Question

Duration of daylight in western Pennsylvania. Refer to the Naval Oceanography Portal data on number of minutes of daylight per day in Sharon, PA, Exercise 7.43 (p. 335). An estimate of the mean number of minutes of daylight per day was obtained using data collected for 12 randomly selected days (one each month) in a recent year.

a. Determine the number of days that need to be sampled in order to estimate the desired mean to within 45 minutes of its true value with 95% confidence.

b. Based on your answer, part a, develop a sampling plan that will likely result in a random sample that is representative of the population.

c. the data is listed below in section C.

d. Use the data, part c, to construct a 95% confidence interval for the desired mean. Does your interval have the desired width?

DATE,HOURS,MINUTES,TOTAL (minutes)

Jan-25,9,51,591 Feb-6,10,18,618 Mar-11,11,44,704 Apr-27,13,51,831 May-31,14,56,896 Jun-21,15,9,909 Jul-30,14,25,865 Aug-11,13,59,839 Sep-15,12,28,748 Oct-13,11,12,672 Nov-20,9,43,583 Dec-1,9,25,565

C Section data:

Date,Sunrise,Sunset,Daylength,Minutes,

1/1/19,7:13,16:22,9:09,549, 1/2/19,7:13,16:23,9:10,550, 1/3/19,7:13,16:24,9:11,551, 1/4/19,7:13,16:25,9:11,551, 1/5/19,7:13,16:26,9:12,552, 1/6/19,7:13,16:27,9:14,554, 1/7/19,7:13,16:28,9:15,555, 1/8/19,7:13,16:29,9:16,556, 1/9/19,7:13,16:30,9:17,557, 1/10/19,7:12,16:31,9:18,558, 1/11/19,7:12,16:32,9:20,560, 1/12/19,7:12,16:33,9:21,561, 1/13/19,7:12,16:34,9:23,563, 1/14/19,7:11,16:35,9:24,564, 1/15/19,7:11,16:37,9:26,566, 1/16/19,7:10,16:38,9:28,568, 1/17/19,7:10,16:39,9:29,569, 1/18/19,7:09,16:40,9:31,571, 1/19/19,7:08,16:41,9:33,573, 1/20/19,7:08,16:43,9:35,575, 1/21/19,7:07,16:44,9:37,577, 1/22/19,7:06,16:45,9:39,579, 1/23/19,7:06,16:46,9:41,581, 1/24/19,7:05,16:48,9:43,583, 1/25/19,7:04,16:49,9:45,585, 1/26/19,7:03,16:50,9:47,587, 1/27/19,7:02,16:51,9:49,589, 1/28/19,7:02,16:53,9:51,591, 1/29/19,7:01,16:54,9:53,593, 1/30/19,7:00,16:55,9:56,596, 1/31/19,6:59,16:57,9:58,598, 2/1/19,6:58,16:58,10:00,600, 2/2/19,6:57,16:59,10:03,603, 2/3/19,6:56,17:00,10:05,605, 2/4/19,6:54,17:02,10:07,607, 2/5/19,6:53,17:03,10:10,610, 2/6/19,6:52,17:04,10:12,612, 2/7/19,6:51,17:06,10:15,615, 2/8/19,6:50,17:07,10:17,617, 2/9/19,6:48,17:08,10:20,620, 2/10/19,6:47,17:10,10:22,622, 2/11/19,6:46,17:11,10:25,625, 2/12/19,6:45,17:12,10:28,628, 2/13/19,6:43,17:13,10:30,630, 2/14/19,6:42,17:15,10:33,633, 2/15/19,6:41,17:16,10:35,635, 2/16/19,6:39,17:17,10:38,638, 2/17/19,6:38,17:19,10:41,641, 2/18/19,6:36,17:20,10:44,644, 2/19/19,6:35,17:21,10:46,646, 2/20/19,6:33,17:22,10:49,649, 2/21/19,6:32,17:24,10:52,652, 2/22/19,6:30,17:25,10:55,655, 2/23/19,6:29,17:26,10:57,657, 2/24/19,6:27,17:27,11:00,660, 2/25/19,6:26,17:29,11:03,663, 2/26/19,6:24,17:30,11:06,666, 2/27/19,6:23,17:31,11:08,668, 2/28/19,6:21,17:32,11:11,671, 3/1/19,6:19,17:34,11:14,674, 3/2/19,6:18,17:35,11:17,677, 3/3/19,6:16,17:36,11:20,680, 3/4/19,6:15,17:37,11:23,683, 3/5/19,6:13,17:38,11:25,685, 3/6/19,6:11,17:40,11:28,688, 