Suppose systolic blood pressure (the higher of the two blood pressure readings) of a large group of individuals is normally distributed. If the mean is 122 and the standard deviation is 20, then
Question incomplete. Need to be more specific.
If the mean systolic blood pressure is 122 and the standard deviation is 20, we can use this information to answer various questions about the distribution of blood pressure in the group.
For example, if we want to know the probability that a randomly selected individual has a systolic blood pressure less than a certain value, we can use the Z-score formula and the standard normal distribution. The Z-score formula is given by:
Z = (X - μ) / σ
where Z is the Z-score, X is the value we are interested in, μ is the mean, and σ is the standard deviation.
To find the probability that a randomly selected individual has a systolic blood pressure less than a certain value, we first calculate the Z-score for that value using the formula above. Then, we look up the corresponding probability on the standard normal distribution table or use a Z-score calculator.
For example, let's say we want to find the probability that a randomly selected individual has a systolic blood pressure less than 130. We calculate the Z-score as follows:
Z = (130 - 122) / 20
Z = 0.4
Next, we look up the probability corresponding to a Z-score of 0.4 on the standard normal distribution table or use a Z-score calculator. This will give us the probability that a randomly selected individual has a systolic blood pressure less than 130.
Additionally, we can use the Z-score formula to calculate the value of systolic blood pressure that corresponds to a certain probability. This is useful when we want to find the blood pressure value below which a certain percentage of individuals fall.
To do this, we rearrange the Z-score formula to solve for X:
X = Z * σ + μ
where X is the blood pressure value we want to find, Z is the Z-score corresponding to the desired probability, σ is the standard deviation, and μ is the mean.
For example, if we want to find the blood pressure value below which 85% of individuals fall, we need to find the Z-score that corresponds to 85% probability. Let's assume the Z-score is 1.036 (we can find this value from the standard normal distribution table or use a Z-score calculator).
Using the formula, we can calculate the blood pressure value as follows:
X = 1.036 * 20 + 122
X = 145.72
Therefore, we can conclude that 85% of individuals in the group have a systolic blood pressure below 145.72.
By utilizing the mean and standard deviation of a normally distributed variable, we can answer various questions about the distribution and estimate probabilities or values based on the Z-score formula and the standard normal distribution.