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solved assignment statistics 3447
part 2 Q1(b),2(a),3(a),4(a),5(a)
Q1(b): For a sample of 60 women taken from a population of over 5,000 enrolled in a weight-reducing program a nationwide chain of health spas, the sample mean
diastolic blood pressure is 101 and the sample standard deviation is 42. At a significance level of 0.02 on the average, did the women enrolled in the program
have diastolic blood pressure that exceeds the value of 75?
ANS: GIVEN:
n = 60 ; d = 42
x = 101 ; a = 0.02
SOLUTION:
Null & alternate hypothesis
Ho : µ = 75
H1 : µ > 75
Level of significance = a = 0.02
Test statistic
t = (x - µ)/ d/vn
t = (101- 75)/ 42/v60
t = 4.79
CRITICAL REGION:
t > ta(n-1)
t > t 0.02(60-1)
t > t 0.02(59)
t > 2.390 (from table)
CONCLUSION:
Since Ho is rejected ,therefore the womens in program have
blood pressure significally higher than given level which is 75.
Q.2 a) Explain with examples the difference between:
i) Simple Hypothesis and Composite Hypothesis
ii) Acceptance Region and Rejection Region
iii) Statistic and Test Statistic
iv) Type-I Error and Type-II Error
ANS: i) SIMPLE HYPOTHESIS:
The weight of evidence that is necessary to find the defendant guilty depends on the type of trial. In a criminal trial that is stated standard is that
the prosecution must prove that the defendant is guilty beyond any reasonable doubt.In civil trials, the burden of proof may be the intermediate level of clear and
convincing evidence or the lower level of the preponderance of evidence.
COMPOSITE HYPOTHESIS:
If a random sample is taken from a distribution with parameter O , a hypothesis is said to be a simple hypothesis. If that hypothesis uniquely specifies the
distribution of population from which the sample is taken.Any hypothesis that is not a simple hypothesis called a composite hypothesis.
Ho : O = Oo
Ha : O Oo
Ha : O > Oo
Ha : O < Oo
ii) Acceptance Region and Rejection Region:
All possible values which a test statistic may assume can be divided into two mutually exclusive groups : One group consisting of values which appear
to be consistent with the null hypothesis and the other having values which appear to be unlikely to occur if Ho is true.The first group is called acceptance
region and second set of values known as rejection region.
iii) Statistic and Test Statistic:
A test statistic is a standarized value that is calulated from sample data during a hypothesis test.You can use test statistic to determine whether to
reject the null hypothesis.The sampling distribution of the test statistic under the null hypothesis is called null distribution.And the value you found from
the table is called statistic.
iv) Type-I Error and Type-II Error:
Statisticans use specific definitions and symbols for the concept. Rejecting a null hypothesis when it is true is called a type 1 error and it,s
probability is symbollized as a(alpha) . Alternatively , accepting a null hypothesis . When it is false is called a type 2 error : and it,s probability is symbollized
as ß(beta).
Q. 3 a) Define and explain the terms regression and correlation. Describe the properties of least square regression line.
ANS: REGRESSION:
Regression analysis involves identifying the relationship between a dependent variable and one or more independent variables. A model of the relationship is
hypothesized, and estimates of the parameter values are used to develop an estimated regression equation. Various tests are then employed to determine if the model
is satisfactory. If the model is deemed satisfactory, the estimated regression equation can be used to predict the value of the dependent variable given values for
the independent variables.
Regression model.
In simple linear regression, the model used to describe the relationship between a single dependent variable y and a single independent variable x is
y = a0 + a1x + k. a0 and a1 are referred to as the model parameters, and is a probabilistic error term that accounts for the variability in y that cannot
be explained by the linear relationship with x.
Correlation:
Correlation and regression analysis are related in the sense that both deal with relationships among variables. The correlation coefficient is a measure of linear
association between two variables. Values of the correlation coefficient are always between -1 and +1. A correlation coefficient of +1 indicates that two variables
are perfectly related in a positive linear sense, a correlation coefficient of -1 indicates that two variables are perfectly related in a negative linear sense.
Neither regression nor correlation analyses can be interpreted as establishing cause-and-effect relationships. They can indicate only how or to what
extent variables are associated with each other.
Properties of Least-squares regression line:
The most common method of choosing the line that best summarises the linear relationship (or linear trend) between the two variables in a linear regression analysis,
from the bivariate data collected.
Of the many lines that could usefully summarise the linear relationship, the least-squares regression line is the one line with the smallest sum of the squares of
the residuals.
