Respuesta :
Answer:
Lower limit: 117.3
Upper limit: 122.7
Step-by-step explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 120, \sigma = 4[/tex]
Middle 50%
So it goes from X when Z has a pvalue of 0.5 - 0.5/2 = 0.25 to X when Z has a pvalue of 0.5 + 0.5/2 = 0.75
Lower limit
X when Z has a pvalue of 0.25. So X when [tex]Z = -0.675[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 120}{4}[/tex]
[tex]X - 120 = -0.675*4[/tex]
[tex]X = 117.3[/tex]
Upper limit
X when Z has a pvalue of 0.75. So X when [tex]Z = 0.675[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 120}{4}[/tex]
[tex]X - 120 = 0.675*4[/tex]
[tex]X = 122.7[/tex]
The upper and lower limits of the blood pressure a person must have to qualify for the study are;
Lower limit: 117.3
Upper limit: 122.7
We are given;
Mean; μ = 120
Standard deviation; σ = 4
Pressure must be in the middle range which is 50%.
This means that;
Lower range = 25% = 0.25
Upper range = 75% = 0.75
This means;
Upper limit will be the sample mean at the z-value of the p-value of 0.25
Lower limit will be the sample mean at the z-value of the p-value of 0.25
From online z-score from p-value calculator, we have;
z at p-value of 0.25; z = -0.6745
z at p-value of 0.75; z = 0.6745
Formula for z-score is;
z = (x' - μ)/σ
Making x' the subject;
x' = zσ + μ
Thus;
Upper limit = (0.6745 * 4) + 120
Upper limit ≈ 122.7
Lower limit = (-0.6745 * 4) + 120
Lower limit ≈ 117.3
Read more about normal distribution at; https://brainly.com/question/25800303
