In a random sample of 112 State College students, 41 students reported being current smokers.
a) Calculate the 95% confidence interval for the proportion of State College students who smoke.
To calculate the 95% confidence interval for the proportion of State College students who smoke, you will need to use the formula for the confidence interval for a proportion. Here are the steps:
Step 1: Find the sample proportion (p-hat).
In this case, the sample proportion of State College students who smoke is 41/112 = 0.366.
Step 2: Find the margin of error (E).
The margin of error is calculated using the formula E = z * sqrt((p-hat * (1-p-hat)) / n), where z is the z-score associated with the desired confidence level, p-hat is the sample proportion, and n is the sample size.
For a 95% confidence interval, the z-score is approximately 1.96 (which corresponds to the 0.025 tail on each side of the normal distribution).
E = 1.96 * sqrt((0.366 * (1-0.366)) / 112) = 0.0637 (rounded to four decimal places).
Step 3: Calculate the lower and upper limits of the confidence interval.
To calculate the lower limit, subtract the margin of error from the sample proportion:
Lower limit = p-hat - E = 0.366 - 0.0637 = 0.3023 (rounded to four decimal places).
To calculate the upper limit, add the margin of error to the sample proportion:
Upper limit = p-hat + E = 0.366 + 0.0637 = 0.4297 (rounded to four decimal places).
Therefore, the 95% confidence interval for the proportion of State College students who smoke is (0.3023, 0.4297).