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Normal

For all problems below, use excel to do the calculation and show the Excel functions to get full credit.

1. Given that z is a standard normal random variable, compute the following probabilities.

a) P(-1.85≤z≤0.66)

b) P(z≥-0.25)

2. Given that z is a standard normal random variable, find z for each situation

a) the area between -z and z is 0.88.

b) the area to the left of z is 0.966.

3. A company’s workers spend an average of 20 hours on the Internet while at work. Assume that the time is normally distributed with a standard deviation of 8 hours.

a) What is the probability that a worker spends at most 30 hours on the Internet?

b) What is the probability that a worker spends between 20 to 30 hours on the Internet?

c) A person is classified as a heavy user if he or she is in the upper 20% of the usage. How many hours must a worker have logged on to the Internet to be considered as a heavy user?

4. Assume that the height of American adult males is normally distributed with a mean of 176 cm and a standard deviation of 9.3 cm.

a) What's the probability that an American adult male is taller than 160 cm?

b) What's the probability that a randomly selected American adult male is between 170 cm and 180 cm tall?

c) What's the maximum height for an merican adult male to b considered as in the bottom 5% in height?

Sampling 1

For all problems below, use excel to do the calculation and show the Excel functions to get full credit.

1. Assume that we want to identify a simple random sample of 10 of 495 students in a school.

a) What's the principle for selecting a simple random sample for a finite population? How about infinite population?

b) How many different simple random samples of size 10 can be selected from this population?

c) Use the 6th row of the five digit random numbers in the table below to identify the 10 students. Begin with 418. * (No need to use Excel function to answer this question.)

2. 10% of the items produced by a machine are defective. A random sample of 50 items is selected and checked for defects.

a) What's the sampling distribution of the sample proportion?

b) What's the probability that the sample will contain at least 8 defective units?

