Basic Concepts of Hypothesis Testing Introduction
A hypothesis is a statement or a tentative theory that may or may not be true, but is initially assumed to be true until new evidence suggests otherwise. It may be proposed from a preliminary observation, a guess or based from previous experiences. In hypothesis testing problem, the researcher has in mind a specific notion concerning the characteristics of the population under study before the sample data are gathered. Then investigate the sample information to examine how consistent the data with the hypothesis in questioned. questioned. If the sample information deviate much from the stated hypothesis, then researcher tend to disbelieved and reject the the proposed statement. Although the proposed statement may may be true, it is expected that any single sample (or samples) will differ slightly from the true characteristic of the population and other will not, because of the sampling variation, have the same exact value as the population parameter. Hence, differences between the sample information and the population under study might be due chance. The procedure of statistical test will provide the basis in deciding whether differences between the sample observation and the hypothesized value could be due to sampling variation alone, or are so large enough as to make the proposed statement untenable. Types of Hypothesis
Null hypothesis – the null hypothesis is denoted by Ho, it is the hypothesis of “no difference” and usually formulated for the purpose of being rejected. Alternative Alternative hypothesis hypothesis – the alternativ alternative e hypothesis hypothesis is
denoted denoted by Ha or H1. This is the hypothesis hypothesis that that contradicts contradicts the hull hull hypothesis. hypothesis.
If the null hypothesi hypothesiss is rejected, rejected, the
alternative is being supported. The alternative hypothesis is the operational operational statement of the experimenter’s research hypothesis. Types of Errors •
Type I error – usually committed if the Ho is rejected when the Ho is true
•
Type II error – usually committed if the Ho is accepted when the Ho is false
In actual situation a given table below summarizes the kind of action and the type of error when one accepts or rejects a hypothesis. Status of Hypothesis
Ho is True
Accept Ho
Correct Decision
Ho is False
Reject Ho
Type I error
Type II error
Correct Decision
One-sided and Two-sided Test Directional/Nondirectional Directional/Nondirectional Test
In one-sided test the Ha specifies that the unknown test. While the
population parameter is entirely above or entirely below the specified
two- sided test, the Ha specifies specifies that the unknown unknown population population
value of the Ho. It is called one-tailed or a directional
parameter parameter can lie on either side of the value specified specified by Ho. It is
called called
two-tailed two-tailed or
nondirectional test. The following are classified as one-tailed test Ho : µ = 100
Ha : µ < 100
Ho : µ = 100
Ha : µ > 100
The Ha is either entirely below 100 or entirely above 100 The following are classified as two-tailed test Ho : µ = 100 Ho : µ1 = µ2
Ha : µ ‡ 100 Ha : (µ1 - µ2) ‡ 0
The Ha can fall in either side of the Ho Note :
For a directional test the inequality inequality symbol of the Ha is less than (<) or greater than (>) while while a non-directional test the symbol symbol is not equal ( ‡ ).
