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Testing of a hypothesis based on a single sample

The 3 problems are from the topic "Test of Hypothesis Based on a Single Sample" of Probability and Statistic. I need assistance with these problems. Need to find out the correct solution and answers.

9. 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 versus H_a:p≠.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 ≤ 7 or x ≥ 18}, R_2= {x: x ≤ 8},
R_3 = {x: x ≥ 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 = .3, again when p = .4, and also for both p = .6 and p = .7.

e) Using the selected region, what would you conclude if 6 of the 25 queried favored company 1?

10. A mixture of pulverized fuel ash and Portland cement to be used for grouting should have a compressive strength of more than 1300 KN/m^2. The mixture will not be used unless experimental evidence indicates conclusively that the strength specification has been met. Suppose compressive strength for specimens of this mixture is normally distributed with o = 60. Let u denote the true average compression strength.

a) What are the appropriate null and alternative hypotheses?

b) Let xbar denote the sample average compressive strength for n = 20 randomly selected specimens. Consider the test procedure with test statistic xbar and rejection region xbar ≥ 1331.26. What is the probability distribution of the test statistic when H_0 is true? What is the probability of a type I error for the test procedure?

c) What is the probability distribution of the test statistic when u = 1350? Using the test procedure of part (b), what is the probability that the mixture will be judged unsatisfactory when in fact u = 1350 (a type II error)?

d) How would you change the test procedure of part (b) to obtain a test with a significance level of .05? What impact would this change have on the error probability of part (c)?

e) Consider the standardized test statistic Z = (Xbar - 1300)/(o/√n) = (Xbar) - 1300)/13.42. What are the values of Z corresponding to the rejection region of part (b)?

13. Let X_1,...,X_n denote a random sample from a normal population distribution with a known value of o.

a) For testing the hypothesis H_0: u = u_0 versus H_a: u > u_0 (where u_0 is a fixed number), show that the test with test statistic Xbar and rejection region of xbar ≥ u_0 + 2.33o/√n has significance level .01.

b) Suppose the procedure of part (a) is used to test H_0: u ≤ u_0 versus H_a: u > u_0. If u_0 = 100, n = 25, and o = 5, what is the probability of committing a type I error when u = 99? When u = 98? In general, what can be said about the probability of a type I error when the actual value of u is less than u_0? Verify your assertion.

Solution Preview

Hello!

Question 9
a. Look at the alternative hypothesis. It is that p is different from 0.5, or equivalently, that p is either higher or lower than 0.5. With this in mind, it's clear that R1 is the appropiate region. It can't be R2: this rejection region doesn't reject the hypothesis that p=0.5 when x is, for example, 25 (when ALL the people in the sample favor the 1st company!). For analogous reasons, R3 can't be correct either.

b. A type I error would be to reject the hypothesis that p=0.5, when the truth is that p=0.5. A type II error would be to fail to reject p=0.5 when the truth is that p is actually not equal to 0.5.

c. The test statistic X follows a binomial distribution, with parameters p=0.5 and n=25 (the sample size).

We would make a type I error if we rejected that p=0.5. Given that we chose R1, we'll reject that hypothesis if x<=7 or if x>=18. Therefore, we will make a type I error if x<=7 or if x>=18. Therefore, we must find:

Prob(Type I error) = Prob(X <= 7) + Prob(X >= 18)

Since X is binomial, it can be quite tedious to find those probabilities. We can use either a binomial calculator or a binomial probability table, which can be found in most Statistics books. I will use an online calculator, which you can find at

http://cnx.rice.edu/content/m11024/latest/
[scroll down to find it]

Plug N=25 and p=0.5 in this calculator. Then choose "less than or equal to" and write "7". This will give that Prob(X <= 7) = 0.0216. Now, since p=0.5, the probability that 7 people like the first company (x<=7) is the same as the probability that 7 people like the 2nd company (x>=18) (you can check this in the calculator by choosing "greater than or equal to" and writing "18"). Therefore, Prob(X >= 18) = 0.0216. We conclude that the prob of a type I error is 0.0432

d. Now we have to find the ...

Solution Summary

Expert provides in-depth demonstration of how to test a hypothesis using a single sample. Text explanation is provided along with appropriate calculations.

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