An investigator used t-test to compare two groups of students on verbal aptitude. He repeated his experiment 20 times and obtained significant difference 19 times. On the basis of this he decided to reject the null hypothesis. The probability of committing type I error was
0.01
0.02
0.05
0.10
Answer (Detailed Solution Below)
Option 3 : 0.05
Testing of Hypothesis MCQ Question 1 Detailed Solution
t-test : t- test is a statistical term which is used to define the significant difference between two groups, the differences are measured in terms of means of the groups.
Hypothesis testing
It uses sample data to make inferences about the population.
It gives tremendous benefits by working on random samples, as it is practically impossible to measure the entire population.
Hypothesis testing is a procedure that assesses two mutually exclusive theories about the properties of a population.
For Hypothesis testing, the two hypotheses are as follows:
Null Hypothesis
Alternative hypothesis
There are two errors defined, both are for null hypothesis condition
Type-I error corresponds to rejecting H0 (Null hypothesis) when H0 is actually true, and a Type-II error corresponds to accepting H0 (Null hypothesis)when H0 is false. Hence four possibilities may arise:
The null hypothesis is true but the test rejects it (Type-I error).
The null hypothesis is false but the test accepts it (Type-II error).
The null hypothesis is true and the test accepts it (correct decision).
The null hypothesis is false and test rejects it (correct decision)
Significant difference: Significant differences in statistics means measurable differences.
Given:
Error Type: Type I error
Experiment done: 20 times
Significant difference: 19 times
To find: Probability of type I error
Terms:
Formula:Probability = No of favourable outcome/Total no of outcome
Calculation:
Probability= No of favourable outcome/Total no of outcome
Since we want to find the probability of type I error, we will calculate the total number of cases (= 20) and number of cases in favour of type I error or which not significant (20-19=1)
Probability= No of favourable outcome(in favour of type I error) /Total no of outcome (cases)
= 1/20
= 0.05
Hence, The probability of committing a type I error was 0.05
Using equivalent samples, a researcher obtained a significant correlation 95 times out of 100 trials. He/She decided to reject the null hypothesis. The alpha level would be :
.01
.0.2
.05
.001
Answer (Detailed Solution Below)
Option 3 : .05
Testing of Hypothesis MCQ Question 2 Detailed Solution
t-test : t- test is a statistical term which is used to define the significant difference between two groups, the differences are measured in terms of means of the groups.
Hypothesis testing
It uses sample data to make inferences about the population.
It gives tremendous benefits by working on random samples, as it is practically impossible to measure the entire population.
Hypothesis testing is a procedure that assesses two mutually exclusive theories about the properties of a population.
For Hypothesis testing, the two hypotheses are as follows:
Null Hypothesis
Alternative hypothesis
There are two errors defined, both are for null hypothesis condition
Type-I error corresponds to rejecting H0 (Null hypothesis) when H0 is actually true, and a Type-II error corresponds to accepting H0 (Null hypothesis)when H0 is false. Hence four possibilities may arise:
The null hypothesis is true but the test rejects it (Type-I error). The probability of making a type I error is α, which is the level of significance you set for your hypothesis test.
The null hypothesis is false but the test accepts it (Type-II error). The probability of making a type II error is β, which depends on the power of the test.
The null hypothesis is true and the test accepts it (correct decision).
The null hypothesis is false and test rejects it (correct decision)
Significant difference: Significant differences in statistics means measurable differences.
Given:
Error Type: Type I error
Experiment done: 100 times
Significant difference: 95 times
To find: alpha level (α) i.e. probability of type I error
Terms:
Formula:Probability = No of favourable outcome/Total no of outcome
Calculation:
Probability= No of favourable outcome/Total no of outcome
Since we want to find the probability of type I error, we will calculate the total number of cases (= 20) and number of cases in favour of type I error or which not significant (100-95=5)
Probability= No of favourable outcome(in favour of type I error) /Total no of outcome (cases)
= 5/100
= 0.05
Hence, The probability of committing a type I error i.e. alpha level was 0.05
t-test : t- test is a statistical term which is used to define the significant difference between two groups, the differences are measured in terms of means of the groups.
Significant difference: Significant differences in statistics means measurable differences.
Hypothesis testing
It uses sample data to make inferences about the population.
It gives tremendous benefits by working on random samples, as it is practically impossible to measure the entire population.
