Question

4) Blood calcium levels are measured in mg/dL. In a group of patients over 30 you find the following: μ = 9.3 mg/dL, and σ = 2.35 mg/dL.

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

(Hint/comment: if you want the middle 50%, how many percent go in each tail?)

b) Give the blood calcium values for the middle 70% of patients.

c) Give the 85th percentile.

d) Are you surprised by the answers to (b) and (c)? Explain. If you're not sure what's going on, draw some pictures of the normal curves.

Answer 1

Solution:

We are given

μ = 9.3 mg/dL, and σ = 2.35 mg/dL.

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

For the middle 50% of the patients, critical z values are given as below:

Lower z value = -0.67448975

Upper z value = 0.67448975

(by using z-table or excel)

Lower value = μ + Z* σ = 9.3 - 0.67448975*2.35 = 7.714949088

Upper value = μ + Z* σ = 9.3 + 0.67448975*2.35 =10.88505091

Required blood calcium values = 7.71 mg/dL and 10.89 mg/dL

b) Give the blood calcium values for the middle 70% of patients.

For the middle 70% of the patients, critical z values are given as below:

Lower z value = -1.036433389

Upper z value = 1.036433389

(by using z-table or excel)

Lower value = μ + Z* σ = 9.3 - 1.036433389*2.35 = 6.864381536

Upper value = μ + Z* σ = 9.3 + 1.036433389*2.35 =11.73561846

Required blood calcium values = 6.86 mg/dL and 11.74 mg/dL

c) Give the 85th percentile.

For the 85th percentile, critical z value is given as below:

Z = 1.036433389

(by using z-table or excel)

85th percentile = μ + Z* σ = 9.3 + 1.036433389*2.35 =11.73561846

85th percentile =11.74

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

Yes, the answers for part c and upper value for part b are same. For the both part, we need to find the z value for same upper tail of 15% area or lower tail of 85% area.