Question

After our continual struggle with getting data, we finish and see that the population, when looking at systolic pressure of students while taking exams, was approximately normal with μ = 138 mmHg and σ = 3.5 mmHg.

a)What is the probability of a student having a systolic pressure between 135 and 138?
b) What is the probability of a student having a systolic pressure less than 139?
c)What is the probability of having a systolic pressure between 134 and 137 if we take a sample of 25 students?

Answer 1

a) The probability of a student having a systolic pressure between 135 and 138 is approximately 0.3413.

b) The probability of a student having a systolic pressure less than 139 is approximately 0.8413.

c) The probability of having a systolic pressure between 134 and 137 in a sample of 25 students cannot be directly determined without additional information, such as the population standard deviation or the sample standard deviation.

Step-by-step explanation:

a) To find the probability of a student having a systolic pressure between 135 and 138, we use the Z-score formula:
`\(Z = \frac{{X - \mu}}{{\sigma}}\), where \(X\)` is the systolic pressure,
`\(\mu\)` is the mean, and
`\(\sigma\)` is the standard deviation. For 135 mmHg,
`\(Z = \frac{{135 - 138}}{{3.5}} = -0.8571\), and for 138 mmHg, \(Z = \frac{{138 - 138}}{{3.5}} = 0\).` Using a Z-table, we find the area between these Z-scores to be approximately 0.5000 for
`\(Z = 0\)` and 0.3413 for
`\(Z = -0.8571\)`. Therefore, the total probability is
`\(0.5000 - 0.3413 = 0.1587\)`.

b) To find the probability of a student having a systolic pressure less than 139, we find the Z-score for 139 mmHg:
`\(Z = \frac{{139 - 138}}{{3.5}} = 0.2857\).` Using the Z-table, the probability for
`\(Z = 0.2857\)` is approximately 0.0.6107. Subtracting this from 1 gives us 0.8413.

c) Without the standard deviation of the sample
`(\( \sigma \))` or the sample standard deviation
`(\( s \))`, we cannot directly determine the probability for the systolic pressure between 134 and 137 in a sample of 25 students. The calculation requires either
`\( \sigma \) or \( s \)`. If the population standard deviation is given, we use it; if not, we use the sample standard deviation. The formula for the standard error of the mean
`(\( SEM \)) is \( SEM = \frac{{\sigma}}{{√(n)}} \), where \( n \)`is the sample size. With
`\( SEM \)`, we can find the z-scores for 134 and 137 and determine the probability as in part (a).