Hypothesis testing is a procedure that assesses two mutually exclusive theories about the properties of a population.
For Hypothesis testing, the two hypotheses are as follows:
Null Hypothesis
Alternative hypothesis
In null hypothesis significance testing, the p-value is the probability that an observed difference could have occurred just by random chance when it is assumed that the null hypothesis is correct.
Important Points
The significance level for a given hypothesis test is a value for which a P-value less than or equal to is considered statistically significant. E.g. p-values are 0.1, 0.05 and 0.01.
If its given t-value to be significant at 0.1 means chances are 1 out of 100 that the difference between means has occurred due to sampling errors.
From above explanation, Chances are 5 out of 100 that the difference between means has occurred due to sampling errors.
Hence, A researcher used t-test to compare two means based on independent samples and found the t-value to be significant at 0.05 level. This means that Chances are 5 out of 100 that the difference between means has occurred due to sampling errors.
Ans: Option 1
Additional Information:
Two samples are independent if the samples from one population is not related or cannot be paired with sample from another group. Independent samples are randomly selected.
Errors in hypothesis testing: There are two errors defined, both are for null hypothesis condition
Type-I error corresponds to rejecting H_{0} (Null hypothesis) when H0 is actually true, and a
Type-II error corresponds to accepting H0 (Null hypothesis)when H0 is false. Hence four possibilities may arise:
The null hypothesis is true but the test rejects it (Type-I error).
The null hypothesis is false but the test accepts it (Type-II error).
The null hypothesis is true and the test accepts it (correct decision).
The null hypothesis is false and the test rejects it (correct decision)
Type-I error corresponds to rejecting H0 (Null hypothesis) when H0 is actually true, and a Type-II error corresponds to accepting H0 (Null hypothesis)when H0 is false. Hence four possibilities may arise.
The null hypothesis is true but the test rejects it (Type-I error).
The null hypothesis is false but the test accepts it (Type-II error).
The null hypothesis is true and the test accepts it (correct decision).
The null hypothesis is false and test rejects it (correct decision)
Important Points
1) Type-I Error:
In a hypothesis test, a Type-I error occurs when the null hypothesis is rejected when it is in fact true. That is, H0 is wrongly rejected. For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average than the current drug. That is, there is no difference between the two drugs on average.
A Type-I error would occur if we concluded that the two drugs produced different effects when in fact there was no difference between them. A Type-I error is often considered to be more serious, and therefore more important to avoid than a Type-II error.
The exact probability of a Type-I error is generally unknown. If we do not reject the null hypothesis, it may still be false (a Type-I error) as the sample may not be big enough to identify the falseness of the null hypothesis (especially if the truth is very close to the hypothesis).
Additional Information
2) Type-II Error
In a hypothesis test, a Type-II error occurs when the null hypothesis, H0, is not rejected when it is in fact false.
For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average than the current drug; that is Ho: there is no difference between the two drugs on average.
A Type-II error would occur if it was concluded that the two drugs produced the same effect, that is, there is no difference between the two drugs on average, when in fact they produced different effects.
A Type-II error is frequently due to sample sizes being too small.
The probability of a Type-II error is symbolized by â and written: P (Type-II error) = â (but is generally unknown).
A Type-II error can also be referred to as an error of the second kind.
Hence, An investigator commits type I error in testing hypothesis when he/she rejects null hypothesis when it is true.
Research is an organized, systematic, and scientific inquiry into a subject to discover facts, theories or to find answers to a problem. It involves several steps including identification of a problem, review of literature, formulation of hypothesis, research design, data collection, analysis, and interpretation, etc.
Hypothesis: The word hypothesis consists of two words, where ‘hypo’ means tentative or subject to verification and ‘thesis’ implies a statement about the solution of a problem. One of the primary functions of a hypothesis is to state a specific relation between two or more variables in such a manner that it is possible to empirically test them.
Characteristics of a Good Hypothesis:
The hypothesis should be empirically testable: A researcher should take utmost care that his/her hypothesis embodies concepts or variables that have clear empirical correspondence and not concepts or variables that are loaded with moral judgments or values.
The hypothesis should be simple: The researcher must ensure to state the hypothesis as far as possible in most simple terms so that the same is easily understandable by all concerned. It should not be complex in nature.
