Question

James Short was a maker of optical instruments, particularly telescopes. He was born in Edinburgh, Scotland and entered the University of Edinburgh in 1726, where he attended lectures by the mathematician Colin Maclaurin. Short is perhaps best remembered for his observations of the transit of Venus on June 6, 1761. His profession made him quite a rich man, by the way. The attached table (see the txt file) provides James Short's measurements of the parallax of the sun (in seconds of a degree), based on the 1761 transit of Venus. There were two samples of observations collected (under two different conditions). The total number of cases is 53 and the "true" value is 8.798. The parallax of the sun is the angle subtended by the earth, as seen from the surface of the sun.

a) Construct the 95% confidence interval for μ based on the combined sample. Is the true value of the parallax of the sun in this interval?

b) Construct the 95% confidence interval for σ based on the combined sample.

c) Test a hypothesis that the parallax of the sun differs from 9 seconds of a degree, based on the combined sample at the 0.1 significance level.

d) For two samples, verify if the true means of the populations they are taken from differ insignificantly at the 0.05 significance level.

e) For two samples, verify if the ratio of the true variances of the populations they are taken from differ insignificantly at the 0.05 significance level.

f) Write a short summary of your analysis. Show your work. Write clearly all assumptions.

txt file is below,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
Parallax

Sample 1
8.5
8.5
7.33
8.64
9.27
9.06
9.25
9.09
8.5
8.06
8.43
8.44
8.14
7.68
10.34
8.07

Sample 2
8.36
9.71
8.65
8.35
8.71
8.31
8.36
8.58
7.8
7.71
8.3
9.71
8.5
8.28
9.87
8.86
5.76
8.44
8.23
8.5
8.8
8.4
8.82
9.02
10.57
9.11
8.66
8.34
8.6
7.99
8.58
8.34
9.64
8.34
8.55
9.54
9.07

Answer 1

a) Construct the 95% confidence interval for μ based on the combined sample.

where

Then,

The "true" value is 8.798. We see that true value of the parallax of the sun is in this interval.

b) Construct the 95% confidence interval for σ based on the combined sample. For that, we define the confidence interval for variance

c) Test a hypothesis that the parallax of the sun differs from 9 seconds of a degree, based on the combined sample at the 0.1 significance level. The null and alternative hypothesis is

Test statistic is

p-value is 9.57E-05

THe p-value is less than 0.1, we reject the null hypothesis and conclude that he parallax of the sun differs from 9 seconds of a degree.

d) For two samples, verify if the true means of the populations they are taken from differ insignificantly at the 0.05 significance level. The null and alternative hypothesis is

Test statistic is

where

P-value is 0.820 with degree of freedom is 31.

The p-value is greater than 0.05, we fail to reject the null hypothesis and conclude that true means of the populations they are taken from differ is insignificantly.