Sampling 2

Manager	Annual Salary	Training Program	3. A company has 300 managers. Their annual salary is in this data set.
1	55769.50	No	a) Calculate the population mean, standard deviation (*use the function for population standard deviation), and % of managers that participated in the training program.
2	50823.00	Yes	b) Draw a random sample of 30 managers using Excel.
3	48408.20	No	c) Calculate the sample mean, standard deviation, and % of managers that participated in the training program.
4	49787.50	No	d) For a sample of size 30, what's the sampling distribribution of sample mean? State the expected value, standard error and shape of the distribution. (Hint: Check the sample size for finite population.)
5	52801.60	Yes	e) What's the probability that a randomly selected sample has a sample mean of at least what you reported in part c)?
6	51767.70	No
7	58346.60	Yes
8	46670.20	No
9	50246.80	Yes
10	51255.00	No
11	52546.60	No
12	49512.50	Yes
13	51753.00	Yes
14	53547.10	No
15	48052.20	No
16	44652.50	Yes
17	51764.90	Yes
18	45187.80	Yes
19	49867.50	Yes
20	53706.30	Yes
21	52039.50	Yes
22	52973.60	No
23	53372.50	No
24	54592.00	Yes
25	55738.10	Yes
26	52975.10	Yes
27	52386.20	Yes
28	51051.60	Yes
29	52095.60	Yes
30	44956.50	No
31	51968.10	Yes
32	52245.20	No
33	52614.70	No
34	55339.40	Yes
35	50113.40	No
36	55392.80	Yes
37	49558.00	Yes
38	53062.60	Yes
39	46940.10	Yes
40	50131.20	No
41	57279.20	Yes
42	55295.90	Yes
43	46482.10	Yes
44	54276.00	No
45	49879.60	Yes
46	43754.70	Yes
47	56117.20	Yes
48	50487.80	Yes
49	46833.50	No
50	49285.50	Yes
51	48523.40	Yes
52	49782.20	No
53	54854.70	No
54	44457.10	No
55	52475.50	Yes
56	50135.50	No
57	52496.60	No
58	50237.80	Yes
59	45433.40	Yes
60	52725.40	Yes
61	45356.80	No
62	49569.50	No
63	54164.80	No
64	51673.10	Yes
65	57655.80	Yes
66	59978.60	No
67	48689.10	No
68	51009.40	Yes
69	54400.40	Yes
70	50792.00	No
71	51384.10	Yes
72	53665.50	No
73	49149.80	Yes
74	49958.20	Yes
75	59026.40	Yes
76	52097.10	Yes
77	56918.00	Yes
78	52042.60	No
79	48331.10	Yes
80	55276.50	Yes
81	51477.10	Yes
82	52034.90	Yes
83	46150.00	No
84	52030.90	Yes
85	54049.30	No
86	44834.90	No
87	49082.80	No
88	58829.10	Yes
89	51814.20	Yes
90	51036.00	No
91	48712.00	Yes
92	52264.00	No
93	51738.90	No
94	59284.60	Yes
95	49793.10	Yes
96	54155.40	Yes
97	51657.80	Yes
98	48522.10	Yes
99	50788.80	Yes
100	48806.20	No
101	49144.10	No
102	46089.70	Yes
103	49129.50	Yes
104	52304.10	Yes
105	48568.50	Yes
106	51357.60	Yes
107	51694.80	Yes
108	52755.50	Yes
109	54594.70	No
110	54975.40	No
111	57995.70	Yes
112	52210.80	No
113	51960.50	No
114	50911.60	No
115	42638.00	Yes
116	55591.20	Yes
117	57856.00	Yes
118	50879.80	Yes
119	42104.00	No
120	51754.50	No
121	55069.40	No
122	47560.80	No
123	43364.50	No
124	56716.60	Yes
125	50105.60	No
126	48537.60	No
127	44119.60	Yes
128	55179.30	Yes
129	51949.80	No
130	51548.90	No
131	44406.70	Yes
132	54034.40	Yes
133	48968.80	Yes
134	41198.80	Yes
135	56366.50	Yes
136	55670.20	Yes
137	48555.60	No
138	50141.90	Yes
139	58609.40	Yes
140	48242.70	Yes
141	47254.80	Yes
142	63538.70	No
143	51759.10	No
144	52707.50	Yes
145	44647.90	Yes
146	60036.20	Yes
147	53166.20	Yes
148	57582.70	Yes
149	51618.50	Yes
150	58639.70	Yes
151	58058.30	Yes
152	59098.20	Yes
153	56035.80	No
154	46858.80	No
155	52658.70	Yes
156	58154.60	Yes
157	53872.00	No
158	46943.30	Yes
159	46718.40	No
160	56740.90	No
161	49943.80	Yes
162	50374.50	No
163	60859.10	Yes
164	52306.90	Yes
165	46572.20	No
166	49134.10	Yes
167	52935.30	Yes
168	55796.70	Yes
169	47988.50	No
170	53467.40	Yes
171	58382.90	Yes
172	44924.30	Yes
173	52927.40	Yes
174	47648.10	Yes
175	51553.80	No
176	44715.40	No
177	49221.10	Yes
178	51362.20	Yes
179	55109.70	Yes
180	49402.20	Yes
181	53576.00	Yes
182	51858.90	No
183	49589.60	Yes
184	58070.40	No
185	52461.90	Yes
186	55504.70	No
187	47434.60	No
188	45891.30	No
189	56954.90	No
190	59049.00	Yes
191	59496.10	Yes
192	50602.60	Yes
193	57737.20	No
194	48368.30	Yes
195	46951.10	No
196	47911.10	Yes
197	48789.50	Yes
198	46739.20	Yes
199	55186.30	Yes
200	50296.00	Yes
201	48565.50	No
202	59115.40	No
203	44606.40	No
204	48398.50	No
205	55479.30	No
206	59126.40	No
207	47787.90	No
208	56565.60	Yes
209	46062.10	No
210	53635.30	Yes
211	51012.90	No
212	57657.60	Yes
213	50484.90	No
214	55286.20	No
215	52949.00	Yes
216	52689.00	Yes
217	56300.90	No
218	52184.50	Yes
219	53014.90	No
220	47212.10	No
221	55170.30	Yes
222	47047.00	Yes
223	45365.80	No
224	54838.60	No
225	50190.00	No
226	52185.50	Yes
227	53055.90	Yes
228	52183.90	No
229	48565.00	Yes
230	51368.10	No
231	51173.20	No
232	54164.80	No
233	60893.10	Yes
234	47129.30	Yes
235	59355.20	Yes
236	52722.20	Yes
237	53509.00	Yes
238	54897.00	Yes
239	47616.20	No
240	48161.40	No
241	48117.30	Yes
242	58127.60	Yes
243	57115.60	Yes
244	46174.30	Yes
245	42905.30	No
246	58035.30	Yes
247	56412.90	No
248	49187.70	Yes
249	53492.00	No
250	56233.10	Yes
251	41556.10	Yes
252	57114.20	Yes
253	44355.10	Yes
254	54880.70	Yes
255	49139.50	Yes
256	58246.10	Yes
257	57790.80	Yes
258	53581.40	No
259	59564.50	Yes
260	50705.30	Yes
261	44348.10	Yes
262	54644.10	Yes
263	45022.80	Yes
264	55690.00	Yes
265	53637.00	Yes
266	56642.80	No
267	50385.60	Yes
268	53307.60	No
269	54857.60	Yes
270	60683.20	Yes
271	52211.10	No
272	56803.20	No
273	53643.40	Yes
274	43031.30	No
275	52247.40	No
276	45297.50	Yes
277	45401.50	Yes
278	51564.70	No
279	44435.40	Yes
280	52351.00	Yes
281	54489.90	Yes
282	56986.60	Yes
283	50458.60	No
284	54220.40	Yes
285	56749.30	Yes
286	43257.00	No
287	54314.00	Yes
288	55962.70	Yes
289	50625.10	Yes
290	58371.50	No
291	46201.70	No
292	52192.50	Yes
293	54662.80	Yes
294	45130.30	No
295	55008.70	Yes
296	52836.60	No
297	50811.70	No
298	53036.30	Yes
299	48574.90	No
300	53427.00	No

