11. NDER3204 Topic 7 - Webnode · 2016-02-28 · probability and inferential statistics....

� INTRODUCTION

We have discussed in the last topic about descriptive statistics which are useful for describing or summarising data. However, in this topic we will look at the probability and inferential statistics. Inferential data provides a framework for drawing conclusion about a population from the data obtained. The researcher normally works with a sample on which judgement or generalisation of the study is made. In order to understand inferential statistics we need to know about sampling distributions.

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77 � Statistics �

Probability and Inferential Statistics

LEARNING OUTCOMES By the end of this topic, you should be able to:

1. Explain the terms Ânormal distribution of dataÊ and Âstandard normal distributionÊ;

2. Identify the common inferential statistics with a view to accepting or rejecting a hypothesis;

3. Differentiate between simple regression and correlation;

4. Describe the assumptions associated with the parametric and non-parametric statistics;

5. Describe a common method of qualitative analysis; and

6. Apply the use of Excel software package in data analysis.

TOPIC 7 STATISTICS – PROBABILITY AND INFERENTIAL STATISTICS � 95

PROBABILITY AND NORMAL DISTRIBUTION

Before learning Normal Distribution of data, you should have some idea about probability. PProbability (P) is a numerical measure of the likelihood that a specific event will occur. Let us discuss a few rules of probability.

(a) PProbability of Mutually Exclusive Events For example, the probability of a pregnant mother delivering a baby boy is half and a baby girl is also half. Here, giving birth to boy and girl are two mutually exclusive events. That means the offspring could be either a boy or a girl, not both at the same time. P [Boy] = �; P [Girl] = � Now you can see that P [Boy or Girl] is � + � = 1.

From the above deduction we can derive two properties of probability:

(i) The sum of the probabilities of all possible outcomes of mutually exclusive events is 1.

(ii) The probability of an event always lies between 0 and 1; � 0 p � 1.

(b) PProbability oof Independent Events For example, we know that the probability of a new born being affected by DownÊs syndrome is 1 out of 10,000. Then what is the probability of having a baby boy affected with DownÊs syndrome? Here the birth of a baby boy and the incidence of DownÊs syndrome are two independent events.

P [Baby boy with DownÊs syndrome] = 1/2 � 1/10,000 = 1/20,000.

In case of independent events such as above, the probability of their joint occurrence is the product of the occurrence of two events.

(c) MMarginal and Conditional Probabilities Suppose all 100 nurses of a hospital were asked whether they are in favour of or against paying high salaries to Senior Nurses without any increase in their own salaries. A classification of this response is given in Table 7.1.

Table 7.1: Two-way Classification of the Responses of 100 Nurses

In Favour Against

Male 5 5

Female 14 76

7.1

� TOPIC 7 STATISTICS – PROBABILITY AND INFERENTIAL STATISTICS

96

Table 7.1 shows the distribution of 100 nurses by gender and whether they are in favour of the proposal or against it. In Table 7.1, note that there are four boxes which contain the numbers indicating the responses. Each of these boxes is a cell and therefore, there are four cells. This Table is an example of a contingency table. For example, 5 nurses in this group possess two characteristics: ÂmaleÊ and Âin favour of paying high salaries to Senior NursesÊ. Similar interpretations can be made for the other cells. By adding the row and column totals, we can come up with a new table, Table 7.2 as shown below.

Table 7.2: Two-way Classification of NursesÊ Responses with Totals

In Favour Against Total

Male 5 5 10

Female 14 76 90

Total 19 81 100

Suppose we select one nurse from the above 100 nurses. This nurse could be either a female or a male, and if we consider one character at a time, the selected nurse may be in favour or against the proposal. The probability of each of these events or characteristics is called mmarginal probability or simple probability. Hence, marginal probability is the probability of a single event without considering any other event. This marginal probability is calculated by dividing the number in each row or in column by the grand total. Thus, the marginal probability of obtaining a male, if we select randomly one individual from the above example, is:

