The State of the Current Affairs - Assignment Example

Question a) The null hypothesis is a ment which is taken to be the of the current affairs. It is an assumption which is thought to be true and which controls an experiment. On the other hand, if the current state of affairs is not true then the alternative option is called the alternative hypothesis. For example, if you take two samples say x1 and x2, the assumption that the sample means of the two populations (µ1 and µ2) are the same is the null while the alternative that they are different is the alternative hypothesis. The hull hypothesis is denoted by H0 while the alternative is denoted by H1. i.e. H0: µ1 = µ2 Vs. H1: µ1 ≠ µ2 b) Significance level is the criterion used in rejecting and accepting of the null hypothesis (Craft, J.L. 1990). Normally, 95% id used as the significance level in many statistical tests. In this case the significance level is said to be .05 (also referred to as “alpha”). According to Gore, S.M. & Altman, D.G. (1992), a test statistic is calculated and is then compared to the table value (value read from statistical tables). If the test statistic lies in the acceptance region, then the null is accepted and if in the critical region, the alternative is accepted and the null rejected. c) According to Diamond, I. & Jefferies, J. (2001), a sampling distribution is a probability distribution from which all the measurements of a certain populations are based; the mean, standard deviation and the variance. While calculating the test statistics, an assumption about the population from which the data is derived from is made. This assumption cannot be made in the absence of a distribution. For example, the normal distribution is assumed in many cases since this allows the data to be standardized so that some properties can be assumed to be as those of a normal distribution (Selvin, S. 1991). d) The p-value is also called the significance value (Newton, R.R., & Rudestam, K.E. 1999). The p-value forms the basis of whether the null hypothesis is to be rejected or accepted (Diamond, I. & Jefferies, J. 2001). The p- value is compared with the significance level (the alpha) to determine whether a rejection or an acceptance. If p>alpha, then the null hypothesis is accepted and if p µ2 or H1: µ1 < µ2. Note that H1 has both signs and not = or ≠ signs. This means that, the acceptance region is on one side of the sampling distribution and the rejection is on the other side as shown below. Acceptance region In a two tailed test, the acceptance region lies in between two rejection regions. In this case using the tow samples as above, H0: µ1 = µ2 Vs. H1: µ1 ≠ µ2. Acceptance region f) Type one error is the rejecting of the null hypothesis when it is actually true (Craft, J.L. 1990). For example, a person who is not guilty may end up in cell. Type two error amounts to the accepting of the null hypothesis when it is actually false (Craft, J.L. 1990). According to Gore, S.M& Altman, D.G. (1992), type I error is denoted by the Greek letter alpha (α) while the Type II error is denoted by the Greek letter beta (ß). g) To reject or to accept the null hypothesis is based on the following; If the p-value is greater than the α, then null hypothesis is accepted. If p-value is less than α, then the null hypothesis is rejected. If the test statistic is greater than the tabulated value, the null is rejected and the alternative accepted, If the test statistic lies in the acceptance region, the null is accepted and the alternative rejected. h) Acceptance of the null hypothesis is not the same as proving the null hypothesis. Accepting the null hypothesis is based on the test statistic and the p-value. On the other hand, proving the null hypothesis is based on carrying out an analysis of the available data and proving whether the assumed hypothesis is true or false. Thus, proving a hypothesis starts with an assumption first and then carrying out the analysis to see whether it is true or not. Accepting the null is based on the p-value. Question 2 Part one Let the mean hours for the single house holds to be X1 and the means hours for multiple house holds to be X2. Then, the hypothesis to be tested here are, a) H0: X1 = Xµ2 vs. H1: X1 ≠ X2. The null is that, the number of hours spent watching television by the single households is the same as the number of hours spent watching television by the multiple households. The alternative states that, the number of hours spent watching television by the single households is different from the number of hours spent watching television by the multiple households. b) A pooled t-test is the best to be used here. Using the below formulas, T = (X1-X2) Sp [(1/n1) + (1/n2)]1/2 Where, n1 = sample size for single households and n2=sample size for multiple households. Sp = (n1-1) S12 + (n2-1) S22 n1+n2-2 And s1 is standard deviation for single house hold and s2 is standard deviation for multiple households. This test is valid in that, with a sample more than 30, the distribution is assumed to be normal. c) The assumptions made here are; Both n1 and n2 are greater than 40 i.e. n1+n2>40, They are independent observations, They both have equal variances, The variances are unknown. d) The test is two tailed, H0: X1 = Xµ2 vs. H1: X1 ≠ X2 Part two Breaking down the data into 4 categories, Single households (1), Households with a couple and no children (2), Households with one parent and one or more children (3), Households with two parents and one or more children (4). This means that the above test cannot be still used in such a case. A test like least squares difference (LSD) is used in such a case. In this test, each category is compared with the other categories and if there is any difference in means then the two are said to be different (Freed, M. et al. 1991). Below is a table showing how the difference is used. Variables Difference Significance 1 2 3 4 -- -- -- -** - - 2 1 3 4 -- -- -- - -** - 3 1 2 4 -- -- -- -** -** - 4 1 2 3 -- -- -- - - - Here variable 1 is compared to variable 2, 3 and 4. Likewise, variable 2 is compared with variable 1, 3 and 4. 3 is compared with 1, 2 and 4 while 4 is compared with 1, 2 and 3. If the difference is significant, then ** are used to denote this and the two variable are said to be significant. Part three Correlation analysis is used to show whether two variables are related with each other (Weiss, N.A., & Hassett, M.J. 1991). It is used to show the strength of the relationship between variables. For example, Number of hours Size of TV Number of hours Pearson Correlation 1 .99** Sig. (2-tailed) - .000 N - - Size of TV Pearson Correlation .99** 1 Sig. (2-tailed) .000 -. N - - ** Correlation is significant at the 0.01 level (2-tailed). This shows that, the number of hours spent watching a TV and its size have a significant relationship. Part four a) To test whether multiple households are more likely to have satellite TVs than the single households, the test to be used is the t-test. This is so because, a t-test with greater than 30 observations is assumed to be normal. In this case the sample size is greater than 30. b) The assumptions made are; Observations are independent, The variance is unknown, The data is normal. c) Assuming that the mean houses with satellite TV in multiple household is µ1 and that of single household is µ2, then; the test here is one tailed with the following hypothesis; H0: µ1 = µ2 Vs. H1: µ1 > µ2 Reference Craft, J.L. (1990). Statistics and Data Analysis for Social Workers. 2nd Edition. Itasca, IL: F.E. Peacock Publishers Diamond, I. & Jefferies, J. (2001). Beginning Statistics. An Introduction for Social Scientists. CA: Sage Publications Freed, M. et al. (1991). Handbook of Statistical Procedures and Their Computer Applications to Education and the Behavioral Sciences. NY: American Council on Education, Macmillan Publishing Company Gardner, M.J. & Altman, D.G. (1989). Statistics with Confidence - Confidence Intervals and Statistical Guidelines. London: British Medical Journal Gore, S.M. & Altman, D.G. (1992). Statistics in Practice. London: British Medical Association Hinton, P.R. (1995). Statistics Explained. A Guide for Social Science Students. NY: Routledge Newton, R.R., & Rudestam, K.E. (1999). Your Statistical Consultant. Answers to Your Data Analysis Questions. CA: Sage Publications Selvin, S. (1991). Statistical Analysis of Epidemiologic Data. NY: Oxford University Press Belle, G. (2002). Statistical Rules of Thumb. MA: Wiley-Interscience Weiss, N.A., & Hassett, M.J. (1991). Introductory Statistics. NY: Addison-Wesley Publishing Co Read More