Testing Level of Significance Significance
The probability of committing type I error error is called the level of significance or margin of error of of the test and it is denoted by α (alpha), and one minus the level of significance is called confidence level. The probability of committing type II error is denoted by ß, and (1– ß) is called the power of the test. This will indicate the ability of the test statistics to determine correctly that the Ho is false, hence it should be rejected. Rather compute the actual chance of committing type I error, the researcher conventionally establish the level of significance before hand by considering the consequences of committing type I error. There are 2 most commonly used level of significance significance 0.05 and 0.01. At 0.05 level, the researcher is willing willing to accept a 5% chance of being wrong decision when Ho is rejected. At 0.01 level, the researcher is willing to accept a 1% chance of being wrong when Ho is rejected. If Ho is rejected at 0.05 level, then it is usually labeled as "significant”, otherwise the result is labeled “not significant”. If Ho is rejected at 0.01 level, then the result is labeled “Highly significant”. For a fixed sample size n, decreasing one type of error would mean increasing the other type of error. The only way to decrease both type of errors simultaneously is by increasing the sample size. The Critical Region
The level of significance determines which values would be considered improbable or probable if the hypothesis were true. Thus, the range of possible values (sampling distribution) is divided into two sections or regions, the acceptance region (the probable values) and the rejection region (improbable values). The size of both region is completely specified by the level of significance. The acceptance region is equal to (1 (1 – α) and the critical region or the rejection region is equal to α. The experimenter will decide to reject the null hypothesis only if the probability of observing of such an observed value is equal to or less than α. The size of the critical region is being determined by α, in general the location of the critical region region is determined by the nature of the alternative hypothesis. The difference in the location of the critical region differentiates the statistical hypothesis into one-tailed or two-tailed test. Critical region is the set of all values of the test statistics that would would cause to reject the null hypothesis. Basic Steps in Hypothesis Testing
The following steps should be properly observed so as to make sure that the thinking is logical. 1. State the Ho and Ha, decide what data to collect and under what conditions 2. Specify the level of significance µ and the sample size size n 3. Find the sampling distribution of the test statistics under under the assumption that Ho is true 4. Establish the critical region for the test statistics statistics 5. Computation of the test statistics, for a sample size n 6. Decision. Z-test Statistics
Suppose that X1, X2, . . . Xn are sample observations from a normal populations with unknown mean and known variance. The The appropriate test statistics in comparing the sample mean and the population is
which is normally distributed with mean 0 and variance 1. t-test Statistics
The population variance is generally unknown and is estimated by the variance of the random samples. The sampling variability of the sample variance may be affected if the sample size is small (n less than or equal to 30). Hence, the unavailability of the population variance must take into consideration and to be estimated by the sample variance. If the given observations X1, X2, . . ., Xn is a random sample from a normal distribution, but the population variance is unknown, then the test statistics
has a t-distribution with (n-1) degrees degrees of freedom. A t-distribution is symmetric probability distribution centered at zero, and looks similar but more variable (spread (spread out) than the normal distribution. The t-distribution becomes more and more similar with the normal distribution as the number of degree of freedom increases. Sample Problems
1. A study shows that the the average score of the applicants who took the entrance examination was 45 with a standard deviation of 5.15. Is there a reason to believe that that the present examinees is better than the previous results results if a random sample of 36 applicants showed an average score of 47.34, use 0.01 level of significance. significance. ANSWER to Sample Problem 1
1.
Ho :
2.
α
3.
µ = 45
= 0.01,
Ha :
µ > 45
n = 36
Test Statistics :
Z-test
4.
Critical Region : Reject the Ho if Zc > 2.33
5.
Computations: (47.36 – 45) V36 Zc = ———————— = 2.75 5.15
6.
Decision: Since Zc > 2.33, therefore reject the Ho hypothesis and conclude that the new batch of applicants is better than the previous one.
2. A high school school principal claims claims that the average performance performance of his graduating graduating class in math was 83. To test this claim, 25 students were randomly randomly selected from from the recent recent graduating class with the following results:
87, 85, 76, 83, 78, 90, 89, 85, 82, 77, 79, 76, 86, 83, 93, 88, 84, 76, 79, 85, 82, 81, 85, 84, 80. Test the principal's claim
using 0.05 level of significance. ANSWER to Sample Problem 2
1. Ho : 2.
α
µ = 83
= 0.05,
Ha :