P [Male] =

Number

of MalesGrandTotal

= 10100

= 0.1

While, the marginal probability of a female, if we select one individual is,

P [Female] = 90

100= 0.9


Marginal probability of nurses in favour of the proposal is P [In Favour] = 19/100 = 0.18 While, the marginal probability of nurses against the proposal is P [Against] = 81/100 = 0.81 Now suppose we select one nurse out of the 100 nurses and we assume that the selected nurse is a male nurse. This means that the nurse selected is attached with a condition i.e. the nurse is a male. Now, what is the probability that the selected nurse is in favour of paying high salaries to senior nurses? This is a cconditional probability where we are trying to find the probability that the nurse selected is in favour, given that the nurse is a male. This Conditional Probability is defined as the probability that an event will occur given that another event has already occurred. This is written as: P[ A � B ] = P [In favour, male] = P (AB) / P (B), where AB is a joint event (in favour, male ) and B is the total number of males. In the above example (Table 3), this is: Number of males in favour � Total number of males = P (AB) / P (B)

= (5/100) / (10 / 100) = 5

10 = 0.5

Similarly, conditional probability that the nurse selected against the proposal,

given that the nurse is a female is 7690

= 0.84.


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CONTINUOUS RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTION

The idea of probability is often used in statistics. If we do a statistical test we always refer to the significance of the test with regard to a particular probability. For example when comparing two sample means, we may obtain the mean of a sample A that is significantly higher than the mean of sample B. This level of significance is related to probability.

7.2

1. In a sample survey of 1800 senior citizens, they were asked whether they have ever been vaccinated against flu (irrespective of whether they have vaccinated once or many times). The following table gives the responses by age group:

Vaccinated Not vaccinated

Age 50-59 106 698

60-69 145 447

70 or over 61 343

If one person is selected at random from the above 1800 senior citizens, find the following probabilities:

(a) P (Have been vaccinated)

(b) P (Not vaccinated)

(c) P (Vaccinated among the age group 50-59)

(d) P (Not vaccinated among the age group 70 or over)

(e) P (Vaccinated among the age groups 50-69) 2. What is the difference between dependent and independent

variables? 3. Why is the joint probability of A and B equals to zero, when A and B

are mutually exclusive events? 4. What is the difference between marginal and conditional probability?

SELF-CHECK 7.1


7.2.1 Normal Distribution

Now let us get an idea of probability using a normal distribution of data. We will look at various statistical tests in the later sections. The normal distribution is an important symmetric, bell-shaped distribution with a single peak. To describe a normal distribution, two parameters must be mentioned: � (mean) where the peak of the distribution occurs and � (standard deviation) which indicates the spread of the distribution. Different values of mean and standard error show different types of normal distribution. Figure 7.1 shows the percentage of area within various standard deviations.

Figure 7.1: Normal distribution showing areas at different points

Source: www.uog.ca In general: 68% of the observations fall within 11 standard deviation of the mmean, that is,

between � � 1� and � + 1s 95% of the observations fall within 2 standard deviations of the mmean, that is,

between � � 2s and � + 2s 99.7% of the observations fall within 33 standard deviations of the mmean, that is,

between � � 3s and � + s Thus, for a normal distribution, almost all values lie within 3 standard deviations of the mean. Example: Let us go through an example to describe a normal distribution curve (see Figure 7.2). We are usually interested in finding the probability that a continuous random variable is within some interval. Supposing x is the number of hours a nurse works during a week. If we get the data of hours worked by


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nurses in a hospital and assume that the data is normally distributed with � = 55 and�� = 5 hours, then how could we find the probability that a particular nurse has worked for more than 65 hours a week? We can give an approximate answer to this problem at this time. However, later we will be able to determine the correct answer by using a standardised normal distribution curve i.e. through a z-score. We know that 95% of working hours of the nurses will be within two standard deviations of the mean. The mean in the above example is 55 hours and two standard deviations give us 5 � 2 = 10 hours. When we use 55 for the mean, and then add and subtract the 10, we get the value of 45 and 65. So, 95% of the hours worked are in the interval of 45 and 65. This leaves approximately 5% of the area under the bell-shaped curve outside the interval of 45 and 65. The areas outside of the interval are called the tails of the curve. The area under the right tail of the curve, pointed by an arrow gives the probability of x � 65 and is one half of the 5%, and we can write: P (x � 65) = 2.5% Similarly, the area under the left tail of the curve is one half of 5%, and we can write: P(x � 45) = 2.5%

Figure 7.2: Normal distribution curve with 0.5% area on the left of � 3� and 0.5% area on

the right of 3s. Arrows on the left and right side indicates 0.5% of area on the left and right extremity.


