While shopping in France, Johnny notices the cost of apples is 3.20€/kg. Johnny isn't familiar with Euros or kilograms, and need
2 answers:
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Answer:
a) 0.18% of cold sufferers experience fewer than 4 days of symptoms
b) 64.40% of cold sufferers experience symptoms for between 7 and 10 days.
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean and standard deviation
, the zscore of a measure X is given by:
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
a. What proportion of cold sufferers experience fewer than 4 days of symptoms?
This is the pvalue of Z when X = 4. So
has a pvalue of 0.0018.
So 0.18% of cold sufferers experience fewer than 4 days of symptoms.
b. What proportion of cold sufferers experience symptoms for between 7 and 10 days?
This is the pvalue of Z when X = 10 subtracted by the pvalue of Z when X = 7. So
X = 10
has a pvalue of 0.9812.
X = 7
has a pvalue of 0.3372.
So 0.9812 - 0.3372 = 0.644 = 64.40% of cold sufferers experience symptoms for between 7 and 10 days.
(a) By DeMoivre's theorem, we have
On the LHS, expanding yields
Matching up real and imaginary parts, we have for (i) and (ii),
(b) By the definition of the tangent function,
(c) Setting
, we have 
and 
. So
At the given value of
, the denominator is a non-zero number, so only the numerator can contribute to this reducing to 0.
Remember, this is saying that
If we replace
with a variable 
, then the above means is a root to the quadratic equation,
Also, if
, then 
and 
. So by a similar argument as above, we deduce that 
is also a root to the quadratic equation above.
(d) We know both roots to the quadratic above. The fundamental theorem of algebra lets us write
Expand the RHS and match up terms of the same power. In particular, the constant terms satisfy
But
for all 
, as is the case for 
and 
, so we choose the positive root.
On the LHS, expanding yields
Matching up real and imaginary parts, we have for (i) and (ii),
(b) By the definition of the tangent function,
(c) Setting
At the given value of
Remember, this is saying that
If we replace
Also, if
(d) We know both roots to the quadratic above. The fundamental theorem of algebra lets us write
Expand the RHS and match up terms of the same power. In particular, the constant terms satisfy
But