**Systolic Blood Pressure**Assume that the mean systolic blood pressure of normal adults is 120 millimeters of mercury (mm Hg) and the standard deviation is 5.6. Assume the variable is normally distributed.a. If an individual is selected, find the probability that the individual's pressure will be between 120 and 121.8 mm Hg.b. If a sample of 30 adults is randomly selected, find the probability that the sample mean will be between 120 and 121.8 mm Hg.c. Why is the answer to part a so much smaller than the answer to part b?**Explanation:**This problem explores the concept of normal distribution and probability calculations. It involves understanding individual versus sample data, highlighting how the sample size affects probability estimation.

a. Consider that the mean and standard deviation of a random variable X are µ and σ, respectively. Thus, the z-score of that random variable X is z = (X – µ)/σ.Define a random variable X, that is the systolic blood pressure of normal adults. According to the given information, X have follows normal distribution with mean µx = 120 mm Hg and the standard deviation of σ x = 5.6 mm Hg. If an individual is selected, the probability that the individual’s pressure will be between 120 and 121.8 mm Hg is, Thus, if an individual is selected, the probability that the individual’s pressure will be between 120 and 121.8 mm Hg is 0.1260. b. Central limit theorem: If the population follows normal distribution or the sample size is large enough (>30), then the sampling distribution of the sample mean follows normal distribution with the expectation and standard deviation explained as follows: E(x̅) = µ and SD(x̅) = σ/√n. Here, for the random variable x the population mean is µ, the population standard deviation is σ, and the sample size is n. Denote that x̅ is the sample mean. Now, for sample of 30, the mean of the sampling distribution of the sample mean is, µx̅ = 120. Now, for sample of 30, the standard deviation of the sampling distribution of the sample mean is, σx̅ = 5.6/√30 = 1.022415441. Thus, the sampling distribution of x̅ is approximately normal with mean µx̅ = 120 and standard deviation σx̅ = 1.022415441. If a sample of 30 adults is randomly selected, the probability that the sample mean will be between 120 and 121.8 mm Hg is, Thus, if a sample of 30 adults is randomly selected, the probability that the sample mean will be between 120 and 121.8 mm Hg is 0.4608. c. The standard deviation of the sample mean is smaller than that of the observations As a result of the this the distribution of sample mean is more conceited to the mean and the corresponding z score is larger. Hence, the area under the curve within two large z scores is greater than the area under the curve within two comparatively smaller z scores. Hence, the answer in part a is so much smaller than the answer to part b.