Questions: According to the American Red Cross, 10.6% of all Connecticut residents have Type B blood. A random sample of 25 Connecticut residents is taken. X the number of CT residents that have Type B blood, of the 25 sampled. What is the standard deviation of the random variable X ? √2.3691 √1.84 √2.2099 √2.229975 √2.427975 √2.269975

According to the American Red Cross, 10.6% of all Connecticut residents have Type B blood. A random sample of 25 Connecticut residents is taken.
X the number of CT residents that have Type B blood, of the 25 sampled.
What is the standard deviation of the random variable X ?
√2.3691
√1.84
√2.2099
√2.229975
√2.427975
√2.269975

Transcript text: According to the American Red Cross, $10.6 \%$ of all Connecticut residents have Type B blood. A random sample of 25 Connecticut residents is taken. $X$ the number of CT residents that bave Type B blood, of the 25 sampled. What is the standard deviation of the random variable $X$ ? $\sqrt{2.3691}$ $\sqrt{1.84}$ $\sqrt{2.2099}$ $\sqrt{2.229975}$ $\sqrt{2.427975}$ $\sqrt{2.269975}$

Solution

Solution Steps

Step 1: Define the Problem

We are tasked with finding the standard deviation of the random variable $ X $, which represents the number of Connecticut residents with Type B blood in a random sample of 25 residents. The probability of a resident having Type B blood is $ p = 0.106 $.

Step 2: Calculate the Mean

The mean $ \mu $ of a binomial distribution is given by: \[ \mu = n \cdot p \] Substituting the given values: \[ \mu = 25 \cdot 0.106 = 2.65 \]

Step 3: Calculate the Variance

The variance $ \sigma^2 $ of a binomial distribution is calculated as: \[ \sigma^2 = n \cdot p \cdot q \] where $ q = 1 - p $. Thus: \[ q = 1 - 0.106 = 0.894 \] Substituting the values: \[ \sigma^2 = 25 \cdot 0.106 \cdot 0.894 = 2.3691 \]

Step 4: Calculate the Standard Deviation

The standard deviation $ \sigma $ is the square root of the variance: \[ \sigma = \sqrt{npq} = \sqrt{2.3691} \approx 1.5392 \]