µ ≠ 83
n = 25
3. Test Statistics :
t-test
4. Critical Region : Reject the Ho if tc > 2.064 5. Computations:
X = 82.92
S = 4.6
n = 25
__ (82.92– 83) V25 tc = ———————— = 2.75 4.6 6.
Decision: Since tc < 2.064, therefore accept the Ho hypothesis and conclude that the claim of the principal is is valid and the performance of new graduating graduating class in Math is not
significantly different from the previous graduates. Comparing Two Dependent Samples
In comparing two dependent samples, the data are considered as a paired values. This is a result of data being obtained from a certain “before and after” studies, a result from a pairs of observation from two different populations, or a result from matching two subjects of similar characteristics to form a matched pairs. This pairs of observations are compared directly to one another by using their observed differences. The purpose of using correlated or dependent samples is to eliminate or remove the effect of uncontrolled factors which are not part but might influence the outcome of the study. Matched pairs or pairing of observations will assure the researcher that the observed differences of the two samples was really due to the influences of the factors under study. Test Statistics
The difference between two populations when dependent or correlated samples was used is almost the same with the t-test with singles samples in the previous module. The only difference is that in dependent samples we will be dealing with the difference of two ob served values rather than the original values.
Sample Problem/s
1. The following following data represent the weights (in lbs.) of 10 obese participants before and after undergoing a two-weeks weight reducing training program. program. Participants
1
2
3
4
5
6
7
8
9
10
Before
275
25 0
235
21 2
2 15
263
289
2 58
215
249
After
250
21 5
218
19 8
2 00
245
26 0
2 50
202
230
Test the hypothesis that the mean weight loss loss of obese participants after the training program is 10 lbs. against the alternative alternative that the loss
weight is greater than than 10 lbs. Use 0.01
level of significance.
2. A researcher wishes to determine if there is systematic difference between the readings of the two digital weighing scales. The following data were obtained: Sample No.
1
2
3
4
5
6
7
8
Scale A
50.0
82.5
53.8
85.4
75.4
63.5
35.8
25.3
Scale B
49.9
82.7
53.8
85.3
75.4
63.7
35.7
24.9
Use 0.05 level of significance to test whether there is no significant difference between the readings of the two scales. (the weights are express in grams).
Comparing Two Independent Samples
Sometimes it is impossible to design a study or an experiment
which utilizes a matched or related related samples in comparing comparing two populations. Because it is difficult difficult to look for the
subjects or units with more or less the same characteristics before the intervention or application of treatments. treatments. The researcher may use the two independent samples by randomly selecting the samples from from the two populations or by randomly assigning the subjects or units to the different groups or treatments. treatments. The samples size in an independent population need not necessarily be of the same size.
The parametric tests for for analyzing data from two independent samples are the z-test and t-test. These test statistics assumes that the data in both samples are normally distributed and are usually in an interval scale of measurement.
Z-test Statistics
Suppose that the observations X1, X2, ..., Xn1 and Y1, Y2, . . . , Yn2 are random samples from two independent populations. To test the hypothesis that the two population means are equal. Then the appropriate test statistics is the Z-test. The formula of the test statistics is given by
which is normally distributed with mean 0 and variance 1.
However, if the population variances are unknown but the sample sizes are assumed to be large. The unknown population variances can be estimated by their corresponding sample variances. Hence the Z-test statistics is given by
t-test Statistics Equal Variance
Hypothesis testing about the differences of means between two independent populations will involved the t-distribution and the t-test statistics. When the two samples are independent, and the two independent populations are normally distributed and the population variances are unknown but assumed to be equal. The test statistics
has a t-distribution with (n-2) degrees of freedom.
Sample Problems
1. A sample study was made by the Office Office of Student Affairs of the weekly allowances of the student of a certain university. If 60 students in the main campus averaged 250.00 pesos with a standard standard deviation deviation of 15.75, while 40 off-campus off-campus students averaged averaged 263.65 with a standard deviation deviation of 17.30, test at 0.05 level of significance significance whether whether the difference difference between these two sample means is significance. Answer
