Eco 72 - Economic Statistics

Final Exam

INSTRUCTIONS: This exam consists of thirteen short-answer questions, worth ten points each. You have two hours to complete the exam. Write all answers on this sheet.

1. A random sample of 400 police departments from 2002 finds that the average department purchased 52 bulletproof dog vests, with a standard deviation of 24 dollars per department.

a. (5 points) Write down the formula for the standard error of the number of bulletproof dog vests purchased by the average police department in 2002, and calculate the result.

b. (5 points) What is a 95 percent confidence interval for the number of bulletproof dog vests purchased by the average police department in 2002? (Show the formula for this.)

2. In a survey of 231 randomly chosen C.W. Post students, 55 percent of the students own pets, while a survey of 225 randomly chosen students at Old Westbury finds that 57 percent of them own pets. Test the hypothesis that the same fraction of C.W. Post students owns pets as of Old Westbury students.

a. (3 points) Write down the sample statistics and population parameters as a set of variables, and state the null hypothesis in terms of these variables.

b. (3 points) Write down the formula for the test statistic in terms of the above variables.

c. (2 points) Calculate the test statistic.

d. (2 points) State the critical value for this statistic at the one percent level. At that level, can we reject the hypothesis that the same fraction of C.W. Post students owns pets as of Old Westbury students?

3. We take a random sample of 371 police precincts in Tapparstan, and find that there were reports of police brutality at 112 of them. Test the hypothesis that there were reports of police brutality at 35 percent of police precincts in Tapparstan last year.

a. (3 points) Write down the sample statistics and population parameters as a set of variables, and state the null hypothesis in terms of these variables.

b. (3 points) Write down the formula for the test statistic in terms of the above variables.

c. (2 points) Calculate the test statistic.

d. (2 points) State the critical value for this statistic at the ten percent level. At that level, can we reject the hypothesis that there were reports of police brutality at 35 percent of police precincts in Tapparstan last year?

4. A group 5 transfer students between Sangamon State and Sandusky State is observed at each college. We have determined the number of songs they have downloaded from the internet per semester while at each school. The results are as follows:

Student
1 2 3 4 5
Sangamon 20 36 36 30 35
Sandusky 50 46 39 52 45

a. (2 points) What are the mean and standard deviation of the number of songs a given student downloads per semester while at Sangamon State and the number of songs the same student downloads per semester while at Sandusky State for this sample, using the population variance formula?

b. (2 points) What is the covariance of the number of songs a given student downloads per semester while at Sangamon State and the number of songs the same student downloads per semester while at Sandusky State for this sample? (Show your work)

c. (2 points) What is the correlation between the number of songs a given student downloads per semester while at Sangamon State and the number of songs the same student downloads per semester while at Sandusky State for this sample?

d. (2 points) What is the value of the test statistic for the hypothesis that the number of songs a given student downloads per semester while at Sangamon State and the number of songs the same student downloads per semester while at Sandusky State are uncorrelated? (Show the formula).

e. (2 points) What critical value for your test statistic should you look at if you are using the ten percent significance level? Should you reject the null hypothesis at that level?

5. We randomly select 111 Gadhanese donkeys and measure their heights. The average Gadhanese donkey in our group has a height of 29 inches, with a standard deviation of 17 inches per donkey. We then randomly select 53 Bailanese oxen and measure their heights. The average Bailanese ox in our group has a height of 35.5 inches, with a standard deviation of 11 inches per ox. Test the hypothesis that Gadhanese donkeys and Bailanese oxen have the same average height.

a. (3 points) Write down the sample statistics and population parameters as a set of variables, and state the null hypothesis in terms of these variables.

b. (3 points) Write down the formula for the test statistic in terms of the above variables.

c. (2 points) Calculate the test statistic.

d. (2 points) State the critical value for this statistic at the one percent level. At that level, can we reject the hypothesis that Gadhanese donkeys and Bailanese oxen have the same average height?

