XIIL.—SIMPLE SAMPLING OF ATTRIBUTES, 275
4. The proportion of successes in the data of Qu. 1is 0°5097. Find the stand-
ard-deviation of the proportion with the given number of throws, and state
whether you would regard the excess of successes as probably significant of bias
in the dice.
5. In the 4096 drawings on which Qu. 2 is based 2030 balls were black
and 2066 white. Is this divergence probably significant of bias?
6. If a frequency-distribution such as those of Questions 1, 2, and 3 be given,
show how =» and p, if unknown, may be approximately determined from the
mean and standard-deviation of the distribution.
Find 2 and p in this way from the data of Qu. 1 and Qu. 3.
7. Verify the following results for Table VI. of Chapter IX. p. 163, and
compare the results of the different grouping of the table on p. 263. In
calculating the actual standard-deviation, use Sheppard’s correction for
grouping (p. 212).
Actual Standard-
Row or Rows. Mean, Standard- deviation *
deviation s. of Sampling s,.
5082 1160 11°18
5095 6:79 6°45
510°0 528 5-00
4 5111 5:03 422
G 5102 367 373
6, 7 5097 4°13 3°24
8,9,10,'11 5087 3°10 2°69
12, 13, 14 5084 255 2:25
15 and upwards. 508-2 2°13 1-85
8. In a case of mice-breeding (see reference given in § 11) the harmonic
mean number in a litter was 4735, and the expected proportion of albinos
50 per cent. Find the standard-deviation of simple sampling for the pro-
portion of albinos in a litter, and state whether the actual standard-deviation
(21°63 per cent. ) probably indicates any real variation, or not.
9. (Data from Report i., Evolution Committee of the Royal Society, p. 17.)
In breeding certain stocks 408 hairy and 126 glabrous plants were obtained.
If the expectation is one-fourth glabrous, is the divergence significant, or might
it have occurred as a fluctuation of sampling ?
10. (Data of Example ix. and Qu. 5, Chap. IIL.) Is the association in
either of the following cases likely to have arisen as a fluctuation of simple
sampling ?
(a) (4B)=47 (48)=12 (eB)=21 (aB)=3
(®) (4.B)=309 (48)=214 (aB)=132 (aB)=119
11. The sex-ratio at birth is sometimes given by the ratio of male to female
births, instead of the proportion of male to total births. If Z is the ratio, 7.e.
Z=plq, show that the standard error of Z is approximately (1 +a2n/ Z
n
n being large, so that deviations are small compared with the mean. (The
student may find it useful to refer to § 8, Chap. XI.)
* Based on the mid-value of the class-interval for single rows, or the
harmonic mean of the mid-values for groups of rows.