The hypothesis should be specific/objective: No vague terms should be used in the formulation of a hypothesis. It should specifically state the posited relationship between the variables.
The hypothesis should be conceptually clear: The concepts used in the hypothesis should be clearly defined, not only formally but also, if possible, operationally. The formal definition of the concepts will clarify what a particular concept stands for, while the operational definition will leave no ambiguity about what would constitute the empirical evidence.
The hypothesis should be related to a body of theory or some theoretical orientation: A hypothesis, if tested, helps to qualify, support, correct or refute an existing theory, only if it is related to some theory or has some theoretical orientation. Thus, the exercise of deriving hypothesis from a body of theory may also lead to a scientific leap into newer areas of knowledge.
Type-I error corresponds to rejecting H_{0} (Null hypothesis) when H_{0} is actually true, and a Type-II error corresponds to accepting H_{0} (Null hypothesis)when H_{0} is false. Hence four possibilities may arise.
The null hypothesis is true but the test rejects it (Type-I error).
The null hypothesis is false but the test accepts it (Type-II error).
The null hypothesis is true and the test accepts it (correct decision).
The null hypothesis is false and test rejects it (correct decision)
1) Type-I Error:
In a hypothesis test, a Type-I error occurs when the null hypothesis is rejected when it is in fact true. That is, H_{0} is wrongly rejected. For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average than the current drug. That is, there is no difference between the two drugs on average.
A Type-I error would occur if we concluded that the two drugs produced different effects when in fact there was no difference between them. A Type-I error is often considered to be more serious, and therefore more important to avoid than a Type-II error.
The exact probability of a Type-I error is generally unknown. If we do not reject the null hypothesis, it may still be false (a Type-I error) as the sample may not be big enough to identify the falseness of the null hypothesis (especially if the truth is very close to the hypothesis).
Important Points
2) Type-II Error
In a hypothesis test, a Type-II error occurs when the null hypothesis, H_{0}, is not rejected when it is in fact false.
For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average than the current drug; that is Ho: there is no difference between the two drugs on average.
A Type-II error would occur if it was concluded that the two drugs produced the same effect, that is, there is no difference between the two drugs on average, when in fact they produced different effects.
A Type-II error is frequently due to sample sizes being too small.
The probability of a Type-II error is symbolized by â and written: P (Type-II error) = â (but is generally unknown).
A Type-II error can also be referred to as an error of the second kind.
Hence, An investigator commits type II error when he/she accepts a null hypothesis when it is false.
Population: Population is our required pool of people or items or events showing similar characteristics or feature for our study.
Sample:
Sample is a part of our interested population. Sample is representation of population and used instead of population for convenience.
For eg. We want to find impact of lockdown on school students in Mumbai city, but if its not feasible to collect data from all students, so we will select students from few schools which are well representative.
Normally distributed data: Data which when plotted on graph forms a bell shape, it shows symmetric along the y axis at mean and the data near the mean are more repeated or frequent than the data which are far away from mean.
we generally make use of parametric and non-parametric tests for making inferences about various population values (parameters). Many methods and techniques are used in statistics. These have been grouped under parametric and non-parametric statistics.
Parametric
Non-Parametric Statistics
It involves data expressed in absolute numbers or values rather than ranks.
An example is the Student’s t-test.
The parametric statistical test operates under certain conditions.
Since the conditions are not ordinarily tested, they are assumed to hold valid.
Tests like t, z, and F are called parametrical statistical tests.
Multiple Regression, Two-way ANOVA.
In this, the statistics are based on the ranks of observations.
It does not depend on any distribution of the population.
It deals with small sample sizes.
It is assumed that the data of the population are not normally distributed, so These are not bound by any assumptions.
These are user friendly compared with parametric statistics and economical in time.
Tests like chi-square test, Kruskal-Wallis test,median test, Man-Whitney U test, sign test, and Wilcoxon matched-pairs signed-ranks test tests are examples of non-parametrical statistics tests.
Descriptive statistics:
Descriptive statistics describe the trend or a feature or a characteristic or phenomena of the data.
This can be normally distributed.
This involves central tendency(mean, median, mode), Spread chart(histogram, bar graphs, etc.), Range, variance, etc.
For e.g. Average height of grade 9th students in XYZ school.
Hence, When the nature of the population from which samples is drawn is not known to be normally distributed the data can be analyzed with the help of Non-parametric statistics.