6. We survey 4 employees of Wutzit Corp who take the company's entrance exam twice -- once when they join the company, and once ten years later. The results are as follows:

Employee
1 2 3 4
First Score 35 30 37 38
Second Score 38 25 47 36

Test the hypothesis that employees' entrance exam scores are on average the same when taken at the start of their job and when taken with ten years' seniority.

a. (2 points) What variable do you use to test the null hypothesis? What are the values of that variable in this sample?

b. (2 points) Write down the formula for the test statistic you will use to test your null hypothesis.

c. (2 points) What is the sample mean of the variable in (a)? What is the standard deviation?

d. (2 points) Calculate the test statistic in (b).

e. (2 points) What critical value for your test statistic should you look at if you are using the one percent significance level? Should you reject the null hypothesis at that level?

7. State, in order, the five steps of conducting a hypothesis test.

8. In a random sample of 202 dreams, the 71 of the dreams weigh more than an ounce.

a. (5 points) Write down the formula for the standard error of the fraction of dreams that weighs more than an ounce, and calculate the result.

b. (5 points) What is a 95 percent confidence interval for the fraction of dreams that weighs more than an ounce? (Show the formula for this.)

9. The average caller at ExquisiteCorp technical support spends 54 minutes on hold, with a standard deviation of 6 minutes for any given caller. Suppose we take a sample of 121 callers.

a. (5 points) What is the standard error of the sample mean?

b. What is the probability that the average caller in the sample spends at least {xbar} minutes on hold?

10. Data from a random sample of 24 C.W. Post students tell us that the average C.W. Post student in the sample exaggerates the amount of money he or she spends on clothes by 55 dollars per month, with a standard deviation of 6 dollars for any given student.

a. (5 points) Write down the formula for the standard error of the number of dollars by which the mean student exaggerates his or her clothing expenditures, and calculate the result.

b. (5 points) What is a 95 percent confidence interval for the number of dollars by which the mean student exaggerates his or her clothing expenditures? (Show the formula for this.)

11. State the three requirements data must satisfy for us to use the central limit theorem.

12. In a sample of 361 Americans, the average person owns 56 gizmats, with a standard deviation of 24 gizmats per person. Test the hypothesis that the average American owns 57.84 gizmats.

a. (3 points) Write down the sample statistics and population parameters as a set of variables, and state the null hypothesis in terms of these variables.

b. (3 points) Write down the formula for the test statistic in terms of the above variables.

c. (2 points) Calculate the test statistic.

d. (2 points) State the critical value for this statistic at the one percent level. At that level, can we reject the hypothesis that the average American owns 57.84 gizmats?

13. We survey 18 randomly selected children in Skudsdale, and find that the average one spent 28 dollars on candy last week, with a standard deviation of 3 dollars for any given child. Test the hypothesis that the child in Skudsdale spends 25.99 dollars on candy per week.

a. (3 points) Write down the sample statistics and population parameters as a set of variables, and state the null hypothesis in terms of these variables.

b. (3 points) Write down the formula for the test statistic in terms of the above variables.

c. (2 points) Calculate the test statistic.

d. (2 points) State the critical value for this statistic at the one percent level. At that level, can we reject the hypothesis that the child in Skudsdale spends 25.99 dollars on candy per week?

