What does noncollinear
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The half-range sine series is the expansion for 
with the assumption that is considered to be an odd function over its full range, 
. So for (a), you're essentially finding the full range expansion of the function
with period 2 so that
for 
and integers 
.
Now, since is odd, there is no cosine series (you find the cosine series coefficients would vanish), leaving you with
where
In this case,
, so
The half-range sine series expansion for is then
which can be further simplified by considering the even/odd cases of, but there's no need for that here.
The half-range cosine series is computed similarly, this time assuming is even/symmetric across its full range. In other words, you are finding the full range series expansion for
Now the sine series expansion vanishes, leaving you with
where
for
. Again, . You should find that
Here, splitting into even/odd cases actually reduces this further. Notice that when is even, the expression above simplifies to
while for odd, you have
So the half-range cosine series expansion would be
Attached are plots of the first few terms of each series overlaid onto plots of. In the half-range sine series (right), I use 
terms, and in the half-range cosine series (left), I use 
or 
terms. (It's a bit more difficult to distinguish from the latter because the cosine series converges so much faster.)
with period 2 so that
Now, since
where
In this case,
The half-range sine series expansion for
which can be further simplified by considering the even/odd cases of
The half-range cosine series is computed similarly, this time assuming
Now the sine series expansion vanishes, leaving you with
where
for
Here, splitting into even/odd cases actually reduces this further. Notice that when
while for odd
So the half-range cosine series expansion would be
Attached are plots of the first few terms of each series overlaid onto plots of
Answer:
1) No, there is not enough evidence to support the claim that the vaccine was effective (P-value=0.104) .
The null and alternative hypothesis are:
being the subindex 1 for the experimental group and subindex 2 for the control group.
2) Test statistic z=-1.26
Step-by-step explanation:
This is a hypothesis test for the difference between proportions.
The claim is that the vaccine was effective.
Then, the null and alternative hypothesis are:
The significance level is 0.01.
The sample 1 (experimental group), of size n1=100000 has a proportion of p1=0.0002.
The sample 2 (control group), of size n2=100000 has a proportion of p2=0.0003.
The difference between proportions is (p1-p2)=-0.0001.
The pooled proportion, needed to calculate the standard error, is:
The estimated standard error of the difference between means is computed using the formula:
Then, we can calculate the z-statistic as:
This test is a left-tailed test, so the P-value for this test is calculated as (using a z-table):
As the P-value (0.104) is bigger than the significance level (0.01), the effect is not significant.
The null hypothesis failed to be rejected.
There is not enough evidence to support the claim that the vaccine was effective.