Statistics for Economics

1. The mean volume of sales for a sample of 100 sales representatives is $25,350 per month. The sample standard deviation is $7,490. The vice president for sales would like to know whether this result is significantly different from $24,000 at a 95 percent confidence level. Set up the appropriate null and alternative hypotheses, and perform the appropriate statistical test.
2. Larry Bomser has been asked to evaluate sizes of tire inventories for retail outlets of a major tire manufacturer. From a sample of 120 stores, he has found a mean of 310 tires. The industry average is 325. If the standard deviation for the sample was 72, would you say that the inventory level maintained by this manufacturer is significantly different from the industry norm? Explain why. (Use a 95 percent confidence level.)
3. Twenty graduate students in business were asked how many credit hours they were taking in the current quarter. Their responses are shown as follows (c2p3):

Student Number	Credit Hours
1	2
2	7
3	9
4	9
5	8
6	11
7	6
8	8
9	12
10	11
11	6
12	5
13	9
14	13
15	10
16	6
17	9
18	6
19	9
20	10

1. Determine the mean, median, and mode for this sample of data. Write a sentence explaining what each means.
   1. It has been suggested that graduate students in business take fewer credits per quarter than the typical graduate student at this university. The mean for all graduate students is 9.1 credit hours per quarter, and the data are normally distributed. Set up the appropriate null and alternative hypotheses, and determine whether the null hypothesis can be rejected at a 95 percent confidence level.
2. Arbon Computer Corporation (ACC) produces a popular PC clone. The sales manager for ACC has recently read a report that indicated that sales per sales representative for other producers are normally distributed with a mean of $255,000. She is interested in knowing whether her sales staff is comparable. She picked a random sample of 16 salespeople and obtained the following results (c2p4):

Page 88

Person	Sales
1	177,406
2	339,753
3	310,170
4	175,520
5	293,332
6	323,175
7	144,031
8	279,670
9	110,027
10	182,577
11	177,707
12	154,096
13	236,083
14	301,051
15	158,792
16	140,891

At a 5 percent significance level, can you reject the null hypothesis that ACC’s mean sales per salesperson was $255,000? Draw a diagram that illustrates your answer.

• Assume that the weights of college football players are normally distributed with a mean of 205 pounds and a standard deviation of 30.
  • What percentage of players would have weights greater than 205 pounds?
  • What percentage of players would weigh less than 250 pounds?
  • Ninety percentage of players would weigh more than what number of pounds?
  • What percentage of players would weigh between 180 and 230 pounds?
• Mutual Savings Bank of Appleton has done a market research survey in which people were asked to rate their image of the bank on a scale of 1 to 10, with 10 being the most favorable. The mean response for the sample of 400 people was 7.25, with a standard deviation of 2.51. On this same question, a state association of mutual savings banks has found a mean of 7.01.
  • Clara Weston, marketing director for the bank, would like to test to see whether the rating for her bank is significantly greater than the norm of 7.01. Perform the appropriate hypothesis test for a 95 percent confidence level.
  • Draw a diagram to illustrate your result.
  • How would your result be affected if the sample size had been 100 rather than 400, with everything else being the same?
• In a sample of 25 classes, the following numbers of students were observed (c2p7):

Class	Number of students
1	40
2	50
3	42
4	20
5	29
6	39
7	49
8	46
9	52
10	45
11	51
12	64
13	43
14	37
15	35
16	44
17	10
18	40
19	36
20	20
21	20
22	29
23	58
24	51
25	54

• Calculate the mean, median, standard deviation, variance, and range for this sample.
  • What is the standard error of the mean based on this information?

Page 89

• What would be the best point estimate for the population class size?
  • What is the 95 percent confidence interval for class size? What is the 90 percent confidence interval? Does the difference between these two make sense?
• CoastCo Insurance, Inc., is interested in forecasting annual larceny thefts in the United States using the following data (c2p8):

Year	Larceny Thefts
1972	4,151
1973	4,348
1974	5,263
1975	5,978
1976	6,271
1977	5,906
1978	5,983
1979	6,578
1980	7,137
1981	7,194
1982	7,143
1983	6,713
1984	6,592
1985	6,926
1986	7,257
1987	7,500
1988	7,706
1989	7,872
1990	7,946
1991	8,142
1992	7,915
1993	7,821
1994	7,876

• Prepare a time-series plot of these data. On the basis of this graph, do you think there is a trend in the data? Explain.
  • Look at the autocorrelation structure of larceny thefts for lags of 1, 2, 3, 4, and 5. Do the autocorrelation coefficients fall quickly toward zero? Demonstrate that the critical value for r_k is 0.417. Explain what these results tell you about a trend in the data.
  • On the basis of what is found in parts a and b, suggest a forecasting method from Table 2.1 that you think might be appropriate for this series.
• Use exploratory data analysis to determine whether there is a trend and/or seasonality in mobile home shipments (MHS). The data by quarter are shown in the following table (c2p9):

