(1) Gluten sensitivity, which is also known as wheat intolerance, affects approximately 15% of people in U.S. Let p be the proportion in a random sample of 800 individuals who have gluten sensitivity. Find the probability that the value of p is
(a) Within 0.02 of the population proportion
(b) Not within 0.02 of the population proportion
(c) Greater than the population proportion by 0.025 or more
Less than the population proportion by 0.03 or more
0.075
To find the probability in each of these cases, we need to use the concept of sampling distribution. The sampling distribution of p is approximately normal with mean equal to the population proportion and standard deviation equal to the square root of pq/n, where p is the population proportion, q is 1-p (the complement of p), and n is the sample size.
Given that p is the proportion in a random sample of 800 individuals who have gluten sensitivity, we can assume that p is normally distributed with mean 15% (or 0.15) and standard deviation sqrt(0.15*0.85/800) ≈ 0.014.
(a) To find the probability that the value of p is within 0.02 of the population proportion, we need to find the area under the normal curve between 0.15-0.02 = 0.13 and 0.15+0.02 = 0.17. We can use a standard normal table or a calculator to find this probability. For example, using a calculator, we can use the normalcdf function.
P(0.13 < p < 0.17) ≈ normalcdf(0.13, 0.17, 0.15, 0.014)
(b) To find the probability that the value of p is not within 0.02 of the population proportion, we need to find the area under the normal curve outside the interval (0.13, 0.17). This is the complement of the probability found in part (a).
P(p < 0.13 or p > 0.17) = 1 - P(0.13 < p < 0.17)
(c) To find the probability that the value of p is greater than the population proportion by 0.025 or more, we need to find the area under the normal curve to the right of 0.15+0.025 = 0.175.
P(p > 0.175) ≈ 1 - normalcdf(0.175, infinity, 0.15, 0.014)
(d) To find the probability that the value of p is less than the population proportion by 0.03 or more, we need to find the area under the normal curve to the left of 0.15-0.03 = 0.12.
P(p < 0.12) ≈ normalcdf(-infinity, 0.12, 0.15, 0.014)
These probabilities can be calculated using statistical software, a calculator, or by using a standard normal table if you have the corresponding z-scores.