Some properties of the least-squares regression line are:
1. The sum of the residuals is zero.
2. The point with x-coordinate equal to the mean of the x-coordinates.
3. The point with y-coordinate equal to the mean of the y-coordinates of the observations is always on the least-squares regression line.
Q.4 a) What is meant by a random variable? Differentiate between the discrete random variable and continuous random variable.
ANS:
Random Variables
A random variable, is a variable whose possible values are numerical outcomes of a random phenomenon.
There are two types of random variables,
1- Discrete random variable.
2- Continuous random variable.
1-Discrete Random Variables
A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,........ Discrete random variables are
usually (not necessarily) counts. If a random variable can take only a finite number of distinct values, then it must be discrete. Examples are number of children in
a family, the Friday night attendance at a cinema e.t.c.
2- Continuous Random Variables
A continuous random variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height,
weight, the amount of sugar in an orange, the time required to run a mile.
A continuous random variable is not defined at specific values. Instead, it is defined over an interval of values, and is represented by the area under a curve
(in advanced mathematics, this is known as an integral). The probability of observing any single value is equal to 0, since the number of values which may be assumed
by the random variable is infinite.
Q.5 a) What is meant by analysis of variance? Explain the general procedure for it.
ANS:
ANALYSIS OF VARIANCE:
Analysis of variance (often abbreviated as ANOVA) that will enable us to test for the significance of the difference among more tham two sample means. Using
analysis of variance , we will be able to make inferences about whether our samples are from the population having same mean.
Analysis of variance is bassed on a comparison of two different estimates of the variance, of our overall population.
In this case we can calculate one of these estimate by examining the variance among the three samples which are 17, 21 and 19.The other estimate of the population
variance is determined by the variatiom with in the three samples themselves , that is (15, 18, 19, 22, 11) , (22, 27, 18, 21, 17) and (18, 24, 19, 16, 22, 15).
Then we compare these two estimates of the population variance.Because both estimates of thry should be approximately equal in value when the null
hypothesis is true. If the null hypothesis is not true these two estimates will differ considerably.
There are three steps in analysis of variance which are:
1- Determine the estimate of the population variance from the variance among the sample means.
2- Determine the second estimate of population variance from the variance within the sample.
3-Compare these two estimates if they are approx equal in value, accept the null hypothesis.
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solved assignment statistics 3447
part 2 Q1(b),2(a),3(a),4(a),5(a)
Q1(b): For a sample of 60 women taken from a population of over 5,000 enrolled in a weight-reducing program a nationwide chain of health spas, the sample mean
diastolic blood pressure is 101 and the sample standard deviation is 42. At a significance level of 0.02 on the average, did the women enrolled in the program
have diastolic blood pressure that exceeds the value of 75?
ANS: GIVEN:
n = 60 ; d = 42
x = 101 ; a = 0.02
SOLUTION:
Null & alternate hypothesis
Ho : µ = 75
H1 : µ > 75
Level of significance = a = 0.02
Test statistic
t = (x - µ)/ d/vn
t = (101- 75)/ 42/v60
t = 4.79
CRITICAL REGION:
t > ta(n-1)
t > t 0.02(60-1)
t > t 0.02(59)
t > 2.390 (from table)
CONCLUSION:
Since Ho is rejected ,therefore the womens in program have
blood pressure significally higher than given level which is 75.
Q.2 a) Explain with examples the difference between:
i) Simple Hypothesis and Composite Hypothesis
ii) Acceptance Region and Rejection Region
iii) Statistic and Test Statistic
iv) Type-I Error and Type-II Error
ANS: i) SIMPLE HYPOTHESIS:
The weight of evidence that is necessary to find the defendant guilty depends on the type of trial. In a criminal trial that is stated standard is that
the prosecution must prove that the defendant is guilty beyond any reasonable doubt.In civil trials, the burden of proof may be the intermediate level of clear and
convincing evidence or the lower level of the preponderance of evidence.
COMPOSITE HYPOTHESIS:
If a random sample is taken from a distribution with parameter O , a hypothesis is said to be a simple hypothesis. If that hypothesis uniquely specifies the
distribution of population from which the sample is taken.Any hypothesis that is not a simple hypothesis called a composite hypothesis.
