1. Two analysts measure the percentage of ammonia in a chemical process over 9 days and find the following discrepancies between their results:

Investigate the mean discrepancy θ between their results and in particular give an interval in which you are 90% sure that it lies.

2. With the same data as in the previous question, test the hypothesis that there is no discrepancy between the two analysts.

3. Suppose that you have grounds for believing that observations xi, yi for , n are such that and also , but that you are not prepared to assume that the are equal. What statistic would you expect to base inferences about θ on?

4. How much difference would it make to the analysis of the data in Section 5.1 on rat diet if we took instead of .

5. Two analysts in the same laboratory made repeated determinations of the percentage of fibre in soya cotton cake, the results being as shown:

Investigate the mean discrepancy θ between their mean determinations and in particular give an interval in which you are 90% sure that it lies

(a) assuming that it is known from past experience that the standard deviation of both sets of observations is 0.1, and

(b) assuming simply that it is known that the standard deviations of the two sets of observations are equal.

6. A random sample is available from an distribution and a second independent random sample is available from an distribution. Obtain, under the usual assumptions, the posterior distributions of and of .

7. Verify the formula for S1 given towards the end of Section 5.2.

8. The following data consists of the lengths in mm of cuckoo’s eggs found in nests belonging to the dunnock and to the reed warbler:

Investigate the difference θ between these lengths without making any particular assumptions about the variances of the two populations, and in particular give an interval in which you are 90% sure that it lies.

9. Show that if m=n then the expression f21/f2 in Patil’s approximation reduces to

10. Suppose that Tx, Ty and θ are defined as in Section 5.3 and that

Show that the transformation from (Tx, Ty) to (T, U) has unit Jacobian and hence show that the density of T satisfies

11. Show that if then

12. Two different microscopic methods, A and B, are available for the measurement of very small dimensions in microns. As a result of several such measurements on the same object, estimates of variance are available as follows:

Give an interval in which you are 95% sure that the ratio of the variances lies.

13. Measurement errors when using two different instruments are more or less symmetrically distributed and are believed to be reasonably well approximated by a normal distribution. Ten measurements with each show a sample standard deviation three times as large with one instrument as with the other. Give an interval in which you are 99% sure that the ratio of the true standard deviations lies.

14. Repeat the analysis of Di Raimondo’s data in Section 5.6 on the effects of penicillin of mice, this time assuming that you have prior knowledge worth about six observations in each case suggesting that the mean chance of survival is about a half with the standard injection but about two-thirds with the penicillin injection.

15. The undermentioned table [quoted from Jeffreys (1961, Section 5.1)] gives the relationship between grammatical gender in Welsh and psychoanalytical symbolism according to Freud:

Find the posterior probability that the log odds-ratio is positive and compare it with the comparable probability found by using the inverse root-sine transformation.

16. Show that if then the log odds-ratio is such that

17. A report issued in 1966 about the effect of radiation on patients with inoperable lung cancer compared the effect of radiation treatment with placebos. The numbers surviving after a year were:

What are the approximate posterior odds that the one-year survival rate of irradiated patients is at least 0.01 greater than that of those who were not irradiated?

18. Suppose that , that is x is Poisson of mean 8.5, and . What is the approximate distribution of x - y?