Question

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 $10,215$women who drink at least one caffeinated carbonated beverage a day are $25.7$and $5.3$, respectively.
a. At least how many women in the sample have a BMI of between $15.1$and $36.3$ ?
b. Fill in the blanks; At least $89\%$ of the women in the sample have BMIs between ________ and _______.

Short answer

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

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

Step-by-step solution

Part(a) Step 1: Given information

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

Part(a) Step 2: Simplify

According to Chebyshev's rule

$$\begin{gathered} \overline{x}-ks=15.1 \\ 25.7-k\left(5.3\right)=15.1 \\ k=2 \end{gathered}$$

and also,

$$\begin{gathered} \overline{x}+ks=36.3 \\ 25.7+k\left(5.3\right)=36.3 \\ k=2 \end{gathered}$$

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

Therefore,

$10215\times \frac{75}{100}=7661.25\approx 7661$

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

Part(b) Step 1: Given information

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

Part(b) Step 2: Simplify

As given in the question,

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

Therefore,

$$\begin{gathered} \overline{x}-3s=25.7-3\left(5.3\right)=9.8 \\ \overline{x}+3s=25.7+3\left(5.3\right)=41.6 \end{gathered}$$

Where,

$\overline{x}=$Mean of sample

$s=$Standard deviation

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