A recent survey showed that among 675 randomly selected subjects who completed 4 years of college, 138 smoke and 537 do not smoke. Determine a 95% confidence interval for the true proportion of the given population that smokes.
(0.174, 0.2348).
To determine a 95% confidence interval for the true proportion of the given population that smokes, you can use the formula:
Confidence Interval = Sample Proportion ± (Critical Value * Standard Error)
The sample proportion is the proportion of smokers among the randomly selected subjects, which is calculated by dividing the number of smokers by the total number of subjects:
Sample Proportion = Number of Smokers / Total Sample Size
In this case, the number of smokers is 138 and the total sample size is 675, so:
Sample Proportion = 138 / 675 = 0.2044
The critical value is based on the desired level of confidence (95%) and the sample size. Since the sample size is large (675), we can use the standard normal distribution and the z-value associated with a 95% confidence level is approximately 1.96.
Standard Error = √(Sample Proportion * (1 - Sample Proportion) / Sample Size)
Standard Error = √((0.2044 * (1 - 0.2044)) / 675)
Standard Error ≈ 0.0165
Now we can calculate the confidence interval:
Confidence Interval = 0.2044 ± (1.96 * 0.0165)
Confidence Interval ≈ 0.2044 ± 0.0323
Confidence Interval ≈ (0.1721, 0.2367)
Therefore, the 95% confidence interval for the true proportion of the given population that smokes is approximately 0.1721 to 0.2367.
To determine the 95% confidence interval for the true proportion of the given population that smokes, we can use the following formula:
Confidence interval = Sample proportion ± (Z-score) * Standard Error
1. Calculate the sample proportion:
Sample proportion (p̂) = Number of subjects who smoke / Total sample size
p̂ = 138 / 675 = 0.2044
2. Calculate the standard error:
Standard error (SE) = √[(p̂ * (1 - p̂)) / n]
SE = √[(0.2044 * (1 - 0.2044)) / 675]
SE = 0.0164
3. Find the Z-score associated with a 95% confidence level. This can be found in a standard normal distribution table or using statistical software. For a 95% confidence level, the Z-score is approximately 1.96.
4. Calculate the confidence interval:
Confidence interval = 0.2044 ± (1.96 * 0.0164)
Confidence interval = 0.2044 ± 0.032
Therefore, the 95% confidence interval for the true proportion of the given population that smokes is approximately 0.1724 to 0.2364.