the systolic blood pressure of MAAUN under graduate.student is distributed normally with arithmetic mean SD 140mmhg and 20mmHg respectively.if a student was selected at random,
(A).what is the probability his/her SBP is more than 160mmhg
(B)less than 120mmHg
(C): between 130mmhg and 190mmHg
(D):if 200 student were selected at random,how many of them will be SBP between 150mmHg and 190mmHG
To answer these questions, we will use the concept of the standard normal distribution.
Step 1: Standardizing values
To use the standard normal distribution table, we need to convert the given values into z-scores (standardized values). We do this using the formula:
z = (x - μ) / σ
where:
- z is the z-score
- x is the given value
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
In this case:
- μ (mean) = 140 mmHg
- σ (standard deviation) = 20 mmHg
(A) Probability of SBP more than 160 mmHg:
To find the probability of SBP being more than 160 mmHg, we need to find the area under the curve to the right of 160 mmHg. So, we standardize the value.
z = (160 - 140) / 20
z = 20 / 20
z = 1
Using the standard normal distribution table or calculator, we find the probability corresponding to z = 1. The probability value is approximately 0.8413. Therefore, the probability that a student's SBP is more than 160 mmHg is 0.8413.
(B) Probability of SBP less than 120 mmHg:
Similarly, we standardize the value to find the area under the curve to the left of 120 mmHg.
z = (120 - 140) / 20
z = -20 / 20
z = -1
Using the standard normal distribution table or calculator, we find the probability corresponding to z = -1. The probability value is approximately 0.1587. Therefore, the probability that a student's SBP is less than 120 mmHg is 0.1587.
(C) Probability of SBP between 130 mmHg and 190 mmHg:
To find the probability of SBP being between 130 mmHg and 190 mmHg, we need to find the area under the curve between the standardized values of these two values.
z1 = (130 - 140) / 20
z1 = -10 / 20
z1 = -0.5
z2 = (190 - 140) / 20
z2 = 50 / 20
z2 = 2.5
Using the standard normal distribution table or calculator, we find the probability corresponding to z = -0.5 and z = 2.5. The probability value for z = -0.5 is approximately 0.3085, and the probability value for z = 2.5 is approximately 0.9938. Therefore, the probability that a student's SBP is between 130 mmHg and 190 mmHg is 0.9938 - 0.3085 = 0.6853.
(D) Number of students with SBP between 150 mmHg and 190 mmHg:
Since we are dealing with a sample size of 200 students, we can use the probabilities we calculated above to estimate the number of students falling within a certain range.
Out of every 100 students, approximately 68.53% have SBP between 130 mmHg and 190 mmHg. Therefore, out of 200 students, the estimated number would be:
68.53% of 200 = 68.53/100 * 200 = 137.06
Since we cannot have fractional students, we round this value to the nearest whole number. Therefore, approximately 137 students would have SBP between 150 mmHg and 190 mmHg.