Period	MHS		Period	MHS		Period	MHS		Period	MHS
Mar-81	54.9		Dec-84	66.2		Sep-88	59.2		Jun-92	52.8
Jun-81	70.1		Mar-85	62.3		Dec-88	51.6		Sep-92	57
Sep-81	65.8		Jun-85	79.3		Mar-89	48.1		Dec-92	57.6
Dec-81	50.2		Sep-85	76.5		Jun-89	55.1		Mar-93	56.4
Mar-82	53.3		Dec-85	65.5		Sep-89	50.3		Jun-93	64.3
Jun-82	67.9		Mar-86	58.1		Dec-89	44.5		Sep-93	67.1
Sep-82	63.1		Jun-86	66.8		Mar-90	43.3		Dec-93	66.4
Dec-82	55.3		Sep-86	63.4		Jun-90	51.7		Mar-94	69.1
Mar-83	63.3		Dec-86	56.1		Sep-90	50.5		Jun-94	78.7
Jun-83	81.5		Mar-87	51.9		Dec-90	42.6		Sep-94	78.7
Sep-83	81.7		Jun-87	62.8		Mar-91	35.4		Dec-94	77.5
Dec-83	69.2		Sep-87	64.7		Jun-91	47.4		Mar-95	79.2
Mar-84	67.8		Dec-87	53.5		Sep-91	47.2		Jun-95	86.8
Jun-84	82.7		Mar-88	47		Dec-91	40.9		Sep-95	87.6
Sep-84	79		Jun-88	60.5		Mar-92	43		Dec-95	86.4

On the basis of your analysis, do you think there is a significant trend in MHS? Is there seasonality? What forecasting methods might be appropriate for MHS according to the guidelines in Table 2.1?

Page 90

1. Home sales are often considered an important determinant of the future health of the economy. Thus, there is widespread interest in being able to forecast home sales (HS). Quarterly data for HS are shown in the following table in thousands of units (c2p10):

Date	Home sales (000) per Quarter
Mar-89	161
Jun-89	179
Sep-89	172
Dec-89	138
Mar-90	153
Jun-90	152
Sep-90	130
Dec-90	100
Mar-91	121
Jun-91	144
Sep-91	126
Dec-91	116
Mar-92	159
Jun-92	158
Sep-92	159
Dec-92	132
Mar-93	154
Jun-93	183
Sep-93	169
Dec-93	160
Mar-94	178
Jun-94	185
Sep-94	165
Dec-94	142
Mar-95	154
Jun-95	185
Sep-95	181
Dec-95	145
Mar-96	192
Jun-96	204
Sep-96	201
Dec-96	161
Mar-97	211
Jun-97	212
Sep-97	208
Dec-97	174
Mar-98	220
Jun-98	247
Sep-98	218
Dec-98	200
Mar-99	227
Jun-99	248
Sep-99	221
Dec-99	185
Mar-00	233
Jun-00	226
Sep-00	219
Dec-00	199
Mar-01	251
Jun-01	243
Sep-01	216
Dec-01	199
Mar-02	240
Jun-02	258
Sep-02	254
Dec-02	220
Mar-03	256
Jun-03	299
Sep-03	294
Dec-03	239
Mar-04	314
Jun-04	329
Sep-04	292
Dec-04	268
Mar-05	328
Jun-05	351
Sep-05	326
Dec-05	278
Mar-06	285
Jun-06	300
Sep-06	251
Dec-06	216
Mar-07	214
Jun-07	240

• Prepare a time-series plot of THS. Describe what you see in this plot in terms of trend and seasonality.
  • Calculate and plot the first 12 autocorrelation coefficients for HS. What does this autocorrelation structure suggest about the trend?

Page 91

1. Exercise 12 of Chapter 1 includes data on the Japanese exchange rate (EXRJ) by month. On the basis of a time-series plot of these data and the autocorrelation structure of EXRJ, would you say the data are stationary? Explain your answer. (c2p11)

Period	EXRJ
1	127.36
2	127.74
3	130.55
4	132.04
5	137.86
6	143.98
7	140.42
8	141.49
9	145.07
10	142.21
11	143.53
12	143.69
13	144.98
14	145.69
15	153.31
16	158.46
17	154.04
18	153.7
19	149.04
20	147.46
21	138.44
22	129.59
23	129.22
24	133.89