A study of the effect of weathering on the radioactivity of granite was made by
collecting a random sample of weathered rocks & another random sample of freshly
exposed rocks. Measurements were made of the amount of radioactivity emitted in
periods of one minute & the following data was obtained.
Fresh rocks 225 188 165 211 178 171 199
Weathered rocks 177 170 140 132 168 172 170
i) Examine the data for evidence of the effect of weathering & find a 95% confidence
interval for the difference in mean radioactivity.
(Please refer to the EXCEL file for calculation)
The formula for a confidence interval on the difference between means ( M1 - M2) is:
Md+- t*Smd
where Md = M1 - M2=191-161.29=29.71 is the statistic
t depends on the level of confidence desired and on the degrees of freedom=7-1=6,
and the two-tailed t(95%,6)=2.447.
Smd is an estimate of the standard error on the difference between means, which is
computed assuming that the variances in the two populations are equal. If the two
sample sizes are equal (n1 = n2 =7) then Smd is estimated by using the following
formula:
Smd = SQRT ((Var1+Var2)/n) = SQRT ((479+311.57)/7) =10.627
Then the 95%confidence interval is:
Md+- t*Smd = 29.71 +- 2.447*10.627 = (3.706, 55.714)
Requirements:
1. The populations are normally distributed.
2. Variances in the two populations are equal.
3. samples are independent
ii) State any assumptions you made in part i).
Since the 95%confidence interval is ranging between (3.706, 55.714).
However, zero is not included. Therefore, we can assume that the effect of weathering
might not be zero, i.e. there’s some difference between these two groups.
iii) Use an F-Test to justify one of the assumptions in part ii)
we can conduct the F test by one-way ANOVA:
Fresh
Mean=Dm
Grand Mean=Dgm
Variation within groups
Weathered
191
161.286
176.14
2874
1869.43
ANOVA
Source of Variation
SS
df
MS
F
7.818
Between Groups
3,090
1
3,090
Within Groups
4,743
12
395
Total
7,834
13
While the critical F value of 95% is
F*(1,12) = 4.9
Since F^ > F*, we have to reject null hypothesis that there’s no difference between the
two populations, and conclude that there is statistically significant effect of
weathering on the radioactivity.