3/7/19,6:10,17:41,11:31,691, 3/8/19,6:08,17:42,11:34,694, 3/9/19,6:06,17:43,11:37,697, 3/10/19,7:05,18:44,11:40,700, 3/11/19,7:03,18:46,11:43,703, 3/12/19,7:01,18:47,11:46,706, 3/13/19,6:59,18:48,11:48,708, 3/14/19,6:58,18:49,11:51,711, 3/15/19,6:56,18:50,11:54,714, 3/16/19,6:54,18:51,11:57,717, 3/17/19,6:53,18:53,12:00,720, 3/18/19,6:51,18:54,12:03,723, 3/19/19,6:49,18:55,12:06,726, 3/20/19,6:47,18:56,12:09,729, 3/21/19,6:46,18:57,12:11,731, 3/22/19,6:44,18:58,12:14,734, 3/23/19,6:42,18:59,12:17,737, 3/24/19,6:40,19:01,12:20,740, 3/25/19,6:39,19:02,12:23,743, 3/26/19,6:37,19:03,12:26,746, 3/27/19,6:35,19:04,12:29,749, 3/28/19,6:33,19:05,12:32,752, 3/29/19,6:32,19:06,12:34,754, 3/30/19,6:30,19:07,12:37,757, 3/31/19,6:28,19:08,12:40,760, 4/1/19,6:27,19:10,12:43,763, 4/2/19,6:25,19:11,12:46,766, 4/3/19,6:23,19:12,12:49,769, 4/4/19,6:21,19:13,12:52,772, 4/5/19,6:20,19:14,12:54,774, 4/6/19,6:18,19:15,12:57,777, 4/7/19,6:16,19:16,13:00,780, 4/8/19,6:15,19:18,13:03,783, 4/9/19,6:13,19:19,13:06,786, 4/10/19,6:11,19:20,13:08,788, 4/11/19,6:10,19:21,13:11,791, 4/12/19,6:08,19:22,13:14,794, 4/13/19,6:06,19:23,13:17,797, 4/14/19,6:05,19:24,13:19,799, 4/15/19,6:03,19:25,13:22,802, 4/16/19,6:02,19:27,13:25,805, 4/17/19,6:00,19:28,13:28,808, 4/18/19,5:58,19:29,13:30,810, 4/19/19,5:57,19:30,13:33,813, 4/20/19,5:55,19:31,13:36,816, 4/21/19,5:54,19:32,13:38,818, 4/22/19,5:52,19:33,13:41,821, 4/23/19,5:51,19:34,13:44,824, 4/24/19,5:49,19:36,13:46,826, 4/25/19,5:48,19:37,13:49,829, 4/26/19,5:46,19:38,13:51,831, 4/27/19,5:45,19:39,13:54,834, 4/28/19,5:43,19:40,13:57,837, 4/29/19,5:42,19:41,13:59,839, 4/30/19,5:41,19:42,14:02,842, 5/1/19,5:39,19:43,14:04,844, 5/2/19,5:38,19:44,14:07,847, 5/3/19,5:37,19:46,14:09,849, 5/4/19,5:35,19:47,14:11,851, 5/5/19,5:34,19:48,14:14,854, 5/6/19,5:33,19:49,14:16,856, 5/7/19,5:32,19:50,14:18,858, 5/8/19,5:30,19:51,14:21,861, 5/9/19,5:29,19:52,14:23,863, 5/10/19,5:28,19:53,14:25,865, 5/11/19,5:27,19:54,14:27,867, 5/12/19,5:26,19:55,14:30,870, 5/13/19,5:25,19:56,14:32,872, 5/14/19,5:24,19:57,14:34,874, 5/15/19,5:23,19:58,14:36,876, 5/16/19,5:22,20:00,14:38,878, 5/17/19,5:21,20:01,14:40,880, 5/18/19,5:20,20:02,14:42,882, 5/19/19,5:19,20:03,14:44,884, 5/20/19,5:18,20:04,14:46,886, 5/21/19,5:17,20:05,14:47,887, 5/22/19,5:16,20:05,14:49,889, 5/23/19,5:15,20:06,14:51,891, 5/24/19,5:15,20:07,14:53,893, 5/25/19,5:14,20:08,14:54,894, 5/26/19,5:13,20:09,14:56,896, 5/27/19,5:13,20:10,14:57,897, 5/28/19,5:12,20:11,14:59,899, 5/29/19,5:11,20:12,15:00,900, 5/30/19,5:11,20:13,15:02,902, 5/31/19,5:10,20:13,15:03,903, 6/1/19,5:10,20:14,15:04,904, 6/2/19,5:09,20:15,15:06,906, 6/3/19,5:09,20:16,15:07,907, 6/4/19,5:09,20:16,15:08,908, 6/5/19,5:08,20:17,15:09,909, 6/6/19,5:08,20:18,15:10,910, 6/7/19,5:08,20:18,15:11,911, 6/8/19,5:07,20:19,15:12,912, 6/9/19,5:07,20:20,15:12,912, 6/10/19,5:07,20:20,15:13,913, 6/11/19,5:07,20:21,15:14,914, 6/12/19,5:07,20:21,15:15,915, 6/13/19,5:07,20:22,15:15,915, 6/14/19,5:07,20:22,15:16,916, 