7.2.2 Standardised Normal Distribution

Here the normal distribution is standardised to convert the mean of 0. Let X be a continuous random variable from a normal distribution N (�, �2), then X can be transformed to a Z-score of standard normal distribution by using the following formula:

��

XZ

The random score Z is said to have a standard normal distribution with a mean of 0, and a known variance 1. The standard normal curve preserves the same normal properties. Figure 7.3 shows the above transformation.

Figure 7.3: Transformation of X to Z score (extracted from Mohd. Kidin, 2006)

With the above transformation, almost all probability problems can be solved easily by using the Table of Standard Normal Curve N (0, 1).


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REGRESSION AND CORRELATION

So far our discussion has been limited to one variable but many of the problems in statistics involve two variables. The sets of data for these problems contain ordered pairs. When we work with these problems, we say that we are doing a bivariate data analysis. One of the first things that we do with bivariate data is to estimate simple regression of a dependent variable on an independent variable. Also a correlation estimate between two variables is usually obtained for two variables which may be dependent or independent of one another.

7.3.1 Regression Analysis

Regression is defined as the dependence of one variable (Y) on the other (X). Here Y is the dependent variable and X is the independent variable. A regression estimate is used to make predictions about phenomena. For example, we could use a regression estimate:

(a) To predict the bodyweight of school children from their respective body heights.

(b) To predict nursing school achievement on their grades in the STPM or Matriculation examination.

(c) To predict the stress level of some factory workers on the noise level in the factory.

The above examples indicate a classical simple approach of studying the effect of a single dependent variable on a single independent variable. In this topic you will only study this approach but you should remember that in nursing research, the statistics involve multivariate procedures where a single variable is determined by more than one independent variable.

7.3

1. What is the difference between a discrete and continuous random variable?

2. What is the standard normal distribution?

SELF-CHECK 7.2


Basic Linear Regression Equation The basic linear regression is represented by the following equation:

Y = a + bX...........................................................................EEquation 7.1

where Y = Predicted value of variable Y (dependent variable)

a = Intercept constant

B = Regression coefficient (slope of the line)

X = Actual value of variable X (independent variable) You may remember from your high school algebra that the above equation is the equation for a straight line. Linear regression is used to determine a straight line from the given values of X and their respective values of Y.

7.3.2 Correlation Between Two Variables, X and Y

Correlation measures the degree of association between two variables. Usually these two variables are assumed to be independent although correlation estimates can be calculated between one independent and one independent variable.

Correlation estimates are usually obtained between two variables that are logically or biologically related. If we want to correlate estimates between two variables such as the level of oxygen in the water of Klang river and the performance in the Kuala Lumpur stock exchange, this becomes a spurious correlation and is not justified.

Method of estimation of correlation coefficient (r) follows almost the same pattern as the estimation of regression coefficient (b) except the denominator terms in correlation coefficient is different from the regression coefficient.

The correlation coefficient is a tool for quantitatively describing the magnitude and direction of a relationship between two variables.

Correlation coefficient =

2 2

Covariance between two variables, say, X and YSquare root of the product of sum squares of X and Y .

xy

x y.....EEquation 7.2


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Where xy is the covariance between variable X and Y, 2y is the square

root of the sum square of the deviation of each X value from their mean 2y is

the square root of the sum square of the deviation of each Y value from their mean. Correlation coefficient ranges from 0 to 1. From the following graphs (see Figure 7.4), you can see the positive correlation when both X and Y values increase, negative correlation when X increases but Y decreases, and no correlation when X increases but value of Y remains constant. For example, the weight of new born babies are usually positively related to their height. PeopleÊs age and the amount of sleep they get are inversely related. Body size and speech fluency of children are mostly non-related.

