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Chapter 3: Q. 21 (page 152)

Caffeinated Beverages and Diabetes. The objective of the article, "Caffeinated and Caffeine-free Beverages and Risk of Type 2 Diabetes by S. Bhupathiraju et al., was to examine the association between caffeinated beverages and type 2 diabetes risk. The mean and standard deviation of the body mass index (BMI) for a sample of 10,215women who drink at least one caffeinated carbonated beverage a day are 25.7and 5.3, respectively.
a. At least how many women in the sample have a BMI of between 15.1and 36.3 ?
b. Fill in the blanks; At least 89% of the women in the sample have BMIs between ________ and _______.

Short Answer

Expert verified

Part(a) 7661women in the sample have a BMI of between 15.1and 36.3

Part(b) At least 89%of the women in the sample have BMIs between 9.8and 41.6

Step by step solution

01

Part(a) Step 1: Given information

We are given BMIs sample of womens who drink caffinated carbonated drink at least once in a day.

02

Part(a) Step 2: Simplify

According to Chebyshev's rule

x-ks=15.125.7-k5.3=15.1k=2

and also,

x+ks=36.325.7+k5.3=36.3k=2

Hence According to Chebyshev's rule if k=2then 75%of sample lies in given range.

Therefore,

10215脳75100=7661.25鈮�7661

7661women in the sample have a BMI of between 15.1and 36.3

03

Part(b) Step 1: Given information

We are given BMIs sample of womens who drink caffinated carbonated drink at least once in a day.

04

Part(b) Step 2: Simplify

As given in the question,

According to Chebyshev's rule k=3for 89%of sample.

Therefore,

x-3s=25.7-35.3=9.8x+3s=25.7+35.3=41.6

Where,

x=Mean of sample

s=Standard deviation

At least 89%of the women in the sample have BMIs between 9.8 and 41.6

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