Two laboratories, A and B, provide radioactive carbon dating. A researcher suspects that they tend to provide different datings. To investigate this, samples of 8 specimens are split in half. One half is sent to A, and the other half is sent to B for carbon dating. The data x contains radioactive carbon datings (in 100 years) from laboratory A, and the data y contains datings from laboratory B.

x = c(18.0,15.4,24.2,20.9,20.1,22.0,21.6,18.1)

y = c(18.7,17.3,27.0,31.2,15.6,30.4,15.3,16.9)

Data set, cdatings.csv, can be downloaded.

1. Find the mean and the standard deviation by which the test statistic is constructed.

   21.55 and 6.797 20.037 and 2.772 -1.513 and 5.753 -1.513 and 5.191

2. Find the test statistic.

   -0.629 -0.583 -0.744 -1.543

3. Do you reject the null hypothesis or not? Find the critical value, and complete the hypothesis test with significance level 0.01

   Since the critical value is 3.499, the null hypothesis can be rejected. Since the critical value is 2.998, the null hypothesis can be rejected. Since the critical value is 3.499, the null hypothesis cannot be rejected. Since the critical value is 2.998, the null hypothesis cannot be rejected.

4. Find an appropriate confidence interval.

   [ -6.322 , 3.297 ] [ -5.366 , 2.341 ] [ -8.63 , 5.605 ]

5. Choose a concluding sentence of the study from the following:

   (i) I can safely conclude that carbon datings from two laboratories are the same.
   (ii) I can conclude that carbon datings from two laboratories are different.
   (iii) I cannot sufficiently prove that carbon datings from two laboratories are different.

   (i) (ii) (iii)