# MGEB12H3 Lecture Notes - Lecture 2: Type I And Type Ii Errors, Null Hypothesis, Statistical Hypothesis Testing

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MGEB12: Quantitative Methods in Economics-II

Problem Set-2

Chpater-9: 36, 38, 40, 46, 48abc, 50, 54, 58

Supplemental:

1. Which of the following statements are true? Where alpha is the probability of a Type I error and beta is the

probability of a Type II error:

A. Alpha+Power=1

B. Power+Beta=1

C. Alpha+Beta=1

D. All of the above are true

E. None of the above is true

2. A survey was conducted at a movie theater to determine the preference for different kinds of popcorn. The

results of the survey showed that Brand A was preferred by 65 percent of the people with a margin of error of

plus or minus 3 percent. What is meant by the statement “plus or minus 3 percent”?

A. Three percent of the population that was surveyed will change their minds.

B. Three percent of the time the results of such a survey are not accurate.

C. Three percent of the population was surveyed.

D. The true proportion of the population who preferred Brand A popcorn could be determined if 3 percent more

of the population was surveyed.

E. It would be unlikely to get the observed sample proportion of 65 percent unless the actual percentage of

people in the studied population who prefer Brand A is between 62 percent and 68 percent.

3. For Ho: = 15 and H1: 15, testing at the 10% level of significance, the probability of rejecting the null

hypothesis when it is true is:

A. 0.1 and the probability of committing a Type I error is 0.1.

B. 5% and the probability of committing a Type II error is 95%.

C. 0.1 and the probability of committing a Type I error is 0.05.

D. 0.05 and the probability of committing a Type II error is 0.1.

E. 0.05 and the probability of committing a Type I error is 0.1.

4. The alternative hypothesis is that a process is producing more than the maximum allowable proportion of

defective items. A Type I error is to conclude that the process is

A. not producing too many defects when it might not

B. not producing too many defects when it is not.

C. producing too many defects when it actually is.

D. not producing too many defects when it actually is.

E. producing too many defects when it actually isn't.

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5. (Not for Test-1, For Test-2) A sample of size 150 from population 1 has 40 successes. A sample of size 250

from population 2 has 30 successes. The value of the test statistic for testing the null hypothesis that the

proportion of successes in population one exceeds the proportion of successes in population two by 0.05 is:

A. 1.645

B. 2.327

C. 1.960

D. 1.977

E. 2.510

6. If the lower and upper confidence limits of the population proportion p, using a sample of size 1500, are 0.184

and 0.238, respectively, then the lower and upper confidence limits of the total number of successes in the

population, given that the population size is 750,000, are respectively:

A. 276 and 357

B. 137,724 and 178,143

C. 138,000 and 178,500

D. 138,276 and 179,857

E. 148,361 and 189,671

Questions 7-10: A survey claims that more than 8 out of 10 doctors recommend aspirin for their patients with

headaches. To test this claim, a random sample of 100 doctors’ results in 86 who indicate that they recommend

aspirin.

7. The null hypothesis and the alternative hypothesis for this test is:

A. H0: P = 0.80 H1: P # 0.80

B. H0: P = 0.80 H1: P < 0.80

C. H0: P = 0.80 H1: P > 0.80

D. H0: P < 0.80 H1: P > 0.80

E. H0: P = 0.80 H1: P ≥ 0.80

8. The p-value of the test statistic in this problem is approximately equal to:

A. 0.0521

B. 0.0802

C. 0.1231

D. 0.0721

E. 0.0668

9. For a 5% significance level, the power of the test for the alternative of 0.85 is approximately:

A. 0.67

B. 0.53

C. 0.44

D. 0.33

E. 0.47

10. If confidence interval was used to estimate the population proportion. What size sample would be necessary

if we wanted the margin of error to be 0.04 using 95% confidence level?

A. 141

B. 289

C. 302

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D. 331

E. 384

Questions 11-12: A normal population has a population standard deviation of 12. The hypotheses about the

population mean, H0: µ = 40 versus H1: µ > 40 are to be tested. A random sample of 100 observations will be

selected and the sample mean is denoted by

x

.

11. Suppose the p-value is 0.015, the value of

x

must be closest to

A. 40.9

B. 41.5

C. 42.6

D. 43.2

E. 44.2

12. If the true population mean is 43 and the hypotheses are tested at the 5% significance level, the probability

of a Type II error is closest to

A. 0.305

B. 0.273

C. 0.229

D. 0.195

E. 0.187

13. The 90% confidence interval for the proportion of busy signals encountered by customers is [0.05, 0.13].

What is the margin of error for this confidence interval?

A. This calculation requires a standard error.

B. 0.01

C. 0.08

D. 0.04

E. 0.02

14. A sample of 39 statistics students contained 14 females. What size sample would be necessary if we wanted

to estimate the true proportion of female statistics students to within ±0.08 with 95% confidence?

A. 2606

B. 564

C. 1300

D. 139

E. 269

15. A student claims the ability to correctly identify whether a person is a business major or an economics major

by the way the person dresses. Suppose she is presented with one person and considers this to be a hypothesis

test with the null hypothesis being that the person is a business major and the alternative that the person is an

economics major. With reference to this test, what would be a Type II error?

A. Identifying the person as an economics major when the person is an economics major.

B. Identifying the person as an economics major when the person is a business major.

C. Identifying the person as a business major when the person is a business major.

D. Identifying the person as a business major when the person is an economics major.

E. None of the above