Question

​​​​​​​ Given a normally distributed random variable, \( X \), with \( \mu=222.3 \) and \( \sigma=15.7 \), find the probability \( \boldsymbol{x} \) will fall between \( 198.4 \) and 218.3. Round your answer

Answer 1

The probability that X falls between 198.4 and 218.3 is approximately

0.334, rounded to three decimal places.

Step-by-step explanation:

In a normal distribution with μ=222.3 and σ=15.7, to find the probability X falls between 198.4 and 218.3, we use the z-score formula:

Z= x−μ / σ

where x is the value, μ is the mean, and σ is the standard deviation.

x=198.4

Z1 = 198.4−222.3 / 15.7 =−1.521

For x=218.3:

Z2 = 218.3−222.3 / 15.7=−0.254

Using a standard normal distribution table or calculator, find the area between these Z scores:P(−1.521<Z<−0.254) From the standard normal distribution table, the area for Z=−1.521 is approximately 0.0643 and for Z=−0.254 is approximately 0.3980.

Thus, the probability X falls between 198.4 and 218.3 is the difference between these areas:0.3980−0.0643=0.3337

Rounded to three decimal places, the probability is 0.334.