(a) (b) (c)

Figure 7.4: Different types of correlation

Test of Significance for the Correlation Correlation coefficient can be tested to determine whether it is significant or not, by comparing the calculated estimate and a tabular value of the estimate obtained from a correlation table (See Buku Sifir published by Faculty of Science, OUM). You can also see the critical values of Pearson Product-moment Correlation Coefficient from the following table.

To obtain the critical value, you need to fix a level of probability. A common alpha level, �, chosen is usually 0.05 for significance and 0.01 for high significance of correlation coefficient. You also have to determine the degrees of freedom (df) for the test. For testing r, df is two less than the subjects provided in an experiment.

If N= 10 (Table 3.9), then df will be 10 �2 =8.


Therefore, in the example of Table 3.9, you need to see the tabular value at 8 degrees of freedom for � = 0.05 or 0.01. From the table of critical values of r, you can see the tabular values are 0.632 and 0.765 respectively. Since the estimated correlation coefficient, 0.955, was higher than both the values, we may say that correlation coefficient between the mid-term marks and final marks is highly significant. y2 = coefficient of determination which means how much variability in y is determined by x.

INFERENTIAL STATISTICS

So far you have learnt descriptive statistics where you present your data in graphical forms, estimate central tendency and dispersion of a distribution, and estimate relationships between variables in bivariate distributions. Researchers Often use Inferential Statistics

(a) For testing whether a sample statistic drawn from a random sample is representative of the population parameter. For example, can we draw a conclusion from an experiment with 20 samples on body temperature using a newly acquired digital thermometer or do we need to develop a confidential interval around the mean temperature and generalise this for a larger population? (see Figure 7.5)

Take another example. If you have estimated from a random sample of a hospital maternity ward the maternal age of 50 warded females at their first pregnancy, how would you obtain the maternal age of all females at their first pregnancy?

7.4

SELF-CHECK 7.3

What is the difference between simple linear regression and simple correlation?


106

Figure 7.5: Example of sample statistics

(b) FFormulating hypothesis: Hypothesis is formed based on some earlier theory or experimental evidence. Supposing we want to test whether maternity patients can be discharged early from the hospital maternity ward on their maternal competence graded as A and B based on health and mental status. We collect data on sampled maternity patients of two grades and compare the differences between the two grades. A nnull hypothesis (Ho) can be formulated as follows: øA = øB (No difference on discharge date between two competency grades). An aalternative hypothesis (HA) is øA � øB (There is a difference between the two competency grades).

(c) For ttesting hypothesis whether there are differences between blood cholesterol levels between male and female patients with age over 50, a null hypothesis could be that there are no differences whereas an alternative hypothesis would be that there are differences.

(d) Prove that a hhypothesis developed during the conceptual phase of the experiment is correct or not. For example a hypothesis could have been developed by a Malaysian researcher that critically-ill arthritic patients can be cured with a new medicine which has just been approved by the Food and Drugs Administration (FDA). This is a question that can only be solved after proper experimentation with some arthritic patients that were administered with this medicine in different hospitals in Malaysia.

(e) TTypes of error: In making inference about a population, the researcher always face some kind of uncertainty. Whether a null hypothesis, such as there is no difference between the results of two hospitals on the efficacy of a drug on 50 patients from each hospital, or an alternative hypothesis that there is a difference in results between the two hospitals. In setting a significance level for the test of null or alternative hypothesis, the researcher puts a value on the size of that uncertainty. It has been mentioned earlier that the significance level for testing is usually set at 5% or 1%. With a significance level of 5%, we are accepting a 1 in 20 chance of rejecting the null hypothesis when it is true.


There are two types of errors:

(a) TType-1 error: When the null hypothesis is true and by mistake we reject it.

(b) TType-2 error: When the null hypothesis is false but by mistake we accept it.

Often, instead of presenting a Type-2 error, we quote the power of a test. The power is simply 100% minus the probability of a Type-2 error. This is the probability of accepting the null hypothesis when it is false. Level of Significance Some statistical tests are used by researchers to find out whether the experimental result obtained is by chance rather than it could be generalised. The level of significance in hypothesis testing is set prior to experiment and is usually set at P = 0.05 or P = 0.01. A probability of 0.05 means there is 95% probability of accuracy of the findings and 5% risk of error. Likewise, a 0.1% probability means there is 99% probability of accuracy and 1% risk of error. Two categories of inferential statistics are:

(a) Parametric tests; and

(b) Non-parametric tests.

