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. If an individual is selected, find the probability that the individual's pressure will be between 120 and 121.8 mm Hg
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To find the probability that an individual's blood pressure will be between 120 and 121.8 mm Hg, we need to calculate the z-scores for each value and then use the standard normal distribution table (or a calculator) to find the corresponding probabilities.
Step 1: Calculate the z-score for 120 mm Hg
The z-score formula is given by:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
For 120 mm Hg:
z1 = (120 - 120) / 5.6
z1 = 0
Step 2: Calculate the z-score for 121.8 mm Hg
For 121.8 mm Hg:
z2 = (121.8 - 120) / 5.6
z2 = 1.8 / 5.6
z2 ≈ 0.321
Step 3: Use the standard normal distribution table (or a calculator) to find the probability corresponding to the z-scores.
P(120 ≤ X ≤ 121.8) = P(0 ≤ Z ≤ 0.321)
Using the standard normal distribution table or a calculator, the probability that Z is between 0 and 0.321 is approximately 0.3767.
Therefore, the probability that an individual's blood pressure will be between 120 and 121.8 mm Hg is approximately 0.3767, or 37.67 percent.
To find the probability that an individual's pressure will be between 120 and 121.8 mm Hg, we need to calculate the area under the normal distribution curve between those two values.
First, we need to convert the values to standardized z-scores. To do this, we subtract the mean from each value and then divide by the standard deviation:
For 120 mm Hg:
z1 = (120 - 120) / 5.6 = 0
For 121.8 mm Hg:
z2 = (121.8 - 120) / 5.6 ≈ 0.32
Next, we can use a standard normal distribution table or a calculator to find the probabilities associated with these z-scores.
Using a standard normal distribution table, we can find that the probability corresponding to a z-score of 0 is 0.5000. Similarly, the probability corresponding to a z-score of 0.32 is approximately 0.6255.
To find the probability between these two values, we subtract the smaller probability from the larger probability:
P(120 < x < 121.8) ≈ 0.6255 - 0.5000 = 0.1255
Therefore, the probability that the individual's pressure will be between 120 and 121.8 mm Hg is approximately 0.1255 or 12.55%.