1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
jarptica [38.1K]
2 years ago
14

A study was conducted to determine the proportion of American teenagers between 13 and 17 who smoke. Previous surveys showed tha

t 15% percent of all teenagers smoke. A Gallup survey interviewed a nationally representative sample of 785 teenagers aged 13 to 17. Seventy-one teenagers in the survey acknowledged having smoked at least once in the past week. Does the study provide adequate evidence to conclude that the percentage of teenagers who smoke is now different than 15%? Be sure to answer all the questions whether or not the requirements are actually met.
Mathematics
1 answer:
timofeeve [1]2 years ago
7 0

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

W = \frac{X-\mu}{\sigma} = \frac{X-117.75}{10.00437}

The cummulative distribution function of W, which we denote \phi 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 \phi(-x) = 1-\phi(x) for any positive value x.

P(X < 70) = P(\frac{X-117.75}{10.00437} < \frac{70-117.75}{10.00437}) = P(W< -4.772914) = \phi(-4.772914) = 1-\phi(4.772914)

If we look at the table, we will realise that \phi(4.772914) 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).

Download pdf
You might be interested in
If the tip varies directly with the number of guests, which equation represents the relationship between the tip, t, and the num
spin [16.1K]
<span>If the tip, t, varies directly with the number of guests, g, then it means that any change in tip there is a corresponding change in the number of guests. When the tip increases the number of guests increases as well. Therefore,

t </span>α g

To change the it to equal sign, we incorporate a proportionality constant, k.

t =<span> kg
</span>
This equation will represent the relation of t to g
3 0
3 years ago
Read 2 more answers
Help please No links And with explanation please!
victus00 [196]
First: -8 second:17
4 0
2 years ago
If 4.3 plus 1.1 times a number equals the sum of the number and 37.6 what is the number
podryga [215]

4.3 + 11*x = x + 37.6

x = 333

6 0
3 years ago
2 4.2.3 Quiz: Slope Intercept Equation of a Line
lions [1.4K]
Substitute 2 into x in the equation so f(x)= 5x2 -12 = -2 so it’s C = -2
7 0
2 years ago
What is 25-3x=10 broken down
Tema [17]

Answer:

The answer is x = 5

Step-by-step explanation:

The given equation is

25 - 3x = 10

So when we break down this equation, we will have the value of x. Now breaking down the equation and moving variables to the correct positions.

25 - 3x = 10

Moving -3x and 10 to the other sides

25 - 10 = 3x

3x = 25 - 10

3x = 15

Now dividing both sides with 3, we get the following answer

3x/3 = 15/3

x = 5

Breaking down the equation gives us the value of x, i-e 5

5 0
3 years ago
Other questions:
  • write an algebraic equation and use it to solve the problem. A jacket in a clothing store is on a rack marked 40% off. if the di
    10·1 answer
  • Three less than the sum of four times a number and six is eight. Find the number.
    11·2 answers
  • How can you show that two objects are proportional with a table?
    14·1 answer
  • A cargo plane flew to Moscow and back. It took six hours longer to go there than it did to come back. The average speed on the t
    15·1 answer
  • If a florist uses her credit card to pay for a new window display that costs $1,242 and does not pay it on until the Second mont
    6·1 answer
  • PLEASE HELP!!
    5·2 answers
  • Which of the following statements demonstrates the associative property of multiplication?
    13·2 answers
  • Help me with this!! I will mark Brainliest ❤️
    10·1 answer
  • Plz it’s for my finals!!
    9·1 answer
  • Can you convert each standard-form equation to slope-intercept form and type the correct code? Please remember to type in ALL CA
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!