A manufacturer of a low-tar cigarette claims that their cigarettes contain an average of 16.3 mg of tar with a standard deviation of 3.1 mg per cigarette. How many sample cigarettes must the manufacturer observe to be 99% confident that the width of the confidence interval is 4 mg or less? Assume that the tar content of the manufactured cigarettes is normally distributed.
To find the minimum sample size required to achieve a desired level of confidence and maximum width of the confidence interval, we can use the formula:
n = (z * σ / E)^2
Where:
n = sample size
z = desired level of confidence (in terms of the z-score)
σ = standard deviation of the population
E = maximum allowable error (width of the confidence interval)
In this case, the desired level of confidence is 99%, which corresponds to a z-score of 2.576 (obtained from a standard normal distribution table). The standard deviation (σ) is given as 3.1 mg per cigarette, and the maximum allowable error (E) is 4 mg.
Plugging these values into the formula, we get:
n = (2.576 * 3.1 / 4)^2
Calculating this expression will give us the minimum sample size needed to achieve the desired confidence level and maximum width of the confidence interval.
To determine how many sample cigarettes the manufacturer must observe to be 99% confident that the width of the confidence interval is 4 mg or less, we can use the formula:
n = (Z * σ / E)²
where:
- n is the required sample size
- Z is the Z-score corresponding to the desired confidence level (99% confidence level corresponds to a Z-score of 2.576)
- σ is the standard deviation of the tar content (given as 3.1 mg)
- E is the maximum acceptable margin of error (given as 4 mg)
Substituting the given values into the formula, we have:
n = (2.576 * 3.1 / 4)²
n = 2.576² * 3.1² / 4²
n = 6.642176 * 9.61 / 16
n = 63.75286936 / 16
n ≈ 3.98455559
Since the sample size must be a whole number, we round up to the nearest whole number.
Therefore, the manufacturer must observe a minimum of 4 sample cigarettes to be 99% confident that the width of the confidence interval is 4 mg or less.