Question

A study investigated allergic reactions of dental patients to local anesthetics. Based on a survey of dental practitioners, the study reported that the mean number of units (ampoules) of local anesthetics used per week by dentists was 79, with a standard deviation od 23. Suppose we want to determine the percentage of dentists who use less than 102 units of local anesthetics per week.

a. Assuming that nothing is known about the shape of the distribution for the data, what % of dentists use less than 102 units of local anesthetics per week.

b. Assuming that the data has a mound-shaped distribution, what % of dentists use less than 102 units of local anesthetics per week

Answer 1

(a)

Here, μ = 79, σ = 23

μ + a = 102 means 79 + a = 102, so a = 23

P(X ≥ 102) ≤ 23^2 /(23^2 + 23^2) = 0.5

This means the percentage of dentists who use 102 or more units of anesthetics is less than or equal to 50%

So, More than 50% of the dentists use less than 102 units of the anesthetic

(b) z = (x - μ)/σ = (102 - 79)/23 = 1

P(x < 102 units) = P(z < 1) = 0.8413

About 84.13% of the dentists use less than 102 units of the anesthetic.