write the null and alternative hypotheses. Use thealternative hypothesis to identify the type of test.

3. Write down all information from the problem.

4. Determine the appropriate test statistics. Find thecritical value using the tables.

5. Compute the test statistic.

6. Make a decision to reject or fail to reject the nullhypothesis. A picture showing the critical value and teststatistic may be useful.

7. Write the conclusion.

Steps in hypothesis testing


13/43

Bivariate Distribution

Involving two variables

- Significant difference (t-test, z-test, ANOVA)

- Significant relationship (Pearson r,

Spearmans rho, Chi-square and other

correlational techniques)


14/43

Testing significant difference using

parametric test (two groups)

t test

distribution is normal

homogeneous variance

sample std. deviation is knownn < 30

z test

distribution is normal

homogeneous variance

population std. deviation is known

n 30


15/43

t-test & z test

(sample vs population)

1;)(

=

= ndf

s

nxt

nxz )( =


16/43

Decision rules

Observed value < Critical value

OR

p-value > level of significance()

Accept the null hypothesis (Theres not enoughevidence to reject the null hypothesis)

Observed value Critical value

OR

p-value < level of significance()

Reject the null hypothesis


17/43

Sample Problems

1. A certain rice miller claims that the average weightof a cavan of rice is 50 kilograms with a standarddeviation of 5 kilograms. A retailer sampled 20cavans of this rice and got an average weight of

46.6 kilograms. Is the claim of the rice miller validusing 5% level of significance?

2. A standardized test was administered to thousandsof pupils with a mean score of 85 and a standarddeviation of 8. A random sample of 50 pupils were

given the same test and showed an average scoreof 83.20. Is there evidence to show that this grouphas a lower performance than the ones in general at0.05 level of significance?


18/43

t-test & z test

(two-sample groups)

y

y

x

x

yx

nn

yxz22

)()(

+

=

2;

11

2

)1()1(

)()(

22+=

+

+

+

= yx

yxyx

yyxx

yxnndf

nnnn

SnSn

yxt


19/43

Sample Problems

1. A random sample of 20 newly-born baby boys showed an average weight of7.4 pounds while a sample of 25 newly-born baby girls showed a meanweight of 6.5 pounds. If the variance of all newly-born babies is 1.25 pounds,can we say that newly-born baby boys are heavier than newly-born babygirls?

2. Two hamburger stores were compared in terms of the number of orders of

hamburger per day. The results of the ten-day observation were as follows:

Using the 0.05 level of significance, test if there is a significant difference inthe number of orders of hamburger from the two stores.

Day 1 2 3 4 5 6 7 8 9 10

Nutri 148 126 103 169 135 152 144 124 132 128

Deli 150 127 125 152 129 146 153 118 126 119


20/43

t-test for paired observations

(dependent groups)

1;

)1(

22 =

=

ndf

nn

dnd

d

t


21/43

Sample Problem

A certain diet pill was developed by a pharmaceutical company. To testthe efficacy of the said pill, 10 randomly selected individuals were selected.The results of the study are presented in the following table:

Use hypothesis testing to determine whether the diet pill is effective or not.

SUBJECTS WEIGHT BEFORE WEIGHT AFTER

1 148 150

2 142 1393 131 130

4 128 128

5 121 123

6 118 115

7 120 119

8 152 151

9 112 110

10 110 105


22/43

Module 7 - Correlation and Simple

Linear Regression

1. Pearsons product-moment correlation

coefficient (Pearson r)

2. Spearmans rank-order correlation

coefficient (Spearmans )

P d l i


23/43

Pearsons product-moment correlation

coefficient

( ) ( )

( )[ ] ( )[ ]

=

2222yynxxn

yxx ynr


24/43

Sample Problem

Determine if there is a relationship between the number of years of service

and the employees monthly salary based on the data gathered from a

certain company.

No. of yrs. of service Monthly salary (in T)

5 25

7 28

8 29

10 32

12 34

2 18

11 32

15 35

20 40

25 50


25/43

Legend for Pearsons r and

Spearmans rho

0.00 0.3 Little or no positive correlation

0.31 0.5 Low positive correlation

0.51 0.7 Moderately positive correlation

0.71 0.9 High positive correlation

0.91 1.0 Very high positive correlation

Overview


26/43

Testing the significance of the

relationship

2,1

2

2=

=

ndfr

n

rt

Where:

r = the correlation coefficientn = no. of pairs

df = degrees of freedom


27/43

Key concepts to remember

Correlation simply describes a relationship

between two variables. It does not explain

why the two variables are related.

Specifically, a correlation should not andcannot be interpreted as proof of a cause-

and-effect relationship between the two

variables.

The value of a correlation can be affectedgreatly by the range of scores represented in

the data.


28/43


One or two extreme data points, often called

outliers, can have a dramatic effect on the

value of a correlation.

