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. Go to the Web site, http://aa.usno.navy.mil/data/docs/Dur_OneYear.php, and collect the data for Sharon, PA using your sampling plan.

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

Answer 1

(a) we know that:

But we don't have any knowledge regarding (Population variance). Thus we will use s (sample variance) instead of . Now, = 1.96 (for 95% significance level). s=129.56, and E= 45. By putting this values into the equation we can get:

n = ((1.96 * 129.56)/45)^2

n = 31.84 32.

(b) We have obtained that the sample size needed would be 32. Now, we have to develop the sampling plan. The simplest approach is given below.

Since there are 12 months in a year. Thus, average number of days in each month can be covered from sample of 32 is

   32/12 = 2.66 3.

Otherwise, you have to generate the weights as given below:

Month	Number of days	Obs. Proportion	Expected proportion	weight (exp./obs.)
Jan	31	0.0849	0.08333	0.98118
Feb	28	0.0767	0.08333	1.08631
March	31	0.0849	0.08333	0.98118
April	30	0.0822	0.08333	1.01389
May	31	0.0849	0.08333	0.98118
June	30	0.0822	0.08333	1.01389
July	31	0.0849	0.08333	0.98118
August	31	0.0849	0.08333	0.98118
Sept.	30	0.0822	0.08333	1.01389
Oct.	31	0.0849	0.08333	0.98118
Nov.	30	0.0822	0.08333	1.01389
Dec.	31	0.0849	0.08333	0.98118
Total	365

(c) the link is not working. Thus, it is not possible for me to answer the question.

(d) Since (c) is not available. Thus unable to answer the (d) question