Probability Table for the Normal Distribution
.
z	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
.
-3	0.0013	0.0013	0.0013	0.0012	0.0012	0.0011	0.0011	0.0011	0.001	0.001
.
-2.9	0.0019	0.0018	0.0018	0.0017	0.0016	0.0016	0.0015	0.0015	0.0014	0.0014
-2.8	0.0026	0.0025	0.0024	0.0023	0.0023	0.0022	0.0021	0.0021	0.002	0.0019
-2.7	0.0035	0.0034	0.0033	0.0032	0.0031	0.003	0.0029	0.0028	0.0027	0.0026
-2.6	0.0047	0.0045	0.0044	0.0043	0.0041	0.004	0.0039	0.0038	0.0037	0.0036
-2.5	0.0062	0.006	0.0059	0.0057	0.0055	0.0054	0.0052	0.0051	0.0049	0.0048
.
-2.4	0.0082	0.008	0.0078	0.0075	0.0073	0.0071	0.0069	0.0068	0.0066	0.0064
-2.3	0.0107	0.0104	0.0102	0.0099	0.0096	0.0094	0.0091	0.0089	0.0087	0.0084
-2.2	0.0139	0.0136	0.0132	0.0129	0.0125	0.0122	0.0119	0.0116	0.0113	0.011
-2.1	0.0179	0.0174	0.017	0.0166	0.0162	0.0158	0.0154	0.015	0.0146	0.0143
-2	0.0228	0.0222	0.0217	0.0212	0.0207	0.0202	0.0197	0.0192	0.0188	0.0183
.
-1.9	0.0287	0.0281	0.0274	0.0268	0.0262	0.0256	0.025	0.0244	0.0239	0.0233
-1.8	0.0359	0.0351	0.0344	0.0336	0.0329	0.0322	0.0314	0.0307	0.0301	0.0294
-1.7	0.0446	0.0436	0.0427	0.0418	0.0409	0.0401	0.0392	0.0384	0.0375	0.0367
-1.6	0.0548	0.0537	0.0526	0.0516	0.0505	0.0495	0.0485	0.0475	0.0465	0.0455
-1.5	0.0668	0.0655	0.0643	0.063	0.0618	0.0606	0.0594	0.0582	0.0571	0.0559
.
-1.4	0.0808	0.0793	0.0778	0.0764	0.0749	0.0735	0.0721	0.0708	0.0694	0.0681
-1.3	0.0968	0.0951	0.0934	0.0918	0.0901	0.0885	0.0869	0.0853	0.0838	0.0823
-1.2	0.1151	0.1131	0.1112	0.1093	0.1075	0.1056	0.1038	0.102	0.1003	0.0985
-1.1	0.1357	0.1335	0.1314	0.1292	0.1271	0.1251	0.123	0.121	0.119	0.117
-1	0.1587	0.1562	0.1539	0.1515	0.1492	0.1469	0.1446	0.1423	0.1401	0.1379
.
-0.9	0.1841	0.1814	0.1788	0.1762	0.1736	0.1711	0.1685	0.166	0.1635	0.1611
-0.8	0.2119	0.209	0.2061	0.2033	0.2005	0.1977	0.1949	0.1922	0.1894	0.1867
-0.7	0.242	0.2389	0.2358	0.2327	0.2296	0.2266	0.2236	0.2206	0.2177	0.2148
-0.6	0.2743	0.2709	0.2676	0.2643	0.2611	0.2578	0.2546	0.2514	0.2483	0.2451
-0.5	0.3085	0.305	0.3015	0.2981	0.2946	0.2912	0.2877	0.2843	0.281	0.2776
.
-0.4	0.3446	0.3409	0.3372	0.3336	0.33	0.3264	0.3228	0.3192	0.3156	0.3121
-0.3	0.3821	0.3783	0.3745	0.3707	0.3669	0.3632	0.3594	0.3557	0.352	0.3483
-0.2	0.4207	0.4168	0.4129	0.409	0.4052	0.4013	0.3974	0.3936	0.3897	0.3859
-0.1	0.4602	0.4562	0.4522	0.4483	0.4443	0.4404	0.4364	0.4325	0.4286	0.4247
0	0.5	0.496	0.492	0.488	0.484	0.4801	0.4761	0.4721	0.4681	0.4641
.
0	0.5	0.504	0.508	0.512	0.516	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.591	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.648	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.67	0.6736	0.6772	0.6808	0.6844	0.6879
.
0.5	0.6915	0.695	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.719	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.758	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.791	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.834	0.8365	0.8389
.
1	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.877	0.879	0.881	0.883
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.898	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
.
1.5	0.9332	0.9345	0.9357	0.937	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.975	0.9756	0.9761	0.9767
.
2	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.983	0.9834	0.9838	0.9842	0.9846	0.985	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.989
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.992	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
.
2.5	0.9938	0.994	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.996	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.997	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.998	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
.
3	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.999	0.999

