Question

Blood calcium levels are measured in mg/dL. In patients over 30, μ = 9.7 mg/dL, and σ = 2.42 mg/dL.

a) Give the blood calcium values for the middle 50% of patients.

b) Give the blood calcium values for the middle 60% of patients.

c) Give the 80th percentile.

d) Are you surprised by the answers to (b) and (c)? Explain.

*** I literally don't know how to solve this. Could you please go step by step without assuming that I know anything that way I can understand this by the end. Thank you!  

Answer 1

(a)

= 9.7

= 2.42

Middle 50% corresponds to area = 0.50/2 = 0.25 on either side of mid value.

Table of Area Under Standard Normal Curve gives Z = 0.675

Low side:

Z = - 1.645 = (X - 9.7)/2.42

So,

X = 9.7 - (1.645 X 2.42)

   = 9.7 - 3.9809

    = 5.7191

High side:

Z = 1.645 = (X - 9.7)/2.42

So,

X = 9.7 + (1.645 X 2.42)

   = 9.7 + 3.9809

    = 13.6809

So,

Answer is:

(5.7191, 13.6809)

(b)

= 9.7

= 2.42

Middle 60% corresponds to area = 0.60/2 = 0.30 on either side of mid value.

Table of Area Under Standard Normal Curve gives Z = 0.84

Low side:

Z = - 0.84 = (X - 9.7)/2.42

So,

X = 9.7 - (0.84 X 2.42)

   = 9.7 - 2.0328

    = 7.6672

High side:

Z = 0.84 = (X - 9.7)/2.42

So,

X = 9.7 + (0.84 X 2.42)

   = 9.7 + 2.0328

    = 11.7328

So,

Answer is:

(7.6672, 11.7328)

(c)

80th percentile is given by area = 0.80 - 0.50 = 0.30

Table gives Z = 0.84

So,

Z = 0.84 = (X - 9.7)/2.42

So,

X = 9.7 + (0.84 X 2.42)

   = 9.7 + 2.0328

    = 11.7328

So,

Answer is:

11.7328

(d)

We are not surprised by the answers to (b) and (c) that upper bound of middle 60% = 11.7328 coincides with 80% percentile = 11.7328 because mathematically upper bound of middle 60%= 0.50 + 0.30 = 0.80, which is the 80th percentile.