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Hypothesis Testing Flashcards

1
Q

Hypothesis

A

statement about the value of a population parameter developed for purpose of testing a theory or belief

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2
Q

Hypothesis Testing Procedure (picture pg 298 book 1)

A

j

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3
Q

Null Hypothesis

A

Designated Hο, hypothesis that the researcher wants to reject. Actually texted and is basis for selection of test stat. Always “equal to” condition

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4
Q

Alternative Hypothesis

A

Há, what is concluded if there is sufficient evidence to reject the null. Usually alternative hypo trying to assess.

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5
Q

One-tailed Two-tailed

A

one - if return is greater than zero two - if research question is returns different from zero Hο:µ=µο vs Há:µnot=µο

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6
Q

Reject Hο if:

A

Test statistic > upper critical value or Test statistic < lower critical value

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7
Q

Hypo testing - two statistics

A

Test statistic calculated from sample data Critical value of test statistic

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8
Q

Test Statistic =

A

(Sample Stat - Hypo Value) / Standard error of sample stat

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9
Q

Type I Error

A

rejection of null hypo is actually true

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10
Q

Type II Error

A

Failure to reject the null hypothesis when actually false

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11
Q

Significance Level

A

The probability of making a Type I Error and designated alpha letter

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12
Q

Decision Rule

A

Based on the distribution of the test statistic. Must figure out one tail or two tails before decision rule -specific and quantitative -test statistic (greater / less than) X, reject the null

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13
Q

Power of a test

A

probability of correctly rejecting the null when it is false. 1-probability of making a Type II error, or 1-P (Type II error) Rejecting null when false (power of test) equals one minus the probability of NOT rejecting the null when it is false (type II error)

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14
Q

Level significance, power of a test, and type two errors. (pg 303 book 1 picture)

A

…

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15
Q

Decreasing significance level

A

(probability of type II error) will increase probability of failing to reject the null, and decreasing power of test. and vice versa

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16
Q

Confidence Level (formula pg 304 book 1)

A

…

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17
Q

Confidence level interval

A

-critical value ≤ test statistic ≤ +critical value Range within which we fail to reject the null for a two-tailed test at a given level of significance

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18
Q

P-Value

A

Probability of obtaining a test statistic that would lead to a rejection of the null, assuming the null was true. -Smallest level of significance for which null can be rejected One tail- pval is probability that lies above the positive value of the computed test stat for upper tail tests, or below for lower tailed tests. Two tail -pval probability that lies above the positive val of the compound test stat plus the probaility that lies below the negative value of the computed test stat

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19
Q

T-Test Picture pg 307 book 1

A

employes a test stat that is distributed according to a t-distribution. Use if: Variance unknown sample is large (n≥30 Sample is small (less than 30) but distribution of population is normal

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20
Q

Z-Test picture pg 308 book 1

A

when population is normally distributed with a known variance Z val is compared to the significance of test.

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21
Q

Memorize this picture pg 308 book 1

A

…

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22
Q

When samples are independant

A

Use the difference in mean test

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23
Q

When samples are dependant

A

statistic is average difference in (paired) observations divided by the standard error of the average difference

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24
Q

Chi-square test

A

used for hypothesis tests concerning the variance of a normally distributed population. Distribution is asymmetrical and approaches the normal distribution in shape as the degrees of freedom increase

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25
Q

Hypothesis Testing :-Definition

A
  • A Hypothesis is a statement about the value of a population parameter developed for the purpose of testing a theory or belief
  • Hypothesis are stated in terms of the population parameter to be tested like the population mean
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26
Q

Null Hypothesis

A
  • It is the hypothesis that the researcher wants to reject
  • It is the hypothesis that is actually tested and is the basis for the selection of test statistics
  • The null is generally stated as a simple statement about a population parameter
  • The null hypothesis includes the equal the equal to condition
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27
Q

