Data in 1980 showed that about 48% of the adult population had never smoked cigarettes. In 2004, a national health survey interviewed a random sample of 5000 adults and found that 54% had never been smokers. Create a 95% confidence interval for the proportion of adults in 2004 who had never been smokers.

How would you solve this by hand and/or calculator?

To solve this problem, we will use the formula for a confidence interval for a proportion:

Confidence Interval = p̂ ± Z * √(p̂ * (1 - p̂) / n)

Where:
- p̂ is the sample proportion
- Z is the z-value corresponding to the desired confidence level
- n is the sample size

Step 1: Calculate the sample proportion (p̂) in 2004.
p̂ = 54% = 0.54

Step 2: Identify the z-value corresponding to the desired confidence level. For a 95% confidence level, the z-value is approximately 1.96.

Step 3: Calculate the confidence interval.
Confidence Interval = p̂ ± Z * √(p̂ * (1 - p̂) / n)

Using the given values:
Confidence Interval = 0.54 ± 1.96 * √(0.54 * (1 - 0.54) / 5000)

Simplifying this:
Confidence Interval = 0.54 ± 1.96 * √(0.54 * 0.46 / 5000)

Now you can calculate the confidence interval using these values.

To create a confidence interval for the proportion of adults in 2004 who had never been smokers, you can use the formula for calculating a confidence interval for a proportion.

The formula is:
CI = p̂ ± Z * √(p̂ * (1 - p̂) / n)
where:
CI is the confidence interval
p̂ is the sample proportion (54% or 0.54 in this case)
Z is the z-score corresponding to the desired level of confidence (1.96 for a 95% confidence interval)
n is the sample size (5000)

To solve this by hand, follow these steps:

Step 1: Calculate the standard error: SE = √(p̂ * (1 - p̂) / n)
SE = √(0.54 * (1 - 0.54) / 5000)
SE ≈ 0.009206

Step 2: Calculate the margin of error: ME = Z * SE
ME = 1.96 * 0.009206
ME ≈ 0.018048

Step 3: Calculate the lower and upper bounds of the confidence interval:
Lower Bound = p̂ - ME
Lower Bound = 0.54 - 0.018048
Lower Bound ≈ 0.521952

Upper Bound = p̂ + ME
Upper Bound = 0.54 + 0.018048
Upper Bound ≈ 0.558048

Step 4: Write the confidence interval:
The 95% confidence interval for the proportion of adults in 2004 who had never been smokers is approximately (0.521952, 0.558048).

Calculating this using a calculator or statistical software can expedite the process. Many statistics programs, such as Microsoft Excel or R, have functions or formulas that can calculate the confidence interval directly given the necessary inputs.

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