7.4.1 Parametric Tests

There are numerical statistical tests that can be classified as parametric and non-parametric tests. In parametric tests:

(a) One assumes that the variable of interest is normally distributed in the population;

(b) There is at least one parameter measured; and

(c) The dependent variable is measured on an interval level or higher.


108

An example of a parametric test is given below: t-Test Comparison between two means when data is collected from two independent groups: The assumptions in this case are:

(a) Both populations from which samples are drawn for comparison are normally or approximately normally distributed;

(b) Variances for both population are equal; and

(c) Sample size drawn from the two populations are small (less than 30). Suppose that we have chosen two groups of students (an experimental group and a control group) with 10 samples from each group. We want to study the pulse rate of students in the experimental group that have received a special intervention to alleviate stress during the examination period. The students of the control group, also under stress, did not receive such intervention. Data collected on the pulse rates is given in the following table. We want to test whether there is a difference between the mean pulse rates of the two groups given in Table 7.3.

Table 7.3: Pulse Rate of Two Groups of 10 Individuals

Pulse rate of experimental group group after intervention, X1

Pulse rate of control group without intervention, X2

1. 100 2. 86 3. 112 4. 80 5. 115 6. 83 7. 90 8. 94 9. 85 10. 105

1. 1052. 95 3. 120 4. 85 5. 110 6. 100 7. 115 8. 93 9. 107 10. 120

Mean value, 1X 95.0 Variance, S1

2 154.46 n1 10

2X 105.0 S2

2 138.67 n2 10


Here we want to test the NNull hypothesis (H0) that ø1 = ø2 when ø1 and ø2 are the respective mean of the two populations and the alternative hypothesis, (HA), is ø1 � ø2. According to the hypothesis, this should be a two-tail test. A t-statistic to test the above hypotheses is:

1 2

1 2

�

�X XtSz z

In the above formula, the numerator is the difference between two means and the denominator is the standard error of difference between means. The standard error of difference is estimated on the basis of the variances of the two samples. If we assume that the variances between two populations are equal, we can use the pooled variance estimate for the standard error of difference between means as follows.

� � 2 2

1 1 2 2

1 2 1 21 2

1 1 1 12

� � ��

� �

� ��

n s n sS

n n n nz z

where n1 and n2 = number of cases in group 1 and 2. Now we compute t. The computational formula for an independent group t is:

� � 2 21 1 2 2

1 2 1 2

1 2x x

1 1 1 12

�

� � ��

� �

� ��

tn s n s

n n n n

Using the above example, we can compute

t = � � �

95.0 – 105.09 154.6 – 9 138.7

1/10 + 1/1018

= � 1.85 df = 10 + 10 � 2 =18.

This t value has to be compared now with the table value of t for 18 degrees of freedom with significance level of P = 0.05 (95% acceptance rate or 5% rejection rate). From the t-distribution of table 7.3, you can see the table value of t for a 2-tail test with df = 18 for P = 0.05 is 2.10. This is a 2-tail test according to the alternative hypothesis. Since the table value (2.10) is greater than the absolute value of the calculated t statistic (� 1.85), we should retain the null hypothesis and reject the


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alternative hypothesis. We can conclude that there is no difference between the pulse rate of two groups (control and experimental). One tail t-test: Using the same example in the Table 7.3 we can also test a different hypothesis, such as the following: H0 : �1 > �2

HA: �1 < �2 Here the null hypothesis is that the mean of the control group could be equal or higher than the mean of the experimental group. The alternative hypothesis is directional. We want to see whether the mean pulse rate of the experimental group is higher than the control group. Hence, this is a one-tail test. If we want to test at a 5% level, we take the value from the table .05 (1-tail) at 18 degrees of freedom (see Table 7.4). Looking at the t-table below, we can see that the calculated absolute value (1.85) is higher than the tabular value (1.73) with same degrees of freedom and same probability. Therefore, the null hypothesis can be rejected and the alternative hypothesis that the mean of the experimental group is lower than the mean of the control group is accepted.


