E-Cigarettes A New York Times article reported that a survey c...

Question

E-Cigarettes A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.7% of them being regular users of e-cigarettes. Because e-cigarette use is relatively new, there is a need to obtain today’s usage rate. How many adults must be surveyed now if we want a confidence level of 95% and a margin of error of 1.5 percentage points?

c. Does the use of the result from the 2014 survey have much of an effect on the sample size?

c. Does the use of the result from the 2014 survey have much of an effect on the sample size?

Accepted Answer

Hello. In this video, we are told that a 2017 survey found that 28% of US adults regularly shop for groceries online. Researchers now want to estimate the current percentage of adults who shop online for groceries. To obtain a 95% confidence level and a margin of error of 2% points, how many adults must be surveyed now? Does the use of the 2017 survey results significantly affect the acquired sample size? So, in this, in this problem, we are trying to find the amount of people we need to survey in order to conduct, gets a 95% confidence interval. So just listing out some given information, we are told that we want a desired margin of error to be 2%, which can also be rewritten as 0.02. Now we are told that we want to obtain a 95% confidence level. So, we have a confidence level of 95%, which is equivalent to 0.95. But what's unique about this too is that this corresponds to a Z critical value of 0.025, which is equal to 1.96. Now, we want our point estimate to be the percentage of US adults that shop online regularly, and that is 28%. So our point estimate in this case is going to be 0.28%. Now, in order to find the amount of people that will allow us to obtain this confidence level, we want to find the sample size, or we want to use the sample size formula for estimating this proportion. That is going to be defined as the N equal to the critical value squared, multiplied by a point estimate, multiplied by the complement of the point estimate, and this will be defined divided by the wanted margin of error squared. So by using our information, we can rewrite that the amount of people we want is going to be 1.96 squared, multiplied by a point estimate 0.02 0.02, that's the margin of error. Our point estimate, which is 0.28, multiplied by the complement, 1 minus 0.28, and this will be divided by the margin of error squared, which is 0.02 squared. Now, with the help of a calculator, this approximates to be 1,936.17. Now, we will go ahead and round to the nearest whole number, and that is going to give us 1,936. So, now what we want to do is you want to determine what's the effect of using the 2017 values. Well, let's go ahead and check what's the difference between taking this point estimate and taking the maximum point estimate. Now, the maximum point estimate is going to be 0.50. So, if we solve for the amount of people with this point estimate, we get that N is equal to 1.96 squared, multiplied by 0.5, multiplied by 1 minus 0.5, divided by 0.02 squared, and this simplifies to give us 2,401. So, what this means is that if we use the point estimate from the 2017 data, then that reduces the required amount of people that we need to sample in order to obtain. The 95% confidence level. And with that being said, the solution to this problem is going to be a. So I hope these people, I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Sample Size Calculation

Confidence Level

Margin of Error

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