Interval

Salary		1. Given the sample of 70 manage salaries, answer the following:
51009.40
54400.40		a) Use excel to develop a summary statistics for the sample. Find the margin of error for interval estimation at 90% confidence level.
50792.00		b) What's the distribution of sample mean for this problem when the population standard deviation is not known? State the expected value, standard error, and shape of the distribution.
51384.10		c) Develop the 90% confidence interval for the population mean.
53665.50
49149.80		d) If the population standard deviation is known to be 4200, at 97% confidence level, what's the margin of error?
49958.20		e) Develop the 97% confidence interval for the population mean with known sigma = 4200.
59026.40		f) For margin of error to be smaller than 600, what's the minimum size of the sample?
52097.10
56918.00		2. For t distribution with 21 degrees of freedom, find the probability in the following regions:
52042.60		a) to the right of 2.15
48331.10		b) between -1.7 and +1.7
55276.50
51477.10		3. Find the t values for the following cases:
52034.90		a) lower tail area of 0.02 with 33 degrees of freedom
46150.00		b) where 95% of the area falls between -t and +t values with 45 degrees of freedom.
52030.90
54049.30
44834.90
49082.80
58829.10
51814.20
51036.00
48712.00
52264.00
51738.90
59284.60
49793.10
54155.40
51657.80
48522.10
50788.80
48806.20
49144.10
46089.70
49129.50
52304.10
48568.50
51357.60
51694.80
52755.50
54594.70
54975.40
57995.70
52210.80
51960.50
50911.60
42638.00
55591.20
57856.00
50879.80
42104.00
51754.50
55069.40
47560.80
43364.50
56716.60
50105.60
48537.60
44119.60
55179.30
51949.80
51548.90
44406.70
54034.40
48968.80
41198.80
56366.50
55670.20
48555.60

