I want to know the answer and detail step by step Two different companies have applied to provide cable television service in a region. Let p denotethe proportion of all potential subscribers who favor the first company over the second. ConsidertestingHo p= 0.5 vs. Ha : p = 0.5based on a random sample of 25 individuals. Let X denote the number in the sample who favorthe first company and a represent the observed value of X.1.Which of the following rejection regions is most appropriate and why?R₁ = {x ≤ 7 or x ≥ 18}R₂ = {x ≤ 8}R3 = {x ≥ 17}2.Describe the Type-l and Type-Il errors for this problem.3.What is the probability distribution of the test statistic X when Ho is true? Use it to compute theprobability of a Type I error based on the rejection region you chose in part 1.4.Using the selected region, what would you conclude if 6 of the 25 queried favored company 1?

There are two companies who provide cable television service in a region. Let '1' denotes the first company and '2' denotes the second company. 'p' denotes the proportion of all the potential subscribers who favor 1 over 2. The hypothesis to be tested is: H0:=0.5 vs. H1:p≠0.5 The sample size is 25. We define a random variable X which denotes the number of all the potential subscribers who favor 1 over 2.a) Let p^ denotes the sample proportion corresponding to the given distribution. Since we have a two tailed alternative, the test statistic is given by: z = p^-pp1-pn=p^-0.50.5×0.525=p^-0.50.1 1 Since the size of the rejection region is not given, we can easily assume α=0.05 The rejection region is given by: Reject H0 if z<-1.96 or z>1.96 Using (1), we have p^-0.50.1<-1.96 or p^-0.50.1>1.96⇒p^<0.304 or p^>0.696x25<0.304 or x25>0.696 x<7.6 or x>17.4 ⇒x≤7 or x≥18 As x will always be an integer Hence, the most appropriate rejection region would be R1=x≤7 or x≥18b) By definition, the Type 1 error means that the null hypothesis is rejected when actually it is true. So, in this problem the Type 1 error would be to believe that same proportion of subscribers do not favor company 1 over company 2 when in fact the truth is that same proportion of subscribers favor company 1 over company 2 as to company 2 over company 1. Type 2 error means not rejecting the null hypothesis when in fact it is false. So, the Type 2 error is that we believe that that same proportion of subscribers favor company 1 over company 2 as to company 2 over company 1 when in fact in reality the proportion is not same.c) When the null hypothesis is true, the proportion of all the potential subscribers who favor 1 over 2 is p = 0.5 Therefore, X denotes the number of all the potential subscribers who favor 1 over 2 and can take the values 0,1,2,....,25 with probability of success same on each trial. Also, the values of X are random and independent. Therefore, we can say that X is a binomial random variable with parameters n=25 and p=0.5 X ~ B25,0.5 The PMF of X is given by: PX=x=x250.5x1-0.525-x=x250.525 x=0,1,....,25 The rejection region is given by: R1=x≤7 or x≥18 Probability of Type 1 error: α=PReject H0|H0=Px≤7 or x≥18 | p=0.5=PX≤7+PX≥18=0.02164+0.02164=0.04328 where the individual properties are calculated in excel using the Binomial function =BINOM.DIST(7,25,0.5,TRUE) and =BINOM.DIST(25,25,0.5,TRUE)-BINOM.DIST(17,25,0.5,TRUE) respectively. The probability of Type 1 error = 0.04328d) The rejection region is: R1=x≤7 or x≥18 If 6 out of 25 queried favored 1, then the number 6 is less than 7, hence it falls in the rejection region. So, we reject the null hypothesis in this case and conclude that p≠0.5, i.e., unequal proportion of subscribers favor company 1 over 2.