Alternative Hypothesis

A
  • The alternative hypothesis is what is concluded if there is sufficient evidence to reject the null hypothesis
  • Since you can never really prove anything with statistics when the null hypothesis is discredited the implication is that the alternate hypothesis is valid
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28
Q

Hypothesis Testing

Process

A
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29
Q

Hypothesis Testing Process

A

Some test Signify Divine and Collective Decision

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30
Q

Hypothesis Test

A
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31
Q

Hpothesis Test

A
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32
Q

Two Tailed Test

A

A two tailed test for population mean may be structured as

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33
Q

General Rule for 2 Tailed Hypothesis Test

A

Since the alternative hypothesis allows for values above and below the hypothesised parameter a two tailed test uses two critical values (or rejection point)

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34
Q

Structuring of General Rule for 2 Tailed Hypothesis Test

A
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35
Q

Example of 2 Tailed Test

A
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36
Q

One Tailed Test

A
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37
Q

Example One Tailed Test

A
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38
Q

Test Statistics

and

Hypothesis Test

A
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39
Q

Test Statistics

A

Hypothesis Testing involves two statistic

  1. ​The test statistics calculated from the data
  2. **The critical value of the test statistics **

The value of the computed test statistics relative to the critical value is the key step in assessing the validity of hypothesis

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40
Q

How is test statistics calculated

A
  • A test statistics is calculated by comparing the point estimate of the population parameter with the hypothesised value of the parmeter
  • The test statistics is the difference between the sample statistics and the hypothesised value scaled by the standard error of the sample statistics
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41
Q

Standard Error

A

The Standard Error of the sample mean is the adjusted standard deviation of the sample

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42
Q
A

When the sample statistic is the sample mean,x, the standard error of the sample statistic for sample size n is calculated as

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43
Q

When the population standard deviation sigma is known

A
44
Q

When the population standard deviation sigma is not known

A
45
Q

Distribution

A

A test statistics is a random variable that may follow one of several distributions depending on the characteristics of the sample and population
We will study 4 distribution for test statistics

  1. t-distribution
  2. z-distribution
  3. Chi-square distribution
  4. F-distribution
46
Q

Power of test

A
  • It is the probability of correctly rejecting the null hypothesis when it is false
  • It is one minus the probability of making Type II Error or 1- P (Type II Error)
47
Q

Power of Test Depends on

A
  • Sample Size and
  • Choice of significance level (Type I Error Probability ) will together determine the probability of a Type II Error
  • Decreasing the significance level (probability of Type I Error ) from 5% to 1% for example will increase the probability of failing to reject the null hypothesis (type II error ) and therefor reduce the power of test
  • Conversely for a given sample size we can increase the power of test only with the cost that the probability of rejecting a true null (type I error ) increases
  • For a given significance level we can decrease the probability of a type II error and increase the power of test only by increasing the sample size
48
Q

p-value
Significance Level

A
  1. The significance level is the probability of making a Type I Error (rejecting the null hypothesis when it’s true ) and is designated by Greek letter alpha
  2. For instance a significance level of 5% means there is 5% chance of rejecting a true null hypothesis
  3. p-value is the probability of obtaining a test statistics that would lead to a rejection of the null hypothesis assuming null hypothesis is true Type I Error
49
Q

Classification of Test

A
50
Q

Null Hypothesis Test is False

and Accepted

A

Type II Error

51
Q

Null Hypothesis is False

and

Rejected

A

No Error
Power of Test

52
Q

Null Hypothesis is True
Accepted

A

No Error

53
Q

Null Hypothesis is True

A

Rejected
Type I Error

54
Q

Null Hypothesis is True

Rejected

A
55
Q

Null Hypothesis is False

Accepted

A
56
Q

Null Hypothesis is True

A

Your Decision
Null Hypothesis is False

Type I Error Falsely Positive

57
Q

When Null Hypothesis is True

and

Rejected

A
  • Significance Level :- The significance level is the probability of making a Type I Error
  • Rejecting the null hypothesis when it’s true is designated by Greek letter alpha
  • For instance a significance level of 5% means there is 5% chance of rejecting a true null hypothesis
58
Q

