Final answer:
b. The probability that more than 8 have hypertension is approximately 1.43%. c. The probability that fewer than 4 have hypertension is approximately 55.17%. d. It would not be considered unusual if more than 10 of them have hypertension. e. The mean number who have hypertension in a sample of 25 adults is 3.75. f. The standard deviation of the number who have hypertension in a sample of 25 adults is approximately 1.97.
Step-by-step explanation:
b. To find the probability that more than 8 people have hypertension, we need to calculate the cumulative probability of having 8 or fewer people with hypertension and subtract it from 1. Let's assume the probability of having hypertension is p. The mean number of adults in the sample is np, where n is the sample size. The standard deviation is sqrt(np(1-p)). We can use the normal distribution to approximate the probability. Given that n = 25 and p = 0.15 (the probability of having hypertension in the population), we have np = 25 * 0.15 = 3.75 and standard deviation = sqrt(25 * 0.15 * (1 - 0.15)) ≈ 1.97. Now, we can calculate the z-score for 8 or fewer people having hypertension: z = (8 - 3.75) / 1.97 ≈ 2.17. Using a z-score table, we find that the cumulative probability for a z-score of 2.17 is approximately 0.9857. Therefore, the probability of having more than 8 people with hypertension is 1 - 0.9857 = 0.0143, or approximately 1.43%.
c. To find the probability that fewer than 4 people have hypertension, we need to calculate the cumulative probability of having 4 or fewer people with hypertension. Using the same formula as before, we can calculate the z-score for 4 or fewer people having hypertension: z = (4 - 3.75) / 1.97 ≈ 0.13. Using a z-score table, we find that the cumulative probability for a z-score of 0.13 is approximately 0.5517. Therefore, the probability of having fewer than 4 people with hypertension is 0.5517, or approximately 55.17%.
d. To determine if it would be unusual for more than 10 people to have hypertension, we need to compare the probability of having more than 10 people with hypertension to a certain threshold. A commonly used threshold is 5%. If the probability is less than 5%, then it would be considered unusual. Using the same formula, we can calculate the z-score for 10 or fewer people having hypertension: z = (10 - 3.75) / 1.97 ≈ 3.19. Using a z-score table, we find that the cumulative probability for a z-score of 3.19 is approximately 0.9991, which is greater than 0.95. Therefore, it would not be considered unusual for more than 10 people to have hypertension.
e. The mean number of adults who have hypertension in a sample of 25 adults can be calculated using the formula np, where n is the sample size and p is the probability of having hypertension in the population. Given that n = 25 and p = 0.15, the mean is np = 25 * 0.15 = 3.75.
f. The standard deviation of the number of adults who have hypertension in a sample of 25 adults can be calculated using the formula sqrt(np(1-p)), where n is the sample size and p is the probability of having hypertension in the population. Given that n = 25 and p = 0.15, the standard deviation is sqrt(25 * 0.15 * (1 - 0.15)) ≈ 1.97.