Question

Given μ=5
and σ=0.5
, find the probability that a random variable, x, is between 3.6 and 6.1.(1 point)
Responses

−98.4%
negative 98.4 percent

98.6%
98.6%

98.4%
98.4%

1.4%
1.4%

Answer 1

To calculate the probability that a random variable \( x \) is between 3.6 and 6.1, given that the mean \( \mu = 5 \) and standard deviation \( \sigma = 0.5 \), we can standardize the variable using the Z-score formula:

\[
Z = \frac{X - \mu}{\sigma}
\]

1. **Calculate the Z-scores for 3.6 and 6.1:**

- For \( x = 3.6 \):
\[
Z_{3.6} = \frac{3.6 - 5}{0.5} = \frac{-1.4}{0.5} = -2.8
\]

- For \( x = 6.1 \):
\[
Z_{6.1} = \frac{6.1 - 5}{0.5} = \frac{1.1}{0.5} = 2.2
\]

2. **Find the probabilities corresponding to these Z-scores using the standard normal distribution table or a calculator:**

- The probability for \( Z = -2.8 \):
\[
P(Z < -2.8) \approx 0.0026
\] (This value can vary slightly depending on the table used, but it's around this ballpark.)

- The probability for \( Z = 2.2 \):
\[
P(Z < 2.2) \approx 0.9861
\]

3. **Calculate the probability that \( x \) is between 3.6 and 6.1:**
\[
P(3.6 < X < 6.1) = P(Z < 2.2) - P(Z < -2.8) = 0.9861 - 0.0026 = 0.9835
\]

4. **Convert the probability to percentage:**
\[
0.9835 \times 100 \approx 98.35\%
\]

So rounding gives us \( 98.4\% \).

The final response is:

**98.4%**