Table 7.4: The t-Distribution Table

The example above will illustrate the point that you must decide before the computation whether to make a one-tail or two-tail test. The significance of test depends on what is your hypothesis.

7.4.2 Analysis of Variance (ANOVA or ANAVA)

(a) WWhat is Analysis of Variance? It is essentially a process for partitioning a total sum of squares into components associated with recognised sources of variance. The advantage

What is the difference between a 2-tail test and 1-tail test?

SELF-CHECK 7.4


112

of using ANOVA is that it allows us to compare the records of a control group with a few treatments. With ANOVA we test a null hypothesis that all of the treatment means are equal against a null hypothesis that there is at least one mean not equal to the others.

(b) AAssumptions

(i) Treatment effects are additive.

(ii) Errors are normally distributed with zero mean and a common variance.

Assumption on data being normally distributed is not required. In actual practice, sometimes the two above assumptions do not hold.

NON-PARAMETRIC TESTS

Non-parametric tests are used for ordinal or nominal data. The variables measured do not follow a normal distribution usually due to a small sample size or convenience sampling.

7.5.1 Chi Square, �2

In nursing research, the most common non-parametric test used is chi square which is symbolised with an � squared. This test determines whether:

(a) There is a goodness of fit between the observed values in an experiment with the expected values; and

(b) Two variables are independent or related. We will deal with these two tests separately.

(a) TTest for Goodness of Fit This test can be illustrated with the following example. We expect the male nurses between 21 � 30 years and 31 � 40 years to have the same average mark (67) in the Management and Medico-legal course of the Open University Malaysia in the May Semester, 2007. Supposing we have collected the marks of 10 students randomly from the marked examination scripts of these two groups and found the average to be 65 and 70 for the above two groups. We want to test that these observed marks are not different from the expected marks, i.e. the null hypothesis is that the observed marks are same as the expected marks.

7.5


To compute �2, we need to do the following steps: Step 1: Develop a table as follows;

Age Groups Observed values (O) Expected Values (E) O-E (O-E)2 /E

21 � 30 65 67 2.5 .09

31 � 40 70 67 2.5 .09

Step 2: Compute chi square with the following formula: �2 = � ((O-E)2/E), and test whether this is significant at 95% level or (P ��0.05). From the table above, we can compute �2 = 0.09 + 0.09 = 0.18.

Step 3: Compare the calculated �2 value with tabular value from a chi-square table. Degrees of freedom in �2 is n � 1, where n is the number of categories of the single variable. In the above example n = 2, and therefore n � 1= 1. For P = 0.05, the tabular value at 1 df is 3.841 (see chi-square table 3.14). The calculated �2 is much lower than the tabular value. Step 4: Conclude that the observed marks of the two groups are the same as the expected marks. Therefore, we accept the null hypothesis.

(b) TTest of Independence

Often we need to find out whether two or more characteristics of a single population are related. Supposing we have two races in a population where there is a group vaccinated against a new flu vaccine and there is another group not vaccinated. We may like to determine whether the two races are independent of each other with regard to the incidence of flu during a year after the vaccination. Perhaps we take a simpler example to work out the various steps required for the computation of �2. A typical example will be to test if blood cholesterol levels are independent of gender. This involves randomly sampled data of 50 men and 50 women in a workplace. The results obtained from the analysis of blood cholesterol are entered below in a contingency table (see Table 7.5).


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Table 7.5: Cholesterol Level of 100 Individuals in a Normal Workplace

Respondents Above 200 mg/dL (N)

Below 200 mg/dL (N)

Total

Men 20 (17.5) 30 (32,5) 50

Women 15 (17.5) 35 (32,5) 50

Total 35 65 100

Values within the parenthesis ( ) are expected values. Calculation is shown in Step 1. Supposing we want to test a null hypothesis (Ho) that the genders are independent and an alternative hypothesis (HA) that they are dependent with regard to level of cholesterol.

