a smoking researcher is interested in estimating the average age when cigarette smokers first began to smoke. Taking a random sample of 25, she determines a sample mean of 16.8 years and a sample deviation of 1.5 years. Construct a 95% confidence interval to estimate the population mean age of the onset of smoking
83.2
To construct a confidence interval for the population mean age of the onset of smoking, we can use the following formula:
Confidence Interval = Sample Mean ± (Critical Value) * (Sample Standard Deviation / √(Sample Size))
Step 1: Calculate the critical value
Since we want a 95% confidence interval, the critical value corresponds to a 95% confidence level. Looking up the critical value in a standard normal distribution table (or using a calculator), the critical value for a 95% confidence level is approximately 1.96.
Step 2: Calculate the confidence interval
Given:
Sample Mean (x̄) = 16.8 years
Sample Standard Deviation (s) = 1.5 years
Sample Size (n) = 25
Confidence Interval = 16.8 ± (1.96) * (1.5 / √25)
Step 3: Calculate the values within the confidence interval
Lower Bound = 16.8 - (1.96) * (1.5 / √25)
Upper Bound = 16.8 + (1.96) * (1.5 / √25)
Step 4: Calculate the confidence interval
Lower Bound = 16.8 - (1.96) * (0.3)
Upper Bound = 16.8 + (1.96) * (0.3)
Therefore, the 95% confidence interval to estimate the population mean age of the onset of smoking is approximately (16.24, 17.36) years.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.025) and its Z score.
95% = mean ± Z SEm
SEm = SD/√n