STAT200 Final Exam Complete Solution A

**Make sure your answers are as complete as possible and show your work/argument. When there are calculations involved, you should show how you come up with your answers with critical work and/or necessary tables. You must show why you choose a certain answer for true-or-false and multiple choice questions. Answers that come straight from program software packages will not be accepted. **

1. (2 pts) True or False: In a right-tailed test, the test statistic is 1.5. If we know P(X < 1.5) = 0.96, then we reject the null hypothesis at 0.05 level of significance. (Justify for full credit)

2. (2 pts) True or False: If a 99% confidence interval contains 1, then the 95% confidence interval for the same parameter must contain 1. (Justify for full credit)

3. (2 pts) Which of the following could reduce the rate of Type I error? (Justify for full credit)

a. Making the significant level from 0.01 to 0.05

b. Making the significant level from 0.05 to 0.01

c. Increase the β level

d. Increase the power

4. (2 pts) Three hundred students took a chemistry test. You sampled 50 students to estimate the average score and the standard deviation. How many degrees of freedom were there in the estimation of the standard deviation? (Justify for full credit)

a. 50

b. 49

c. 300

d. 299

(For Questions 5&6) Mimi was the 5th seed in 2015 UMUC Tennis Open that took place in August. In this tournament, she won 75 of her 100 serving games.

5. (2 pts) Find a 90% confidence interval estimate of the proportion of serving games Mimi won. (Show work and round the answer to three decimal places)

6. (5 pts) According to UMUC Sports Network, Mimi wins 80% of the serving games in her 5-year tennis career. In order to determine if this tournament result is worse than her career record of 80%.We would like to perform the following hypothesis test:

(a) (2 pts) Find the test statistic. (Show work and round the answer to two decimal places)

(b) (2 pts) Determine the *P*-value for this test. (Show work and round the answer to three decimal places)

(c) (1 pt) Is there sufficient evidence to justify the rejection of at the level? Explain.

7. (5 points) The SAT scores are normally distributed. A simple random sample of 100 SAT scores has a sample mean of 1500 and a sample standard deviation of 300.

(a) (1 pt) What distribution will you use to determine the critical value for a confidence interval estimate of the mean SAT score? Why?

(b) (3 pts) Construct a 95% confidence interval estimate of the mean SAT score.(Show work and round the answer to two decimal places)

(c) (1 pt) Is a 99% confidence interval estimate of the mean SAT score wider than the 95% confidence interval estimate you got from part (b)? Why? [You don’t have to construct the 99% confidence interval]

8. (6 pts) Assume the population is normally distributed with a population standard deviation of 100. Given a sample size of 25, with a sample mean 770, we perform the following hypothesis test.** **

(a) (1 pt) Is this test for the population proportion, mean or standard deviation? What distribution should you apply for the critical value?

(b) (2 pts) What is the test statistic? (Show work and round the answer to three decimal places)

(c) (2 pts) What is the p-value? (Show work and round the answer to two decimal places)

(d) (1 pts) What is your conclusion of the test at the α = 0.10 level? Why? (Show work)

9. (7 points) Consider the hypothesis test given by

Assume the population is normally distributed. In a random sample of 25 subjects, the sample mean is found to be, and the sample standard deviation is

(a) (1 pt) Is this test for the population proportion, mean or standard deviation? What distribution should you apply for the critical value?

(b) (1 pt) Is the test a right-tailed, left-tailed or two-tailed test?

(c) (2 pts) Find the test statistic. (Show work and round the answer to two decimal places)

(d) (2 pts) Determine the *P*-value for this test. (Show work and round the answer to three decimal places)

(e) (1 pt) Is there sufficient evidence to justify the rejection of at the level? Explain.

10. (7 pts) A new prep class was designed to improve SAT math test scores. Five students were selected at random. Their scores on two practice exams were recorded; one before the class and one after. The data recorded in the table below. We want to test if the scores, on average, are higher after the class.

SAT Math Score

Student 1

Student 2

Student 3

Student 4

Student 5

Score before the class

620

700

650

640

620

Score after the class

640

700

670

670

630

(a) (1 pt) Which of the following is the appropriate test and best distribution to use for the test?

(i) Two independent means, normal distribution

(ii) Two independent means, Student’s t-distribution

(iii) Matched or paired samples, normal distribution

(iv) Matched or paired samples, Student’s t-distribution

(b) (1 pt) Let μd be the population mean for the differences in scores (scores after the class –before the class). Fill in the correct symbol (=, ≠, ≥, >, ≤, <) for the null and alternative hypotheses.

(a) H*0*: μd ________ 0

(b) H*a*: μd ________ 0

(c) (2 pts) What is the test statistic? (Show work and round the answer to three decimal places)

(d) (2 pts) What is the p-value? (Show work and round the answer to three decimal places)

(e) (1 pt) What is your conclusion of the test at the α = 0.05 level? Why? (Show work)

Statistics problems

1. (2 pts) True or False: In a right-tailed test, the test statistic is 1.5. If we know P(X < 1.5) = 0.96, then we reject the null hypothesis at 0.05 level of significance. (Justify for full credit)

2. (2 pts) True or False: If a 99% confidence interval contains 1, then the 95% confidence interval for the same parameter must contain 1. (Justify for full credit)

3. (2 pts) Which of the following could reduce the rate of Type I error? (Justify for full credit)

a. Making the significant level from 0.01 to 0.05

b. Making the significant level from 0.05 to 0.01

c. Increase the β level

d. Increase the power

4. (2 pts) Three hundred students took a chemistry test. You sampled 50 students to estimate the average score and the standard deviation. How many degrees of freedom were there in the estimation of the standard deviation? (Justify for full credit)

a. 50

b. 49

c. 300

d. 299

(For Questions 5&6) Mimi was the 5th seed in 2015 UMUC Tennis Open that took place in August. In this tournament, she won 75 of her 100 serving games.

5. (2 pts) Find a 90% confidence interval estimate of the proportion of serving games Mimi won. (Show work and round the answer to three decimal places)

6. (5 pts) According to UMUC Sports Network, Mimi wins 80% of the serving games in her 5-year tennis career. In order to determine if this tournament result is worse than her career record of 80%.We would like...

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