Two laboratories, A and B, provide radioactive carbon dating. A researcher suspects that they tend to provide different datings. To investigate this, samples of 20 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(13.7,12.5,16.4,18.4,12.1,16.1,12.8,13.9,14.5,14.1,14.0,12.8,11.2,11.5,11.3,15.0,13.0,12.4,17.0,37.4)

y = c(20.4,18.4,19.1,23.4,16.2,20.8,14.6,16.6,18.1,16.7,10.6,14.4,14.1,14.2,15.6,10.2,13.8,17.0,21.3,37.7)

Data set, cdatings.csv, can be downloaded.

  1. Find an appropriate hypothesis test regarding the mean carbon dating, $\mu_1$ and $\mu_2$ from laboratory A and B.

    $H_0: \mu_1 = \mu_2$ versus $H_A: \mu_1$    < $\mu_2$ $H_0: \mu_1 \neq \mu_2$ versus $H_A: \mu_1 = \mu_2$ $H_0: \mu_1 = \mu_2$ versus $H_A: \mu_1$    > $\mu_2$

  2. Find the test statistic.

    -2.111 -2.043 -4.199 -1.468

  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 -2.539, the null hypothesis cannot be rejected. Since the critical value is -2.539, the null hypothesis can be rejected. Since the critical value is 2.861, the null hypothesis can be rejected. Since the critical value is 2.861, the null hypothesis cannot be rejected.

  4. Find an appropriate confidence interval.

    $( -4.464 , -0.846 )$ $( -4.261 , \infty )$ $( -\infty , -1.049 )$

  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 cannot sufficiently prove that carbon datings from two laboratories are different.
    (iii) I can conclude that carbon datings from laboratory A is older.
    (iv) I can conclude that carbon datings from laboratory B is older.

    (iv) (iii) (ii) (i)


Department of Mathematics
Last modified: 2024-11-26