	Critical Values for t Distribution
	Confidence Intervals
	80%	90%	95%	98%	99%	99.9%
df	Level of Significance for One-Tailed Test
	0.1	0.05	0.025	0.01	0.005	0.0005
	Level of Significance for Two-Tailed Test
	0.2	0.1	0.05	0.02	0.01	0.001
1	3.078	6.314	12.706	31.821	63.657	636.619
2	1.886	2.92	4.303	6.965	9.925	31.599
3	1.638	2.353	3.182	4.541	5.841	12.924
4	1.533	2.132	2.776	3.747	4.604	8.61
5	1.476	2.015	2.571	3.365	4.032	6.869
6	1.44	1.943	2.447	3.143	3.707	5.959
7	1.415	1.895	2.365	2.998	3.499	5.408
8	1.397	1.86	2.306	2.896	3.355	5.041
9	1.383	1.833	2.262	2.821	3.25	4.781
10	1.372	1.812	2.228	2.764	3.169	4.587
11	1.363	1.796	2.201	2.718	3.106	4.437
12	1.356	1.782	2.179	2.681	3.055	4.318
13	1.35	1.771	2.16	2.65	3.012	4.221
14	1.345	1.761	2.145	2.624	2.977	4.14
15	1.341	1.753	2.131	2.602	2.947	4.073
16	1.337	1.746	2.12	2.583	2.921	4.015
17	1.333	1.74	2.11	2.567	2.898	3.965
18	1.33	1.734	2.101	2.552	2.878	3.922
19	1.328	1.729	2.093	2.539	2.861	3.883
20	1.325	1.725	2.086	2.528	2.845	3.85
21	1.323	1.721	2.08	2.518	2.831	3.819
22	1.321	1.717	2.074	2.508	2.819	3.792
23	1.319	1.714	2.069	2.5	2.807	3.768
24	1.318	1.711	2.064	2.492	2.797	3.745
25	1.316	1.708	2.06	2.485	2.787	3.725
26	1.315	1.706	2.056	2.479	2.779	3.707
27	1.314	1.703	2.052	2.473	2.771	3.69
28	1.313	1.701	2.048	2.467	2.763	3.674
29	1.311	1.699	2.045	2.462	2.756	3.659
30	1.31	1.697	2.042	2.457	2.75	3.646
40	1.303	1.684	2.021	2.423	2.704	3.551
60	1.296	1.671	2	2.39	2.66	3.46
120	1.289	1.658	1.98	2.358	2.617	3.373

Answers to Problems

1. a.

= 24/400^0.5 = 1.2. b. X +/- 1.96 × SE, or 52 - 1.96×1.2 to 52 + 1.96×1.2, or 49.648 to 54.352.

2. a. p₁ = 0.55, p₂ = 0.55, p_c = 55.9868421053,

, n₁ = 231, n₂ = 225. b. z = (p₁ - p₂)/(p_c[1-p_c]/n₁ + p_c[1 - p_c]/n₂)^0.5 c. z = (0.55 - 0.55)/[55.9868421053(1-55.9868421053)/231 + 55.9868421053(1-55.9868421053)/225]^0.5 = 0/NAN = NAN. d. Critical value at 1%: 2.576; Reject: Yes because |NAN| > 2.576

3. a. p = 0.301886792453,

= 0.35, n = 371. b.

c. z = (0.301886792453 - 0.35)/sqrt(0.35*(1-0.35)/371) = -1.943. d. Critical value at 10%: 1.645; Reject: Yes because |-1.943| > 1.645