Additional Information
Inferential Statistics:
Inferential Statistics deals with drawing conclusion about population from sample or about prediction about future events from past events.
For e.g. We can generalize about average height of 9th grade students in Mumbai City from a sample of same grade students.
An investigator used t-test to compare two groups of students on verbal aptitude. He repeated his experiment 20 times and obtained significant difference 19 times. On the basis of this he decided to reject the null hypothesis. The probability of committing type I error was
0.01
0.02
0.05
0.10
Answer (Detailed Solution Below)
Option 3 : 0.05
Testing of Hypothesis MCQ Question 9 Detailed Solution
t-test : t- test is a statistical term which is used to define the significant difference between two groups, the differences are measured in terms of means of the groups.
Hypothesis testing
It uses sample data to make inferences about the population.
It gives tremendous benefits by working on random samples, as it is practically impossible to measure the entire population.
Hypothesis testing is a procedure that assesses two mutually exclusive theories about the properties of a population.
For Hypothesis testing, the two hypotheses are as follows:
Null Hypothesis
Alternative hypothesis
There are two errors defined, both are for null hypothesis condition
Type-I error corresponds to rejecting H0 (Null hypothesis) when H0 is actually true, and a Type-II error corresponds to accepting H0 (Null hypothesis)when H0 is false. Hence four possibilities may arise:
The null hypothesis is true but the test rejects it (Type-I error).
The null hypothesis is false but the test accepts it (Type-II error).
The null hypothesis is true and the test accepts it (correct decision).
The null hypothesis is false and test rejects it (correct decision)
Significant difference: Significant differences in statistics means measurable differences.
Given:
Error Type: Type I error
Experiment done: 20 times
Significant difference: 19 times
To find: Probability of type I error
Terms:
Formula:Probability = No of favourable outcome/Total no of outcome
Calculation:
Probability= No of favourable outcome/Total no of outcome
Since we want to find the probability of type I error, we will calculate the total number of cases (= 20) and number of cases in favour of type I error or which not significant (20-19=1)
Probability= No of favourable outcome(in favour of type I error) /Total no of outcome (cases)
= 1/20
= 0.05
Hence, The probability of committing a type I error was 0.05
Using equivalent samples, a researcher obtained a significant correlation 95 times out of 100 trials. He/She decided to reject the null hypothesis. The alpha level would be :
.01
.0.2
.05
.001
Answer (Detailed Solution Below)
Option 3 : .05
Testing of Hypothesis MCQ Question 10 Detailed Solution
t-test : t- test is a statistical term which is used to define the significant difference between two groups, the differences are measured in terms of means of the groups.
Hypothesis testing
It uses sample data to make inferences about the population.
It gives tremendous benefits by working on random samples, as it is practically impossible to measure the entire population.
Hypothesis testing is a procedure that assesses two mutually exclusive theories about the properties of a population.
For Hypothesis testing, the two hypotheses are as follows:
Null Hypothesis
Alternative hypothesis
There are two errors defined, both are for null hypothesis condition
Type-I error corresponds to rejecting H0 (Null hypothesis) when H0 is actually true, and a Type-II error corresponds to accepting H0 (Null hypothesis)when H0 is false. Hence four possibilities may arise:
The null hypothesis is true but the test rejects it (Type-I error). The probability of making a type I error is α, which is the level of significance you set for your hypothesis test.
The null hypothesis is false but the test accepts it (Type-II error). The probability of making a type II error is β, which depends on the power of the test.
The null hypothesis is true and the test accepts it (correct decision).
The null hypothesis is false and test rejects it (correct decision)
Significant difference: Significant differences in statistics means measurable differences.
Given:
Error Type: Type I error
Experiment done: 100 times
Significant difference: 95 times
To find: alpha level (α) i.e. probability of type I error
Terms:
Formula:Probability = No of favourable outcome/Total no of outcome
Calculation:
Probability= No of favourable outcome/Total no of outcome
Since we want to find the probability of type I error, we will calculate the total number of cases (= 20) and number of cases in favour of type I error or which not significant (100-95=5)
Probability= No of favourable outcome(in favour of type I error) /Total no of outcome (cases)
= 5/100
= 0.05
Hence, The probability of committing a type I error i.e. alpha level was 0.05