6/15/19,5:07,20:23,15:16,916, 6/16/19,5:07,20:23,15:16,916, 6/17/19,5:07,20:23,15:17,917, 6/18/19,5:07,20:24,15:17,917, 6/19/19,5:07,20:24,15:17,917, 6/20/19,5:07,20:24,15:17,917, 6/21/19,5:07,20:25,15:17,917, 6/22/19,5:08,20:25,15:17,917, 6/23/19,5:08,20:25,15:17,917, 6/24/19,5:08,20:25,15:17,917, 6/25/19,5:09,20:25,15:17,917, 6/26/19,5:09,20:25,15:16,916, 6/27/19,5:09,20:25,15:16,916, 6/28/19,5:10,20:25,15:16,916, 6/29/19,5:10,20:25,15:15,915, 6/30/19,5:11,20:25,15:14,914, 7/1/19,5:11,20:25,15:14,914, 7/2/19,5:12,20:25,15:13,913, 7/3/19,5:12,20:25,15:12,912, 7/4/19,5:13,20:24,15:12,912, 7/5/19,5:13,20:24,15:11,911, 7/6/19,5:14,20:24,15:10,910, 7/7/19,5:15,20:24,15:09,909, 7/8/19,5:15,20:23,15:08,908, 7/9/19,5:16,20:23,15:07,907, 7/10/19,5:17,20:22,15:05,905, 7/11/19,5:18,20:22,15:04,904, 7/12/19,5:18,20:21,15:03,903, 7/13/19,5:19,20:21,15:02,902, 7/14/19,5:20,20:20,15:00,900, 7/15/19,5:21,20:20,14:59,899, 7/16/19,5:22,20:19,14:57,897, 7/17/19,5:22,20:18,14:56,896, 7/18/19,5:23,20:18,14:54,894, 7/19/19,5:24,20:17,14:53,893, 7/20/19,5:25,20:16,14:51,891, 7/21/19,5:26,20:15,14:49,889, 7/22/19,5:27,20:14,14:47,887, 7/23/19,5:28,20:14,14:46,886, 7/24/19,5:29,20:13,14:44,884, 7/25/19,5:30,20:12,14:42,882, 7/26/19,5:31,20:11,14:40,880, 7/27/19,5:32,20:10,14:38,878, 7/28/19,5:33,20:09,14:36,876, 7/29/19,5:34,20:08,14:34,874, 7/30/19,5:35,20:07,14:32,872, 7/31/19,5:36,20:06,14:30,870, 8/1/19,5:37,20:04,14:28,868, 8/2/19,5:38,20:03,14:25,865, 8/3/19,5:39,20:02,14:23,863, 8/4/19,5:40,20:01,14:21,861, 8/5/19,5:41,20:00,14:19,859, 8/6/19,5:42,19:58,14:16,856, 8/7/19,5:43,19:57,14:14,854, 8/8/19,5:44,19:56,14:12,852, 8/9/19,5:45,19:54,14:09,849, 8/10/19,5:46,19:53,14:07,847, 8/11/19,5:47,19:52,14:05,845, 8/12/19,5:48,19:50,14:02,842, 8/13/19,5:49,19:49,14:00,840, 8/14/19,5:50,19:48,13:57,837, 8/15/19,5:51,19:46,13:55,835, 8/16/19,5:52,19:45,13:52,832, 8/17/19,5:53,19:43,13:50,830, 8/18/19,5:55,19:42,13:47,827, 8/19/19,5:56,19:40,13:44,824, 8/20/19,5:57,19:39,13:42,822, 8/21/19,5:58,19:37,13:39,819, 8/22/19,5:59,19:35,13:37,817, 8/23/19,6:00,19:34,13:34,814, 8/24/19,6:01,19:32,13:31,811, 8/25/19,6:02,19:31,13:29,809, 8/26/19,6:03,19:29,13:26,806, 8/27/19,6:04,19:27,13:23,803, 8/28/19,6:05,19:26,13:21,801, 8/29/19,6:06,19:24,13:18,798, 8/30/19,6:07,19:22,13:15,795, 8/31/19,6:08,19:21,13:13,793, 9/1/19,6:09,19:19,13:10,790, 9/2/19,6:10,19:17,13:07,787, 9/3/19,6:11,19:16,13:04,784, 9/4/19,6:12,19:14,13:02,782, 9/5/19,6:14,19:12,12:59,779, 9/6/19,6:15,19:11,12:56,776, 9/7/19,6:16,19:09,12:53,773, 9/8/19,6:17,19:07,12:50,770, 9/9/19,6:18,19:05,12:48,768, 9/10/19,6:19,19:04,12:45,765, 9/11/19,6:20,19:02,12:42,762, 9/12/19,6:21,19:00,12:39,759, 9/13/19,6:22,18:58,12:36,756, 9/14/19,6:23,18:57,12:34,754, 9/15/19,6:24,18:55,12:31,751, 9/16/19,6:25,18:53,12:28,748, 9/17/19,6:26,18:51,12:25,745, 9/18/19,6:27,18:50,12:22,742, 9/19/19,6:28,18:48,12:19,739, 9/20/19,6:29,18:46,12:17,737, 9/21/19,6:30,18:44,12:14,734, 9/22/19,6:31,18:42,12:11,731, 9/23/19,6:33,18:41,12:08,728, 