Answer:
Identity (a) can be re-written as
which we already proven in another question, while for idenity (b)
step A is simply expressing each function in terms of sine and cosine.
step B is adding the terms on the LHS while multiplying the one on RHS.
step C is replacing the term on the numerator with the equivalent from the pithagorean identity
Answer:
The study only provides evidence that the percentage of teenagers who smoke is different than 15% but only considering teenagers between 13 and 17.
Step-by-step explanation:
I will assume that when we talk about a teenager, we are talking about a teenager between 13 and 17 years old. We can solve this problem with a hypothesis test. Lets first define the main hypothesis of our test and the alternate hypothesis, note that 70 is way less than 15% of 785 (is less than 10%), thus we can use the following ones
H0: 15% of the teenagers smoke
H1: Less than 15% of teenagers smoke
Lets rewrite H0 and H1 using probabilities. Let X be the amount of teenagers that smoke in a sample with length 785. X is a binomial random variable. If we take H0 to be true, then the probability of success in each individual outcome of X is 0.15. This means that the mean is μ = 0.15*785 = 117.75, and the standard deviation is σ = √(117.75(1-0.15)) = 10.00437.
Since we are working with a sample of length high enough (> 30), then the Central Limit Theorem tells us that X behives pretty similar to a Normal random variable, with similar mean and standard deviation; therefore, we may assume directly that X is normal.
The hypothesis can be rewritten in terms of X this way:
H0: μ = 117.75 (this means that in average 15% of a sample of 785 smoke)
H1: μ < 117.75
We will use a 95% confidence interval. note that if only 70 teenagers smoke, then that means that SX = 70, where SX is the sample we obtain. We will calculate the probability that X is less than (or equal) to 70, if that probability is less than 0.05, then we can say that we have evidence that the percentage of teenagers who smoke is different (in fact, less), than 15%.
In order to calculate P(X < 70), we will use the standarization of X, given by
The cummulative distribution function of W, which we denote has well known values and they can be found in the attached file. Also, since the density function of a standard random normal variable is symmetric, then we have that
for any positive value x.
If we look at the table, we will realise that is practically 1, thus P(X < 70) is practically 0 if we assume that the mean of X is 117.75.
This means that we have evidence that the percentage of teenagers who smoke is no longer 15%, it is less.
However, here we are assuming that the term 'teenager' and teenager between 13 and 17' is the same. Maybe the the study took teenagers with 20 years old or so, and if that happened, then it makes sense that the results here are not the same. Therefore, we conclude that the study only provides evidence that the percentage of teenagers who smoke is different than 15% but only considering teenagers between 13 and 17 (because that is where the sample came from).