Question

A study was conducted to determine the proportion of American teenagers between 13 and 17 who smoke. Previous surveys showed that 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%? Question: Suppose you had been in charge of designing the study. What sample size would be needed to construct a margin of error of 2% with 95% confidence? Use the prior point estimate of p* = 0.15 for this calculation. Round up to the nearest whole number. (For example, 144.1 would round to 145)

          A study was conducted to determine the proportion of American
teenagers between 13 and 17 who smoke. Previous surveys showed that
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%?
Question:
Suppose you had been in charge of designing the study. What
sample size would be needed to construct a margin of error of 2%
with 95% confidence? Use the prior point estimate of p* =
0.15 for this calculation. Round up to the nearest whole
number. (For example, 144.1 would round to 145)

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Added by Luke D.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert T. L.

11/15/2022

Step 1

15 \) (proportion of teenagers who smoke) - \( p = \frac{71}{785} \) (proportion of teenagers in the sample who smoke) - \( n = 785 \) (sample size) Plugging in the values: \[ z = \frac{\frac{71}{785} - 0.15}{\sqrt{\frac{0.15 \times 0.85}{785}}} \] \[ z = Show more…

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A study was conducted to determine the proportion of American teenagers between 13 and 17 who smoke. Previous surveys showed that 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%? Question: Suppose you had been in charge of designing the study. What sample size would be needed to construct a margin of error of 2% with 95% confidence? Use the prior point estimate of p* = 0.15 for this calculation. Round up to the nearest whole number. (For example, 144.1 would round to 145)

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Transcript

00:01 Hey there, welcome to numerade.

00:03 So we have the proportion of american teenagers who smoke that is around 15 % of all teenagers.

00:13 So with this, we have our own sample and the number of teenagers that smoke at least once in the past week.

00:22 So we want to see if this percentage of teenagers who smoke is now different than 15%.

00:27 So this, we're going to start with our null and alternative.

00:31 Hypotheses.

00:34 So our null hypothesis we have our proportion here is equal to 0 .15, and our alternative hypotheses is the opposite where it does not equal to 0 .15.

00:47 So with this, we have our c score equation, taking our sample proportion, which is the 71 teenagers out of the 785.

00:58 This is then subtracted to our population proportion of 0 .1.

01:03 Then entire divided by the square root of the sample proportion of 0 .15 times 1 minus 0 .15 is 0 .85 divided by the square i'm sorry divided by the sample size of 785.

01:27 This would give us a z score that is pretty small of negative 4 .6 .7.

01:35 So what's this small value of our z score, this gives us a really small p value.

01:43 That is basically approximately 0 .000000.

01:51 All right so therefore we have that we compare this to our significance level here of around 0 .05 since we are not given a significance level here 0 .05.

02:10 Therefore, we reject the null hypothesis.

02:15 Since our p value is significantly smaller than our significance level, we reject the null hypothesis, meaning that there is sufficient evidence to conclude that.

02:51 To conclude that our claim is true...

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