Null Hypothesis is False

and

Accepted

A
  • Type II Error Falsely Negative
59
Q

Null Hypothesis is False
and

Accepted

A

Type II Error

60
Q

Standard Error Vs Sample Error

A
  • Sampling Error of the Sample Mean

= Sample Mean - Population Mean

  • Standard Error:-Standard error of the sample mean is the standard deviation of the distribution of the sample means
61
Q

Type I and Type II Error

A
62
Q

When hypothesis testing

the choice between using critical value based on the t-distribution or z-distribution depends on

A
  1. Sample Size
  2. the distribution of the population
  3. Whether or not the variance of population is known
63
Q

The t-Test

A
  • Brief :- The t-test is a widely used hypothesis test that employs a test statics that is distributed according to t-distribution.
  • In the real world the underlying variance of the population is rarely known so the test enjoys widespread application
  • Where to Use:- Use the t-test if the population variance is unknown and either of the following condition is met 1:-The sample size is large > 30 2:-The sample is small less than 30 but thedistribution of the population is normal or approximately normal
64
Q

The t-Test

A

To conduct the t-test the t-statistics is compated to critical t-value at desired level of significance with the appropiate degrees of freedom

Formula

65
Q

The z-Test

A
  • Brief :-The z-test is the appropriate hypothesis test of the population mean when the population is normally distributed with known variance
66
Q

The z-test

Formula

When the population variation is known

A
67
Q

The z-Test

Formula when population variance is not known and the sample size is large enough

A

Note that use of sample standard deviation versus the population standard deviation .

Remember that this is acceptable if sample size is large although the t-statistics is more conservative measure when population deviation is not known.

68
Q

The z-Test

Critical Values

A
69
Q

The t-Test

A
  • There are two distinct hypothesis tests 1) one about the significance of the difference between the means of two populations 2)and one about the significance of mean of the difference between pairs of observations Rules :-
  • The test of the difference of means is used when there are two independent samples
  • A test of the significance of the mean of the difference between paired observations is used when the samples are not independent
70
Q

z-test Numerical

A
71
Q

z-test Numrical Continued

A
72
Q
A
73
Q
A
74
Q
A
75
Q
A
76
Q
A
77
Q

Types of t - tests

A

till now we have studied single population wala t-test only

78
Q

2-Sample t-test

Independent Samples

A
  • There are 2 t-tests that are used to test differences between the means of two populations
  • Application of either of these tests requires that we are reasonably certain that our samples are 1) Independent 2)Taken from two normally distributed population 3) Both of these t-tests can be used when the population variance is unknown
79
Q

2-Sample t-test

Independent Samples

Test 1

Variance Assumed to be equal

A

Test 1
Independent Samples

  • The population variances are assumed to be equal and the sample observations are pooled
  • A pooled variance is used with the t-test for testing the hypothesis that the means of two normally distributed population are equal when the variances of the populations are unknown but assumed to be equal
80
Q

2-Sample t-test

Independent Samples

Test 1

Variance Assumed to be Equal

A

Test Statistics

Since we assume that the variances are equal we just add the variances of the two sample means in order to calculate the standard error in the denominator

81
Q

2-Sample t-test

Independent Samples

Test -2

Variance not assumed to be equal

A

Test 2
Independent Samples

  • No assumption is made regarding the equality between the two population variances and the t-test uses an approximated value for the degrees of freedom
  • The t-test for equality of population means when the population are normally distributed and have variances that are unknown and assumed to be unequal uses the sample variances of both populations
82
Q