4. a. Means are (20 + 36 + 36 + 30 + 35)/5 = 31.4 and (50 + 46 + 39 + 52 + 45)/5 = 46.4; standard deviations are [((20-31.4)² + (36-31.4)² + (36-31.4)² + (30-31.4)² + (35-31.4)²)/5]^0.5 = 6.11882341631 and [((50-46.4)² + (46-46.4)² + (39-46.4)² + (52-46.4)² + (45-46.4)²)/5]^0.5 = 4.49888875168. b. Covariance is

= ((20-31.4)×(50-46.4) + (36-31.4)×(46-46.4) + (36-31.4)×(39-46.4) + (30-31.4)×(52-46.4) + (35-31.4)×(45-46.4) + (20-31.4)×(50-46.4) = -17.96; c. Correlation is

= -17.96/(6.11882341631×4.49888875168) = -0.652428851785; d. Test statistic is

= -0.652428851785×1.73205080757/(1-(-0.652428851785)²) = -1.49111294885; e. Critical value is t₃(10) = 2.35337987285. Reject? No because |-1.49111294885| < 2.35337987285.

5. a. X₁ = 29, X₂ = 35.5, s₁ = 17, s₂ = 17,

, n₁ = 111, n₂ = 53. b.

c. z = (29 - 35.5)/[17²/111 + 11²/53]^0.5 = -6.5/2.21057062125 = -2.94. d. Critical value at 1%: 2.576; Reject: Yes because |-2.94| > 2.576

6. a. Differences are -3, 5, -10, 2; b.

; c. mean is (-3 + 5 + -10 + 2)/4 = -1.5, standard deviation is [( + (-3 - -1.5)² + (5 - -1.5)² + (-10 - -1.5)² + (2 - -1.5)²)/(4-1)]^0.5 = 6.5574385243l d. statistic is -1.5/3.27871926215 = -0.457; e. Critical value is t₁(3) = 5.84090941255. Reject? No because |-0.457| < 5.84090941255.

7. 1. Decide on a null hypothesis and an alternative hypothesis. 2. Decide on a significance level. 3. Decide on a test statistic. 4. Create a decision rule, usually involving a critical value. 5. Collect the data and run the test.

8. SE =

= [0.351485148515×(1-0.351485148515)/202]^0.5 = 0.0335921474564. b. p +/- 1.96 × SE, or 0.351485148515 - 1.96×0.0335921474564 to 0.351485148515 + 1.96×0.0335921474564, or 0.2856445395 to 0.417325757529.

9. a. The standard error is

= 6/121^0.5 = 0.545454545455. b. The z value for a sample mean of 52.7073 is (52.7073 - 54)/0.545454545455 = -2.37. The probability that a Standard Normal variable is at least -2.37 is 0.991105960668.

10. a.

= 6/24^0.5 = 1.22474487139. b. X +/- 2.0686576103× SE (where 2.0686576103 comes from the t table with n-1 = 23 degrees of freedom), or 55 - 2.0686576103×1.22474487139 to 55 + 2.0686576103×1.22474487139, or 52.4664222011 to 57.5335777989.

11. 1. The variable must have a finite population mean and variance. 2. The observations must be independent and identically distributed. 3. The sample size must be at least 30.

12. a. X = 56, s = 24,

= 57.84, n = 361. b.

c. z = (56 - 57.84)/[24/sqrt(361)] = -1.457. d. Critical value at 1%: 2.576; Reject: No because |-1.457| < 2.576

13. a. X = 28, s = ,

= 25.99, n = 18. b.

c. t = (28 - 25.99)/[3/sqrt(18)] = 2.843. d. Critical value at 1%: t₁(17) = 2.89823051627; Reject: No because |2.843| < 2.89823051627

Probability Table for the Normal Distribution

Critical Values for t Distribution

Answers to Problems