Final answer:
To calculate the probability that the sample proportion is greater than 0.28, we use the standard error formula for proportions and the normal distribution. The probability is approximately 0.2949.
Step-by-step explanation:
To calculate the probability that the sample proportion is greater than 0.28, we need to use the standard error formula for proportions and the normal distribution. The standard error is given by the formula:
SE = sqrt(p * (1-p) / n)
where p is the population proportion and n is the sample size. In this case, we have p = 0.26 (the proportion of smokers in the population) and n = 139 (the sample size). Plugging these values into the formula, we get:
SE = sqrt(0.26 * (1-0.26) / 139)
SE = sqrt(0.1924 / 139)
SE ≈ 0.037
Once we have the standard error, we can calculate the z-score using the formula:
z = (sample proportion - population proportion) / SE
In this case, the sample proportion is 0.28, the population proportion is 0.26, and the SE is 0.037. Plugging these values into the formula, we get:
z = (0.28 - 0.26) / 0.037
z ≈ 0.02 / 0.037
z ≈ 0.541
To find the probability that the sample proportion is greater than 0.28, we need to find the area to the right of the z-score on the standard normal distribution. Using a standard normal distribution table or calculator, we find that the probability is approximately 0.2949. Therefore, the probability that the sample proportion is greater than 0.28 is approximately 0.2949.