1. Ho : μ1 = μ2 2.
α
= 0.05,
Ha : μ1 ≠ μ2 n1 = 60
3. Test Statistics :
n2 = 40
Z-test
4. Critical Region : Reject the Ho if IZcI > 1.96 5. Computations: x1 = 250.00
S1 = 15.75
x2 = 263.65
S2 = 17.32
Zc = - 4.00
6. Decision: Since IZcI >1.96, therefore reject the Ho hypothesis and conclude conclude that the two groups of of students have different Allowances 2. A classroom teacher wishes to compare the performance of students in statistics using two methods of teaching. Two independent samples of sizes 15 and 10 were randomly selected. The following following data have been obtained: Method A
85
83
76
78
82
81
86
75
77
83
Method B
85
78
83
90
89
87
84
83
79
82
76
Is there a significant difference between the performance of students in the two methods of teaching statistics? Use 0.05 level of significance.
ANSWER
1. Ho : μ1 = μ2 2.
α
= 0.05,
Ha : μ1 ≠ μ2 n1 = 10
3. Test Statistics :
n2 = 15
t-test (independent samples)
4. Critical Region : Reject the Ho if I tcI > 2.069 5. Computations:
x1 = 80.60
S1 = 3.86
x2 = 83.60
S2 = 4.29
tc = -1.753
84
89
78
87
6. Decision: Since I tcI < 2.069, therefore accept the Ho hypothesis and conclude that the the two methods have the same performance. Comparing More Than Two Samples Introduction
In the preceding modules the statistical procedures being considered are the comparison of two independent or dependent normal populations. This module will considers the procedure extended in comparing more than two independent samples. The statistical procedure is called the Analysis of Variance known as ANOVA that uses a general methodology and gives a test statistics for testing a hypothesis of equality of two or more means from an independent groups. The ANOVA is the general extension of t-test for independent samples. Assumptions of ANOVA
The following are the assumptions of the Analysis of Variance
1. The population from which the samples were drawn are normally distributed. 2. The samples are independent from each other. 3. The variances of the different samples are homogeneous Data Layout Samples
Group 1
Group 2
...
Group p
1
X1
Y1
.
Z1
2
X2
Y2
.
Z2
3
.
.
.
.
.
.
.
.
Xn1
Yn2
.
Znp
T1
T2
.
Tp
TOTALS
Outline ANOVA Table
Sources of Variation De D egree of Freedom
Sum of Square
Mean Square
Fc
Ftab 0.05
Ftab 0.01
Between Groups
p-1
SSB
MSB
Fc
____
____
Within Groups
N-p
SSW
MSW
.
.
.
Total
N-1
SST
.
.
.
.
N - Total Number of Samples
MSB - Between Groups Mean Square
p - Number of Groups to be compared
MSW - Within Groups Mean Square
SST - Total Sum of Squares
Fc - F computed
SSB - Between Groups Sum of Squares
SSW - Within Groups Sum of Squares
Formula for Computations Computations
Hypothesis Testing Procedure
1. Ho : There is no significant difference between means Ha : At least one mean is significantly different 2. Specify the level of significance and the sample sizes 3. Test Statistics : F-test (One-way ANOVA) 4. Critical Region : Reject Ho if :
Fc > Ftab at 0.05 level of significance
5. Computations: Construct an ANOVA ANOVA Table 6. Decision :
Sample Problems
1. The following data represent the scores of a random sample students in each section during the first long examination: Section
Samples
A
87
45
75
65
82
B
78
56
66
49
56
45
C
53
76
59
73
43
32
Is there a significant difference between the performance of of students in the
three sections? Use 0.01 level of significance.
62
2.
An animal science experiment was conducted to determine the effective feed formulation on the growth performance of pigs over a specified period. There were four feed
formulation being considered and the following data were obtained: Feed Formulation
Samples (kgs)
A
112.5
100.3
105.6
99.3
78.9
B
89.2
82.6
100.3
86.8
79.9
C
100.8
99.8
103.9
98.7
89.5
D
75.8
82.1
86.5
89.8
79.6
Is there a significant difference in the average growth growth these pigs for the different feed formulation? Use 0.05 level of significance.
82.7
88.2
96.4