9/24/19,6:34,18:39,12:05,725, 9/25/19,6:35,18:37,12:02,722, 9/26/19,6:36,18:35,12:00,720, 9/27/19,6:37,18:34,11:57,717, 9/28/19,6:38,18:32,11:54,714, 9/29/19,6:39,18:30,11:51,711, 9/30/19,6:40,18:28,11:48,708, 10/1/19,6:41,18:27,11:45,705, 10/2/19,6:42,18:25,11:43,703, 10/3/19,6:43,18:23,11:40,700, 10/4/19,6:45,18:21,11:37,697, 10/5/19,6:46,18:20,11:34,694, 10/6/19,6:47,18:18,11:31,691, 10/7/19,6:48,18:16,11:28,688, 10/8/19,6:49,18:15,11:26,686, 10/9/19,6:50,18:13,11:23,683, 10/10/19,6:51,18:11,11:20,680, 10/11/19,6:52,18:10,11:17,677, 10/12/19,6:54,18:08,11:14,674, 10/13/19,6:55,18:06,11:12,672, 10/14/19,6:56,18:05,11:09,669, 10/15/19,6:57,18:03,11:06,666, 10/16/19,6:58,18:01,11:03,663, 10/17/19,6:59,18:00,11:01,661, 10/18/19,7:00,17:58,10:58,658, 10/19/19,7:02,17:57,10:55,655, 10/20/19,7:03,17:55,10:52,652, 10/21/19,7:04,17:54,10:50,650, 10/22/19,7:05,17:52,10:47,647, 10/23/19,7:06,17:51,10:44,644, 10/24/19,7:08,17:49,10:42,642, 10/25/19,7:09,17:48,10:39,639, 10/26/19,7:10,17:46,10:36,636, 10/27/19,7:11,17:45,10:34,634, 10/28/19,7:12,17:44,10:31,631, 10/29/19,7:14,17:42,10:28,628, 10/30/19,7:15,17:41,10:26,626, 10/31/19,7:16,17:39,10:23,623, 11/1/19,7:17,17:38,10:21,621, 11/2/19,7:19,17:37,10:18,618, 11/3/19,6:20,16:36,10:16,616, 11/4/19,6:21,16:34,10:13,613, 11/5/19,6:22,16:33,10:11,611, 11/6/19,6:24,16:32,10:08,608, 11/7/19,6:25,16:31,10:06,606, 11/8/19,6:26,16:30,10:04,604, 11/9/19,6:27,16:29,10:01,601, 11/10/19,6:29,16:28,9:59,599, 11/11/19,6:30,16:27,9:57,597, 11/12/19,6:31,16:26,9:54,594, 11/13/19,6:32,16:25,9:52,592, 11/14/19,6:34,16:24,9:50,590, 11/15/19,6:35,16:23,9:48,588, 11/16/19,6:36,16:22,9:46,586, 11/17/19,6:37,16:21,9:44,584, 11/18/19,6:39,16:20,9:42,582, 11/19/19,6:40,16:19,9:40,580, 11/20/19,6:41,16:19,9:38,578, 11/21/19,6:42,16:18,9:36,576, 11/22/19,6:43,16:17,9:34,574, 11/23/19,6:45,16:17,9:32,572, 11/24/19,6:46,16:16,9:30,570, 11/25/19,6:47,16:15,9:29,569, 11/26/19,6:48,16:15,9:27,567, 11/27/19,6:49,16:14,9:25,565, 11/28/19,6:50,16:14,9:24,564, 11/29/19,6:51,16:14,9:22,562, 11/30/19,6:52,16:13,9:21,561, 12/1/19,6:54,16:13,9:19,559, 12/2/19,6:55,16:13,9:18,558, 12/3/19,6:56,16:12,9:17,557, 12/4/19,6:57,16:12,9:15,555, 12/5/19,6:58,16:12,9:14,554, 12/6/19,6:59,16:12,9:13,553, 12/7/19,7:00,16:12,9:12,552, 12/8/19,7:00,16:12,9:11,551, 12/9/19,7:01,16:12,9:10,550, 12/10/19,7:02,16:12,9:09,549, 12/11/19,7:03,16:12,9:09,549, 12/12/19,7:04,16:12,9:08,548, 12/13/19,7:05,16:12,9:07,547, 12/14/19,7:05,16:12,9:07,547, 12/15/19,7:06,16:12,9:06,546, 12/16/19,7:07,16:13,9:06,546, 12/17/19,7:08,16:13,9:05,545, 12/18/19,7:08,16:13,9:05,545, 12/19/19,7:09,16:14,9:05,545, 12/20/19,7:09,16:14,9:05,545, 12/21/19,7:10,16:15,9:05,545, 12/22/19,7:10,16:15,9:05,545, 12/23/19,7:11,16:16,9:05,545, 12/24/19,7:11,16:16,9:05,545, 12/25/19,7:12,16:17,9:05,545, 12/26/19,7:12,16:17,9:05,545, 12/27/19,7:12,16:18,9:06,546, 12/28/19,7:13,16:19,9:06,546, 12/29/19,7:13,16:20,9:07,547, 12/30/19,7:13,16:20,9:07,547, 12/31/19,7:13,16:21,9:08,548,

