Question

consider the standard normal distribution for the following question. what is the percentage of the data with z - scores between - 2 and 2?
a 90.83%
b 98.18%
c 95.44%
d 92.20%

Explanation:

Step1: Recall the empirical rule for standard normal distribution

In a standard normal distribution (mean = 0, standard deviation = 1), the empirical rule (also known as the 68 - 95 - 99.7 rule) states that:

• Approximately 68% of the data lies within \( z = - 1\) and \( z=1\)
• Approximately 95% of the data lies within \( z=-2\) and \( z = 2\)
• Approximately 99.7% of the data lies within \( z=-3\) and \( z = 3\)

Mathematically, we can also calculate the probability \( P(-2<Z < 2)\) using the properties of the normal distribution. We know that \( P(-2<Z < 2)=P(Z < 2)-P(Z < - 2)\)

From the standard normal table (or z - table), the value of \( P(Z < 2)\approx0.9772\) and \( P(Z < - 2)=1 - P(Z < 2)=1 - 0.9772 = 0.0228\)

Then \( P(-2<Z < 2)=0.9772-0.0228 = 0.9544\) or 95.44%

Step2: Match with the given options

Looking at the options, option c has the value 95.44%

Answer:

c. 95.44%