2-Sample t-test

Independent Samples

Test -2

Variance not assumed to be equal

A

Test Statistics

83
Q

2 Sample t- test

Dependent Samples

A
  • If the observations in two samples both depend on some other factors we can construct a paired comparison test of the means of the differences between observations for two samples are different
  • Dependence may result from an event that affects conditions both sets of observations for a number of companies Or because for two firms ?
  • Overtime are both influenced by market returns or economic
84
Q

2 Sample t-test

Dependent Samples

A

Hypothesis Testing Equation

85
Q

2 - Sample t-Test

Dependent Samples

A

Test Statistics

86
Q

t-Test

Important Note

A
87
Q

Chi-Square Test

Definition

A
  • Similar to other hypothesis tests the chi square tests compares the test statistics to a critical chi-square values at a given level of significance and n-1 degrees of freedom :- N-1
88
Q

What parameter does t-test and z-test involve ?

A

Means

89
Q

What parameter does Chi-Square and F-test involve ?

A

Variance

90
Q

Chi - Square Test

Properties

A
  • The chi-squared test is used for hypothesis test concerning the variance of a normally distributed population :- V
  • Note that since the Chi-Square test is bounded below by zero chi-square values cannot be negative :- N
  • The chi square distribution is asymmetrical And approaches the normal distribution in shape as the degrees of freedom increases :- A
91
Q

Chi-Square Test

Null Hypothesis and Alternate Hypothesis

A
92
Q

Chi-Square Test

Test Statistics

A
93
Q

Chi-Square

Example

A
94
Q

Chi- Square Test

Example Continued

A
95
Q

Chi-Square Test

Test Statistics

A
96
Q

What is the difference between Chi Square and F Test ?

A

Chi-Square is used when we are dealing with the variance of the single normally distributed population where as F-test is used when we are dealing with the variance of 2 normally distributed population .

97
Q

F-Test

A

Testing the Equality of Variances of Two Normally Distributed Populations

  • Based on Two Independent Random Samples is tested with an F-Distributed Test Statistics

Assumption
Population from which samples are drawn are normally distributed

and

that the samples are independent

98
Q

F-Test

Null and Alternate Hypothesis

A
99
Q

F-Test

Test Statistics

A
100
Q

Chi-Square Test

A

F-Test

101
Q

Parametric Tests

A
  • Parametric Tests rely on the assumptions regarding the distribution of the population and are specific to population parameters
  • For Example z-test relies upon a mean and a standard deviation to define the normal distribution
  • The z-test also requires that either the sample is large relying on the central limit theorem to assure a normal sampling distribution or that the population is normally distributed
102
Q

Non Parametric Tests

A
  • Non Parametric Tests either do not consider a particular population parameter or have few assumption about the population that is sampled
  • Non parametric tests are used when there is concern about other than the parameters of distribution or the assumptions of parametric tests can’t be supported
  • They are also used when data are not suitable for parametric tests (ranked observations )
103
Q

Non-Parametric Test Cont…

A

Situations where a non parametric test is called for

  1. The assumptions about the distribution of the random variable that support a parametric test are not met . An example would be a hypothesis test of the mean value for a variable that comes from a distribution that is normal and is of small size so that neither the t-test nor the z-test are appropiate.
  2. When data are ranks (an ordinal measurement scale ) rather than value.
  3. The hypothesis does not involve the parameter of distribution such as testing whether a variable is normally distributed . We can use a non-parametric test called a run test to determine whether data are random . A run test provides an estimate of the probability that a series of changes are random
104
Q

Spearman Rank Correlation Test

A
  • It can be used when data are not normally distributed
  • Consider the performance ranks of 20 mutual funds for 2 years .
  • The ranks (1 through 20 ) are not normally distributed so a standard t-test is not appropriate
105
Q

Spearman Rank Correlation Test

A
  • A large positive value of the Spearman Rank Correlation such as .85 would indicate a high rank in one year if associated with a high rank in second year
  • Alternatively a large negative rank correlation would indicate that high rank in year one suggest a low rank in year two and vice versa

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