Use the data and confidence level to construct a confidence...

Use the data and confidence level to construct a confidence interval estimate of $p,$ then address the given question. The same study cited in the preceding exercise produced these results after six months for the 198 patients given sustained care: $25.8 \%$ were no longer smoking, and these results were biochemically confirmed, but $40.9 \%$ of these patients reported that they were no longer smoking. Construct the two $95 \%$ confidence intervals. Compare the results. What do you conclude?

Use the data and confidence level to construct a confidence interval estimate of $p,$ then address the given question.
The same study cited in the preceding exercise produced these results after six months for the 198 patients given sustained care: $25.8 \%$ were no longer smoking, and these results were biochemically confirmed, but $40.9 \%$ of these patients reported that they were no longer smoking. Construct the two $95 \%$ confidence intervals. Compare the results. What do you conclude?

Key Concepts

Confidence Interval for a Proportion

This concept involves estimating the range within which the true population proportion (p) lies, given the data obtained from a sample. It uses the sample proportion (p?) along with a margin of error that accounts for sampling variability. The interval is constructed such that there is a specified level of confidence (often 95%) that the interval captures the true proportion, reflecting the uncertainty inherent in the estimation process.

Standard Error of a Proportion

The standard error (SE) for a proportion quantifies the variability of the sample proportion due to random sampling, and it is calculated using the formula SE = sqrt[p?(1 ? p?)/n], where p? is the sample proportion and n is the sample size. This measure is essential for determining the width of the confidence interval, as a larger standard error leads to a wider interval and reflects greater uncertainty.

Z-Critical Value and Confidence Level

This key component relates to the use of the normal distribution in constructing the confidence interval when sample sizes are sufficiently large. For a given confidence level, such as 95%, a corresponding z-critical value (approximately 1.96) is used to determine the margin of error. This value represents the number of standard deviations needed to capture the desired proportion of the distribution under the normal curve.

Comparing Confidence Intervals and Interpretation

When two different confidence intervals are constructed from the same study data—one based on biochemically confirmed results and the other on self-reported results—comparing them involves evaluating the difference in interval estimates for the underlying proportion. This comparison can reveal the impact of measurement error or bias, as discrepancies between the intervals might indicate over-reporting, under-reporting, or validation issues in the methodology used to assess the outcome.

Transcript

00:01 Over here let's start with the people who stopped smoking who stopped smoking so in this case our proportion is 0 .258 that is p -cap sample proportion is 0 .258 what is our n in this case our n is 198 now we want a 95 % confidence level which means our alpha by 2 is 0 .025 so our z alpha by 2 is 1 .96 z alpha by 2 is 1 .96 okay now we are going to use two formulas the first one is for the margin of error which is z alpha by 2 multiplied by root over p cap into 1 minus p cap upon n and the second one is for the confidence interval our confidence interval is going to be p cap minus e to p cap plus e right okay so in the first case we have all the three values that we need so what is our e in the first case our e over here comes out to be 0 .0609 after substituting the value this is 0 .0609 so what is the interval using the second formula our confidence interval for the proportion will be 19 .71 to 31 .89 19 .71 percent to 31 .89 percent.

01:55 Okay now let's move on to the people who reported stopped smoking.

02:03 Okay.

02:05 In the second case, in the second case what is our people? cap 0 .409 okay n is 198 n is equal to 198 and again alpha by 2 is same as before so z alpha by 2 is 1 .96 z alpha by 2 is 1 .96 again we have all the values that we need so putting this in the value for e okay in the formula to get e we get 0 .0.

02:48 0 .0685, 0 .0685.

02:54 Okay.

02:55 Now, how do i get my interval? my interval, we already know how to calculate the interval.

03:04 This is going to be 34 .05 to 47 .75.

03:09 34 .05 to 47 .75 % percent...

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