$$ n=18,8=4 $$ find the area of the following a.) $$ P(x35) $$ c.) $$ P(21

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$$ n=18,8=4 $$ find the area of the following a.) $$ P(x<21) $$ b.) $$ P(x>35) $$ c.) $$ P(21

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To solve the given probability problems, we'll assume that the variable $$ x $$ follows a **normal distribution**. Based on the provided information: - **Mean ($$ \mu $$)**: 18 - **Standard Deviation ($$ \sigma $$)**: 4 We'll use the **standard normal distribution (Z-score)** formula to convert the $$ x $$-values to $$ z $$-scores: $$ z = \frac{x - \mu}{\sigma} $$ Once we have the $$ z $$-scores, we'll use standard normal distribution tables or a calculator to find the corresponding probabilities. ### a.) $$ P(x < 21) $$ 1. **Calculate the Z-score:** $$ z = \frac{21 - 18}{4} = \frac{3}{4} = 0.75 $$ 2. **Find the Probability:** Using a standard normal distribution table or calculator: $$ P(z < 0.75) \approx 0.7734 \text{ or } 77.34\% $$ ### b.) $$ P(x > 35) $$ 1. **Calculate the Z-score:** $$ z = \frac{35 - 18}{4} = \frac{17}{4} = 4.25 $$ 2. **Find the Probability:** A $$ z $$-score of 4.25 is extremely high, and the probability beyond this point is negligible. $$ P(z > 4.25) \approx 0.00001 \text{ or } 0.001\% $$ ### c.) $$ P(21 < x < 35) $$ 1. **Calculate the Z-scores:** - For $$ x = 21 $$: $$ z_1 = \frac{21 - 18}{4} = 0.75 $$ - For $$ x = 35 $$: $$ z_2 = \frac{35 - 18}{4} = 4.25 $$ 2. **Find the Probabilities:** $$ P(z < 4.25) \approx 0.99999 $$ $$ P(z < 0.75) \approx 0.7734 $$ 3. **Calculate the Area Between 21 and 35:** $$ P(21 < x < 35) = P(z < 4.25) - P(z < 0.75) \approx 0.99999 - 0.7734 = 0.2266 \text{ or } 22.66\% $$ ### **Summary of Results:** - **a.)** $$ P(x < 21) \approx 77.34\% $$ - **b.)** $$ P(x > 35) \approx 0.001\% $$ - **c.)** $$ P(21 < x < 35) \approx 22.66\% $$ - **a.)** The probability that $$ x $$ is less than 21 is approximately 77.34%. - **b.)** The probability that $$ x $$ is greater than 35 is approximately 0.001%. - **c.)** The probability that $$ x $$ is between 21 and 35 is approximately 22.66%.

find the area of the following
a.)
b.)
c.)

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