STA1502 EXAM PACK & EXAM PREP 2022.

STA1502 EXAM PACK & EXAM PREP 2022. STA1502 - Statistical Inference I Inference about the difference between two means: Independent samples STUDY Keller Chapter 13 Inference about comparing two populations 13.1 Inference about the difference between two means: independent samples Make sure that you understand figure 13.1 of Keller: Note that we need subscripts to distinguish between the parameters of two different variables! We are now sampling from two independent populations where the means of the populations are our focus. Downloaded by Mandla Msibi () lOMoARcPSD| 2 The derivation of the test statistic is based on the three assumptions: 1. We have two independent populations from which we draw small random samples. 2. Both populations have normal distributions. 3. Both populations have the same variance, i.e. σ 2 1 = σ 2 2 = σ 2 . In statistical notation we summarise this as follows: If we have a random sample of sizen1 from a n(µ1 ; σ2 1 ) population and an independent random sample of size n2 from a n(µ2 ; σ2 2 ) population, σe 2 = Σ(x 1i − x 1) 2 + Σ(x 2i − x 2) 2 n1 + n 2 − 2 = s 2 pooled is the pooled estimate of the unknown common variance assuming that σ 2 1 = σ 2 2 . [ :-) I like to add the subscript "pooled" to remind me that it is a combined/composed variance and not the subscript consisting only of "p" as Keller does!] The test statistic is t (x 1−x 2 ) = (x 1 − x 2) − (µ 1 − µ 2 ) t s 2 pooled u 1 n1 + 1 n2 which has a t-distribution withυ = (n 1+n 2−2) degrees of freedom. [ :-) I like to add the subscript " (x 1 − x 2)" to t to remind me that it is a different t-statistic from what we used in chapter 12 of Keller ] The test statistic can be used directly to perform a hypothesis test or be manipulated to create a lower and an upper bound for the confidence interval. The null hypothesis H0 : (µ1 − µ 2 ) = D 0 may be tested at the α% level of significance against one of the following alternatives: (i) H1 : (µ1 − µ 2 ) = D 0 or (ii) H1 : (µ1 − µ 2 ) < D 0 or (iii) H1 : (µ1 − µ 2 ) > D 0 The symbol D0 implies a known, specified difference under H0 and is usually (mostly!) the value 0, indicating that we are testing H0 : µ1 = µ 2 . Downloaded by Mandla Msibi () lOMoARcPSD| 3 STA1502/1 To obtain a (1 − α)100% confidence interval estimate for the difference between the two populations means, (µ1 − µ 2 ), we compute (x 1 − x 2) ± t α 2 ;(n 1+n 2−2) u s 2 pooled( 1 n1 + 1 n2 ) t α 2 ;(n 1 +n 2−2) is obtained from Table 4 (see Appendix B, Keller) as P (−t α 2 ;(n 1+n 2−2) ≤ t ≤ t α 2 ;(n 1+n 2−2) ) = (1 − α) 0 " /2 tn1+n2 -2 t " /2;n1+n2 -2 -t " /2;n1+n2 -2 t " /2 1-" After you have studied section 13.1 of chapter 13 of the textbook you should try and work through activities 1.1 and 1.2 to enhance your understanding of a hypothesis test for the difference between two population means. Activity 1.1 Say whether the following statements are correct or incorrect and try to rectify the incorrect statements to make them true. (a) s 2 pooled = (n1 − 1)s 2 1 + (n 2 − 1)s 2 2 n1 + n 2 − 2 = Σ(x 1i − x 1) 2 + Σ(x 2i − x 2) 2 n1 + n 2 − 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (b) If we derive a confidence interval for(µ1 − µ 2 ) we use SE = v s 2 1 n1 + s 2 2 n2 but if we test H0 : µ1 = µ 2 we use SE = u s 2 pooled( 1 n1 + 1 n2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Downloaded by Mandla Msibi () lOMoARcPSD| 4 (c) In a one-tailed test for the difference between two population means, (µ1 − µ 2 ), if the null hypothesis is rejected when the alternative hypothesis, H1 : µ1 < µ 2 is false, a Type I error is committed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedback Feedback (a) Correct. With a little algebraic manipulation it follows from the definitions of s 2 1 = Σ(x 1i − x 1) 2 n1 − 1 and s 2 2 = Σ(x 2i − x 2) 2 n2 − 1 that (n1 − 1)s 2 1 = Σ(x 1i − x 1) 2 and that (n2 − 1)s 2 2 = Σ(x 2i − x 2) 2 . (b) Incorrect. We use SE = u s 2 pooled( 1 n1 + 1 n2 ) for both the hypothesis test and the confidence interval! (c) Correct. You will find that in most of the exercises on this section, whether they are for an assignment, the examination or exercises in Keller, the information you have to work with will either be • · raw data for two samples, or • summarised data given in a table format as Population 1 2 Sample size n1 n2 Sample mean x1 x2 Sample variance s 2 1 s 2 2 There could be "variations" on the theme of summarised data where computed sums are given instead of sample statistics, e.g. Σx 1i instead of x1or Σx 2 1i and Σx 1i instead of s 2 1 . In the case of raw data, you must try to have at least a Scientific Pocket Calculator with Statistical Functions that will enable you to compute the sample statistics: Downloaded by Mandla Msibi () lOMoARcPSD| 5 STA1502/1 Activity 1.2 Psychologists have claimed that the scores on a tolerance measurement scale have a normal distribution. Suppose that this scale is administered to two independent random samples of males and females and their tolerance towards other road users is measured. (The higher the score, the more tolerant you are.) The following scores were obtained: Males: Females: 14 12 (a) Test H0 : µmales = µ females against the alternative H1 : µmales = µ females . Use α = 0, 01 and assume that σ 2 1 = σ2 2