Two Different Companies Have Applied to Provide Cable Television Service $H _ { 0 } : p = .5 \text { versus } H _ { \pm} : p \neq .5$

Essay

Two different companies have applied to provide cable television service in a certain region. Let p denote the proportion of all potential subscribers who favor the first company over the second. Consider testing $H _ { 0 } : p = .5 \text { versus } H _ { \pm} : p \neq .5$ based on a random sample of 25 individuals. Let X denote the number in the sample who favor the first company and x represent the observed value of X.
a. Which of the following rejection regions is most appropriate and why? $R _ { 1 } = \{ x : x \leq 7 \text { or } x \geq 18 \} , R _ { 2 } = \{ x : x \leq 8 \} \text {, }$
or $R _ { 3 } = \{ : x \geq 17 \}$
b. In the context of this problem situation, describe what type I and type II errors are.
c. What is the probability distribution of the test statistic X when $H _ { 0 }$
is true? Use it to compute
the probability of a type I error..
d. Compute the probability of a type II error for the selected region when p = .6 and p = .7.
e. Using the selected region, what would you conclude if 6 of the 25 queried favored company 1?