Step 1: Observed values are given in the table. To get the expected values for each cell in the above table, we need to multiply the respective row table with the respective column table, and then divide the product by the total. Expected value of row 1 and column 1 = 50 � 35/100 = 17.5 Expected value of row 1 and column 2 = 50 � 65/100 = 32.5 Expected value of row 2 and column 1 = 50 � 35/100 = 17.5 Expected value of row 2 and column 2 = 50 � 65/100 = 32.5 Step 2: Now we calculate the �2 = � ((O-E)2/E).

2 2 2 22 (20 17.5) (30 32.5) (15 17.5) (35 32.5)

17.5 32.5 17.5 32.5� � � � �

� � �

= 0.357 + 0.192 + 0.357 + 0.192 = 1.098 Step 3: Compare the above �2 value with the critical value of �2 in the chi-square table for one degree of freedom. We consider 1 df because there are two classes here, Men and Women. If they are in classes, df will be n � 1. The table value is 3.84 and is more than the calculated value of 1.098. Hence, we accept the null hypothesis (Ho).


1. The cashier counter of a hospital is open from Monday to Friday, 9 am to 10 pm. The cashier wants to investigate if the payments by patients are the same for each of the five days. He randomly selected one week and counted the payments in RM made on each of the five days during the week. The information obtained is given in the following table.

Day Monday Tuesday Wednesday Thursday Friday

Payments Collected

(RM)

253,000 197,000 204,000 279,000 267,000

At the 1% level, can we reject the null hypothesis that the payments for each of the five days of the week are the same? Assume that this week is typical of all weeks in regard to the payments made by patients.

2. Suppose that there are 500 lung cancer patients in 2006 being attended by the oncology department of UMMC. Of these 500, 285 are men and 215 women. Can we test a null hypothesis that there is a gender difference in the incidence of lung cancer and an alternative hypothesis that there is no gender difference? Test at two probability levels, P= 0.05 and P= 0.01.

3. What is the difference of a test of goodness of fit and a test of independence?

4. What is a contingency table?

5. How does one find degrees of freedom for chi-square test?

6. Find the chi-square value from the following data on the age of the patients in three different hospitals and state your conclusion at � =10%.

Hospital 1 2 3 Total

Under 30 10 30 12 52

30 and over 18 22 37 77

Total 28 52 49 129

SELF-CHECK 7.5


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Table 7.6: Chi-Square Probabilities

The areas given across the top are the areas to the right of the critical value. To look up an area on the left, subtract it from one, and then look it up (i.e.: 0.05 on the left is 0.95 on the right) df 0.995 0.99 0.975 0.95 0.90 0.10 0.05 0.025 0.01 0.005 1 --- --- 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 10.597 3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.838 4 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860 5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750 6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548 7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278 8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955 9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589 10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188 11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757 12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300 13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819 14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319 15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801 16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.267 17 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718 18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156 19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582 20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997 21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401 22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796 23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181 24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559 25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928 26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290 27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645 28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993 29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336 30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672 40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766 50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490 60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952 70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215 80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321 90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299 100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169


QUALITATIVE ANALYSIS

You have learnt that qualitative research is different from quantitative analysis where numerical data are collected through experimentation. Qualitative data focuses on narrative data collected through interviews, note taking, tape recording etc. Basically there are three approaches for qualitative data analysis. (a) PPhenomenology

With this approach, the data is collected by researchers through observation, interviews, videotapes and descriptive notes written by the subjects. The researcher analyses the data and begins as soon as the first set of data is collected. Interviews are usually tape recorded and transcribed in the researchers own words to avoid data loss. Observations made by the researchers include verbal and non-verbal behaviour of the subject, the environment (e.g. poor, medium or good social circumstances) and researcherÊs own analysis of the situation.

Meanings of the data collected are usually expressed using a descriptive approach to identify the essence of behaviour of the subjects. However, there are many software tools available for qualitative data analysis. The following reference may be useful for research based on phenomenology. Qualitative Research: Analysis Types and Software Tools. Falmer Publishing, RR Tesch � London, UK, 1990.