Answer 1

a.

Given:

Margin of error, MoE =45 mins.

Confidence level =95%. So, the Z-score for two-tailed case at 95% confidence level is Z =1.96

The sample std.deviation of given data(in minutes) of 591,618,704,831,896,909,865,839,748,672,583,565 is: s=129.56 minutes (this is an unbiased estimator of population standard deviation, $\sigma$​​​​​​)

MoE =Z. $\sigma/\sqrt{n}$ =1.96(129.56/$\sqrt{n}$​​​​​​) =45

$\implies$​​​​​n =[(1.96*129.56)/45]² =32

Thus, the number of days that need to be sampled in order to estimate the desired mean to within 45 minutes of its true value with 95% confidence is 32 days.

b.

Sampling plan - Systematic sampling:

Select any day randomly and choose next 31 days at regular intervals. Record the mean number of minutes of daylight per day for these 32 days. So, we get 32 values for which mean is calculated.

For example, on 1^st of every month, record the mean daylight duration in minutes for 32 months.

(OR) Record the daylight duration on 5^th January and then every 5^th one such as 10^th, 15^th, 20^th, 25^th,30^th​​​​​​, again 5^th, 10^th and so on till we get 32 values.

c.

The sample of 32 values (every 5^th starting from 1/5/19: ignoring 31 dates and 28 date in February. Taking dates which are multiples of 5 for easy collection of the data and without missing randomness and the sample would be representative of the population) are as follows:

552,558,566,575,585,596,610,622,635,649,663,685,700,714,729,743,757,774,788,802,816,829,842,854,865,876,886,894,902,909,913,916

Sample mean, $\bar{X} =\Sigma X/n$ =23805/32 =743.91​​​​​​ mins

Sample std.deviation, s =$\sqrt{\Sigma (X-\bar{X})^2/(n-1)}$ =123.30 mins

(s is an unbiased estimator of $\sigma$​​​​​​).

d.

95% confidence interval for the desired mean =$\bar{X} \pm Z. \sigma/\sqrt{n}$ =$743.91 \pm[1.96(123.30/\sqrt{32})]$ =(701.19 mins, 786.63 mins).

​​​​​​Width of the interval =786.63 - 701.19 =85.44

Margin of error =width/2 =42.72 minutes.

We wanted margin of error to be within 45 minutes and 42.72 minutes is within 45 minutes. Thus, the interval has the desired width.