Ho : O = Oo
Ha : O Oo
Ha : O > Oo
Ha : O < Oo
ii) Acceptance Region and Rejection Region:
All possible values which a test statistic may assume can be divided into two mutually exclusive groups : One group consisting of values which appear
to be consistent with the null hypothesis and the other having values which appear to be unlikely to occur if Ho is true.The first group is called acceptance
region and second set of values known as rejection region.
iii) Statistic and Test Statistic:
A test statistic is a standarized value that is calulated from sample data during a hypothesis test.You can use test statistic to determine whether to
reject the null hypothesis.The sampling distribution of the test statistic under the null hypothesis is called null distribution.And the value you found from
the table is called statistic.
iv) Type-I Error and Type-II Error:
Statisticans use specific definitions and symbols for the concept. Rejecting a null hypothesis when it is true is called a type 1 error and it,s
probability is symbollized as a(alpha) . Alternatively , accepting a null hypothesis . When it is false is called a type 2 error : and it,s probability is symbollized
as ß(beta).
Q. 3 a) Define and explain the terms regression and correlation. Describe the properties of least square regression line.
ANS: REGRESSION:
Regression analysis involves identifying the relationship between a dependent variable and one or more independent variables. A model of the relationship is
hypothesized, and estimates of the parameter values are used to develop an estimated regression equation. Various tests are then employed to determine if the model
is satisfactory. If the model is deemed satisfactory, the estimated regression equation can be used to predict the value of the dependent variable given values for
the independent variables.
Regression model.
In simple linear regression, the model used to describe the relationship between a single dependent variable y and a single independent variable x is
y = a0 + a1x + k. a0 and a1 are referred to as the model parameters, and is a probabilistic error term that accounts for the variability in y that cannot
be explained by the linear relationship with x.
Correlation:
Correlation and regression analysis are related in the sense that both deal with relationships among variables. The correlation coefficient is a measure of linear
association between two variables. Values of the correlation coefficient are always between -1 and +1. A correlation coefficient of +1 indicates that two variables
are perfectly related in a positive linear sense, a correlation coefficient of -1 indicates that two variables are perfectly related in a negative linear sense.
Neither regression nor correlation analyses can be interpreted as establishing cause-and-effect relationships. They can indicate only how or to what
extent variables are associated with each other.
Properties of Least-squares regression line:
The most common method of choosing the line that best summarises the linear relationship (or linear trend) between the two variables in a linear regression analysis,
from the bivariate data collected.
Of the many lines that could usefully summarise the linear relationship, the least-squares regression line is the one line with the smallest sum of the squares of
the residuals.
Some properties of the least-squares regression line are:
1. The sum of the residuals is zero.
2. The point with x-coordinate equal to the mean of the x-coordinates.
3. The point with y-coordinate equal to the mean of the y-coordinates of the observations is always on the least-squares regression line.
Q.4 a) What is meant by a random variable? Differentiate between the discrete random variable and continuous random variable.
ANS:
Random Variables
A random variable, is a variable whose possible values are numerical outcomes of a random phenomenon.
There are two types of random variables,
1- Discrete random variable.
2- Continuous random variable.
1-Discrete Random Variables
A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,........ Discrete random variables are
usually (not necessarily) counts. If a random variable can take only a finite number of distinct values, then it must be discrete. Examples are number of children in
a family, the Friday night attendance at a cinema e.t.c.
2- Continuous Random Variables
A continuous random variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height,
weight, the amount of sugar in an orange, the time required to run a mile.
A continuous random variable is not defined at specific values. Instead, it is defined over an interval of values, and is represented by the area under a curve
(in advanced mathematics, this is known as an integral). The probability of observing any single value is equal to 0, since the number of values which may be assumed
by the random variable is infinite.
Q.5 a) What is meant by analysis of variance? Explain the general procedure for it.
ANS:
ANALYSIS OF VARIANCE:
Analysis of variance (often abbreviated as ANOVA) that will enable us to test for the significance of the difference among more tham two sample means. Using
analysis of variance , we will be able to make inferences about whether our samples are from the population having same mean.
Analysis of variance is bassed on a comparison of two different estimates of the variance, of our overall population.
In this case we can calculate one of these estimate by examining the variance among the three samples which are 17, 21 and 19.The other estimate of the population
variance is determined by the variatiom with in the three samples themselves , that is (15, 18, 19, 22, 11) , (22, 27, 18, 21, 17) and (18, 24, 19, 16, 22, 15).
Then we compare these two estimates of the population variance.Because both estimates of thry should be approximately equal in value when the null
hypothesis is true. If the null hypothesis is not true these two estimates will differ considerably.
There are three steps in analysis of variance which are:
1- Determine the estimate of the population variance from the variance among the sample means.
2- Determine the second estimate of population variance from the variance within the sample.
3-Compare these two estimates if they are approx equal in value, accept the null hypothesis.
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