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INFERENTIALSTATISTICS

Inferential StatisticsStatisticsThis is testing the significance of the difference between two means, two standard deviations, two proportions, or two percentages.

Comparison between two or more variables often arises in research or experiments and to be able to make valid conclusions regarding the result of the study, one has to apply an appropriate test statistic.

This chapter deals with the discussion of the different test statistics that are commonly used in research studiesStatisticsStatisticsHypothesisA hypothesis is a conjecture or statement which aims to explain certain phenomena in the real world.

The Null HypothesisNull Hypothesis is denoted by Ho. It is the hypothesis that is subjected to testing to determine whether its truth can be accepted or rejected. This hypothesis states that there is no significant relationship or difference between two or more variables.StatisticsStatisticsThe Null HypothesisIn statistical research, the hypothesis should be written in NULL FORM.E.g.There is no significant difference between the effectiveness of method A and method B.StatisticsThe Alternative HypothesisAlternative Hypothesis is denoted by Ha. This is the hypothesis that challenges the NULL HYPOTHESIS. E.g.There is a significant difference between the effectiveness of method A and method B.StatisticsThe Alternative HypothesisThe sample is depending on the type whether the test is either one-tailed or two-tailed.Significance LevelTo test the NULL HYPOTHESIS of no significance in the difference between the two methods in the above example, one must set the level of significance first. This is the probability of having a type I error and is denoted by the symbol a. A type I error is the probability of accepting the alternative hypothesis, StatisticsStatisticsThe Significance LevelHa when, in fact the NULL HYPOTHESIS, Ho is true.The probability of accepting the null hypothesis when, in fact , it is false is called a type II error and is denoted by the symbol B. the most common level of significance is 5%.A test is called a one-tailed test if the rejection region lies on one extreme side of the distribution and two-tailed if the rejection region is located on both ends of the ends of the distribution.

FIG.8.1One-Tailed and Two-Tailed TestsStatisticsStatisticsOne-Tailed and Two-Tailed TestsIn fig.8a ( one-tailed), the rejection region is the area to the right of the vertical line under the bell curve. In fig 8b ( two-tailed), the rejection is the areas to the extreme left and right of the curve marked by the two vertical lines. A test is called a one-tailed test if the rejection region lies on one extreme side of the distribution and two-tailed if the rejection region is located on both ends of the ends of the distribution.

FIG.8.1StatisticsOne-Tailed and Two-Tailed TestsTesting HypothesisStatisticsBelow are the steps when testing the truth of hypothesis.Formulate the null hypothesis. Denote it as Ho and the alternative as Ha.Set the desired level of significance (a).Determined the appropriate test statistics to be used in testing the null hypothesis.Testing HypothesisStatistics4. Compute for the value of the statistic to be used.5. Compute for the degrees of freedom.6. Find the tubular value using the table of values for different tests from the appendix tables.

Testing HypothesisStatistics7. Compare the computed value, CV, to the tubular value, TV.

DECISION RULE: if the CV is less than the TV, Accept the null hypothesis. If the CV is greater than the TV, reject the null hypothesis. Make a conclusion using the result of the comparison.Degree Of Freedom (df)StatisticsThe degree of freedom gives the number of pieces of independent information available for computing variability. For any statistical tool used in testing hypothesis, the number of degrees of freedom required will vary depending on the size of the distribution. For a single group of population, the number of degrees of freedom is N-1, where N is the population.StatisticsFor two groups, the formula for df is: N1 + N2 2 fpr t-Test and N- 2 for pearson r. These test statistics will be discussed later in this chapter.Test Concerning MeansStatistics8.8.1 z-test on the comparison between the population means and the sample meanIf the population mean (u) and the population variance (o) are known, and U will be compared to a sample mean x, use the formula.

Z=(x-u) 0Test Concerning MeansStatistics8.8.1 z-test on the comparison between the population means and the sample meanThe tabular values of z can be obtained from the following table:

Table8.1Critical values of zLEVEL OF SIGNIFICANCETest type0.100.050.0250.01One-tailed test+_ 1.28+_ 1.645+_ 1.96+_ 2.33Two-tailed test+_ 1.645+_ 1.96+_ 2.33+_ 2.53Test Concerning MeansStatistics8.8.1 z-test on the comparison between the population means and the sample meanDecision rule: Reject Ho If 1z1 >_ 1z tabular 1.

Example: A company which makes a battery operated toy cars claims that its products have a mean lifespan of 5 yrs with a standard deviation of 2 yrs.StatisticsTest the null hypothesis that u= 5 yrs against the alternative hypothesis that u=/ 5 yrs if a random sample of 40 toy cars was tested and found to have mean life span of only 3 yrs . Use a 0.05 level of significance.SOLUTION:Ho: the means lifespan of the battery operated toy cars is 5 yrs ( u=5yrs) Ha : the mean lifespan of the battery operated toy cars is not 5 yrs. ( u=/ 5 yrs)2. a= 0.05 ; two- tailed3. Use z-test as test statistics.4. Computation: Statistics5. critical regions : z 1.966. decision: reject the Ho and accept the preposition that the mean lifespan of a toy is not equal to 5 yrs since 1z1 which is 6.32, is greater than 1z tabular 1 , which is 1.96.7. the difference is significant.StatisticsStatisticsCan be used to compare the means when the population mean (u) is known but the population variance (o) is unknown.

When the population standard deviation is unknown but the sample standard deviation can be computed the t-test can also used instead of the z- test the formulation is given below:T-test on the comparison between the population mean and the sample meanStatisticsThe denominator of the formula, s, divided by the n for t is called the standard error of the statistics. It is the standard deviation of the sampling distribution of a statistics for random samples n.

Decision rule : Reject:

T-test on the comparison between the population mean and the sample meanStatisticsExample 3: The average length of time for people to vote using the old procedure during a presidential. election period in precinct A is 55 minutes. Using computerization as a new election method, a random sample of 20 registrants was used and found to have mean length of voting time of 30 mins with a standard deviation of 1.5 mins.T-test on the comparison between the population mean and the sample meanStatisticsSolution:T-test on the comparison between the population mean and the sample meanStatisticsWhen two samples are drawn from naturally distributed populations with the assumption that their variances are equal, the t-test with the given formula should be used.

Example 4:T-test concerning means of independent samplesStatisticsWhen comparing two correlated means, the t-test is the appropriate test statistic. A typical example is when comparing the results of the pre-test and the post-test administered to a group of individuals. The two tests must be the same.T-test on the significance of the difference between two correlated means.StatisticsTo determined if there is a significant difference between proportions of two variables, the z-test will be used.Z-test on the significance of the difference between two independent proportionsStatistics8.11.1 ANALYSIS OF VARIANCE when the variances of two or more independent samples differ the appropriate test statistics to determined the significance of such difference is the analysis of variance (ANOVA) which makes use of the F ratio or variance ratio. The Various groups being compared are assumed to Significance of the difference between variancesStatisticsBelong to a population with a normal distribution, each group randomly selected and independent from the other groups. The variables from each group also have standard devaitions that are approximately equal.

Steps in solving the analysis of variance1 state the null hypothesis2 set the level of significance 3 accomplish the ANOVA table.StatisticsWhen the observations are paired, then they are correlated and their variances are not independent estimates. In this case, the t-test given by the formula below should be used.T-test for samples with correlated variances

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