Hypothesis

Time		1. A marketing company bases charges to a client on the assumption that telephone surveys can be completed in a mean time of 15 minutes or less. If a longer mean survey time is necessary, a premium rate is charged.
17		A sample of 35 surveys provided the survey times shown on the left. The population standard deviation is assumed known with 4 minutes.
11		Based on the sample statistics, is the premium rate justified? (Hint: If the population mean is over 15 minutes, the premium rate is not justified.)
12
23		a) Formulate the null and alternative hypotheses for this application.
20		b) Compute the sample mean.
23		c) Compute the test statistic.
15		d) Test the hypothesis at 5% significance level using the p-value approach. State the p-value and test conclusion.
16
23		2. Consider the following hypothesis test:
22		H0: μ = 10
18		Ha: μ ≠ 10
23		A sample of 81 provides a sample mean of 9.5. The population standard deviation of 1.8.
25
14		a) Compute the value of the test statistic.
12		b) At 1% significance level , test the above hypotheses using p-value approach. State your conclusion.
12		c) At 5% significance level , test the above hypotheses using the critical value approach. State the critical value first.
20		d) Construct the 95% confidence interval for the population mean based on the sample mean.
18		e) Based on your result in d), should you reject the null hypothesis at 5% significance level? Explain.
12
19
11
11
20
21
11
18
14
13
13
19
16
10
22
18
23

Regresion

Week	Sales (dollars)	Profit			The manager is interested in the relationship between Sales and Profit (let X be Sales and Y be Profit). The model is Y = β0 + β1X + ε.
1	$ 37,000.00	$ 4,120.00
2	$ 38,900.00	$ 4,200.00			a) Develop a scatter diagram and trendline. What's the relationship between sales and profit?
3	$ 41,200.00	$ 3,960.00			b) Run the simple linear regression.
4	$ 39,600.00	$ 4,010.00			c) Write the estimated regression equation.
5	$ 40,500.00	$ 3,950.00			d) What is the correlation coefficient? How do you interpret it?
6	$ 42,000.00	$ 3,875.00
7	$ 43,500.00	$ 3,800.00
8	$ 41,500.00	$ 3,910.00
9	$ 44,100.00	$ 3,602.00
10	$ 42,800.00	$ 3,500.00
11	$ 43,700.00	$ 3,680.00
12	$ 47,900.00	$ 3,476.00
13	$ 46,790.00	$ 3,680.00
14	$ 49,000.00	$ 3,367.00
15	$ 46,800.00	$ 3,578.00
16	$ 48,680.00	$ 3,456.00
17	$ 50,100.00	$ 3,345.00
18	$ 46,900.00	$ 3,672.00
19	$ 50,330.00	$ 3,409.00
20	$ 49,890.00	$ 3,256.00
					e) Follow the least square method, develop a table similar to Table 14.2 of the textbook. Calculate b0 and b1 and write the estimated regression equation. (4 points)
					f) Calculate the estimated values of yi (yi_hat), SSE and SST. (2 points)
					g) Calculate the coefficient of determination and interprete the value.
	xi	yi	xi-xbar	yi-ybar	(xi-xbar)(yi-ybar)	(xi-xbar)^2	yi_hat	(yi-yi_hat)^2	(yi-ybar)^2
	$ 37,000.00	$ 4,120.00
	$ 38,900.00	$ 4,200.00
	$ 41,200.00	$ 3,960.00
	$ 39,600.00	$ 4,010.00
	$ 40,500.00	$ 3,950.00
	$ 42,000.00	$ 3,875.00
	$ 43,500.00	$ 3,800.00
	$ 41,500.00	$ 3,910.00
	$ 44,100.00	$ 3,602.00
	$ 42,800.00	$ 3,500.00
	$ 43,700.00	$ 3,680.00
	$ 47,900.00	$ 3,476.00
	$ 46,790.00	$ 3,680.00
	$ 49,000.00	$ 3,367.00
	$ 46,800.00	$ 3,578.00
	$ 48,680.00	$ 3,456.00
	$ 50,100.00	$ 3,345.00
	$ 46,900.00	$ 3,672.00
	$ 50,330.00	$ 3,409.00
	$ 49,890.00	$ 3,256.00