A correlation measures the degree ofrelationship between two variables.

The values ofrrange from -1.00 to +1.00.

The valuer2

is called thecoefficient of

determination because it measures the

proportion of variability in one variable that

can be determined from the relationship with

the other variable.

S k d l ti


29/43

Spearmans rank-order correlation

coefficient (Spearmans )

)1(

6

1 2

2

=

nn

d

Where:n = no. of pairs

d = difference between the ranks of each pair

Statistics to test the


30/43

Statistics to test the

significance of

1= nz


31/43

Sample Problem

Seven instructors are rated by

freshmen and sophomore

students on clarity of

presentation and the results

are tabulated. What is the

Spearman rho for the

following?

Instructor Freshmen Sophomore

1 44 58

2 39 42

3 36 18

4 35 22

5 33 31

6 29 38

7 22 38

Where a is the intercept and b is the slope or the incremental change in Y when X changes by one unit


32/43

Regression Analysis

Regression Analysis is a statistical technique used to

describe relationships among variables. This

relationship is expressed in a form of mathematical

equation. The simplest case of such a relationship is

when there is a single independent variable (X)explaining the dependent variable (Y) in a linear

fashion.

bxay +=

Where a is the intercept and b is the slope or the incremental change in Y when X changes by one unit.

Where a is the intercept and b is the slope or the incremental change in

Y when X changes by one unit.


33/43


Regression analysis is the most widely used technique of

Multivariate Analysis with applications across all types of

problems and all disciplines.

It is a statistical technique that is concerned with describing and

evaluating the relationship between a metric variable called

dependent variable and one or more metric or non-metric

variables called independent variables orregressors.

It attempts to predict the change in the dependent variable as a

result of changes in the independent variables. In addition, the

analysis of the independent variables allows assessment of their

respective explanatory impact on the dependent variable.

NSAT Achievement

G d

SUMMARY OUTPUT


34/43

Grade

78 82

79 83 Regression Statistics

80 82 Multiple R 0.95

92 91 0.90

93 94 Adjusted 0.8866686 85 Standard

Error1.45918

88 87 Observations 8

86 86

ANOVA

df SS MS F Significance F Regression 1 118.7248 118.725 55.7605 0.0003

Residual 6 12.77516 2.1292

Total 7 131.5

Coefficients StandardError

t Stat P-value Lower 95%

Upper95%

Intercept 25.46 8.16 3.12 0.02 5.50 45.42

NSAT 0.71 0.10 7.47 0.00 0.48 0.95

ab


35/43

Using the formula:

Predicted Achievement grade = 25.46 + 0.71 * 70

= 75.38

The value of r-squared indicates the percentage ofrelationship between the NSAT scores and the

achievement grade. Thus, there is 90.29 %

association.


36/43

Module 8 Selected Nonparametric

Statistics

Chi-square test (2)

Mann-Whitney U test

Kruskal-Wallis H test


37/43

Chi-square Test

Significant relationship

Test of goodness-of-fit

Test of independence

( ))1)(1(;

2

2=

= crdf

EF

EFOF

Where:

OF= observed frequency

EF = expected frequency

Sample problem:


38/43

Red Yellow Green Blue Total

Introvert 10 3 15 22 50

Extrovert 90 17 25 18 150

100 20 40 40 200

Sample problem:

Suppose we want to find out if there is a relationship between the students

color preference and personality. The data may be illustrated in the

contingency table below:

Observed frequencies:

Row

totals

(fr)

Column totals (fc)

Grand total (n)


39/43

To determine the expected frequencies for each cell, we

use the formula below:

n

ffef cr

))((=

Whereef = expected frequency

fr = total frequencies of the corresponding row

fc = total frequencies of the corresponding column

n = grand total


40/43

Testing significant difference using

nonparametric test (two groups)

Mann-Whitney U test

+

+= 111

2112

)1(R

NNNNU

+

+= 222

2122

)1(R

NNNNU

Where:

U is the lower value between U1 and U2.


41/43

Example:

Treatment Control

4 20

7 17

1 3

12 15

2 7

2 12

9 18


42/43

Testing significant difference

(3 or more groups)

Parametric test (distribution is normal)

ANOVA (Analysis of variance)

Nonparametric test

Kruskal-Wallis

1),1(3

)1(

12

1

2 =+

+

= =

kdfnR

nn

Hk

i

i


43/43

To compare four bowling balls, a professional bowler bowls five

games with each ball and gets the following scores:

Bowling ball A 221 232 207 198 212

Bowling ball B 202 225 252 218 226

Bowling ball C 210 205 189 196 216

Bowling ball D 229 192 247 220 208

Use the H test at 0.05 level of significance to test the nullhypothesis that on the average the bowler performs equallywell with the four bowling balls.

Documents

Statistics for College Students-part 2