(b) EEthnography

In this type of research, usually a field study is involved which includes extensive note taking, participant observations and interviews. Data analysis is essentially the analysis of the field notes and interviews to gain access to the health beliefs and practices of a culture. For example, a researcher might like to know the indigenous nursing practices for maternity care of different groups of people in Sarawak. The data collected is categorised and relationships are developed between categories and patterns of behaviour are identified. The research findings give the nurses a perspective of the subjects which becomes helpful for providing better nursing care. See the following website to know more about the methodology of ethnography research in nursing

http://herkules.oulu.fi/isbn9514264312/html/c459.html

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(c) CContent aanalysis The qualitative data collected is changed to numerical data for quantitative analysis. The method of analysis reduces the richness of the data and may result in loss of information. To understand more about content analysis of nursing data, read the following article:

Graneheim, U. H., & Lundman, B. (2004, Feb). Qualitative content analysis in nursing research: Concepts, procedures and measures to achieve trustworthiness. In Nurse Education Today, Volume 24, Issue 2, pp. 105 � 112.

DATA ANALYSIS USING EXCEL

Excel is a powerful tool for statistical analysis with both qualitative and quantitative data.

(a) In order to perform your analysis with Excel, open Microsoft Excel and enter your data in the Excel spreadsheet. Indicate the input range (only numerical) to obtain the total of each treatment. For example A$2$D$6. This means your data starts from Column A2 and ends in column D6. To obtain a frequency diagram, bar diagram, histogram and pie diagram, click the icon showing histogram.

(b) When starting with Excel, a screen of empty cells appear, and Sheet 1 � 3 are visible at the bottom of the screen. These sheets constitute a workbook. The screen has the standard tool bars and a formula bar that is used for creating various functions.

(c) Entering and saving the data in each cell is a procedure similar to a Microsoft Word spreadsheet.

(d) With the above points in mind, let us do some data analysis. First let us practice some histogram, a pie diagram showing the total output of growth at different treatments, and then some inferential statistics. While you can build in statistical function/s in the formula bar, it is advised for this course to use the Âadd inÊ function within Excel that offers more powerful data analysis facilities. In order to obtain the various statistics, Open Excel, CClick Tool, and then click Data aanalysis. You will see various Analysis Tools that can be performed using Excel on your computer. Is that not handy?

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(e) Below is the histogram of total output in growth at different treatments (see Figure 7.6).

CControl Treat.1 Treat.2 Treat.4 34 42 42 51 37 45 43 49 41 41 41 41 29 40 46 55 33 39 42 54

Total: 174 207 214 250

Figure 7.6: Histogram of total output in growth at different treatments

Using the total of each treatment, we can draw a pie diagram. Series1 � control; series 2 � treatment 1, series 3 � treatment 2, Series 4 � treatment 3 (see Figure 7.7).

Figure 7.7: Pie Chart


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� Inferential statistics allow a researcher to make inferences about the

characteristics of the population based on data collected in a sample.

� Sampling distributions are the basis for inferential statistics.

� The inferential analysis includes the hypothesis testing of relationships whether it can be accepted or rejected.

� The result of the hypothesis can be statistically significant or non-significant.

Regression analysis

Normal distribution

Correlation analysis

Inferential statistics

Parametric data

Analysis of variance

Non-parametic tests

1. What is the difference between sample statistics and population

parameters? 2. What is hypothesis testing? 3. What is null hypothesis? 4. Which of the following factors should be considered by a nursing

researcher as possibly influencing a non-significant result?

(i) Internal validity

(ii) Sample selection

(iii) Incorrect choice of hypothesis

(iv) Statistical tests


(v) Sample size

(vi) Research design

5. List the following statements as applying to descriptive or inferential statistics.

(i) Used to describe and summarise data.

(ii) Used in hypothesis testing.

(iii) Can demonstrate the variability of measures.

(iv) Based upon data obtained from a random sample.

6. Identify each of the following assumptions as applying to parametric tests or non-parametric tests.

(i) Applied to normal or ordinal data.

(ii) Not based on representation of a population.

(iii) Assume normal distribution of variable in population.

(iv) Less restrictive assumptions regarding shape of distribution.

(v) Estimation of at least on population characteristics.

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