Questions: Using the 2^k >= n rule, determine the number of classes needed for the following data set sizes. a) n=75 b) n=200 c) n=1500 d) n=3000

Transcript text: Using the $2^{k} \geq n$ rule, determine the number of classes needed for the following data set sizes. a) $n=75$ b) $n=200$ c) $n=1500$ d) $n=3000$

Solution

Solution Steps

Step 1: Understand the $2^{k} \geq n$ rule

The $2^{k} \geq n$ rule is used to determine the number of classes (or bins) needed for a data set of size $n$. Here, $k$ represents the number of classes, and the rule ensures that the number of classes is sufficient to cover the data set.

Step 2: Solve for $k$ in each case

To find the number of classes $k$, we solve the inequality $2^{k} \geq n$ for each given $n$.

a) $n = 75$

We solve $2^{k} \geq 75$: \[ 2^{k} \geq 75 \] Taking the logarithm base 2 of both sides: \[ k \geq \log_{2}(75) \] Using a calculator: \[ \log_{2}(75) \approx 6.2288 \] Since $k$ must be an integer, we round up to the next whole number: \[ k = 7 \]

b) $n = 200$

We solve $2^{k} \geq 200$: \[ 2^{k} \geq 200 \] Taking the logarithm base 2 of both sides: \[ k \geq \log_{2}(200) \] Using a calculator: \[ \log_{2}(200) \approx 7.6439 \] Since $k$ must be an integer, we round up to the next whole number: \[ k = 8 \]

c) $n = 1500$

We solve $2^{k} \geq 1500$: \[ 2^{k} \geq 1500 \] Taking the logarithm base 2 of both sides: \[ k \geq \log_{2}(1500) \] Using a calculator: \[ \log_{2}(1500) \approx 10.5507 \] Since $k$ must be an integer, we round up to the next whole number: \[ k = 11 \]