Question

Math

Posted 4 months ago

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A fuel pump at a gasoline station doesn't always dispense the exact amount displayed on the meter. When the meter reads $1.000 \mathrm{~L}$, the amount of fuel a certain pump dispenses is normally distributed with a mean of $1 \mathrm{~L}$ and standard deviation of $0.05 \mathrm{~L}$. Let $X=$ the amount dispensed in a random trial when the meter reads $1.000 \mathrm{~L}$.
Find $P(X<1)$.
You may round your answer to two decimal places.
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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

To find $P(X < 1)$ for a normally distributed variable, we need to standardize the variable using the Z-score formula

step 2

The Z-score is calculated by the formula $Z = \frac{X - \mu}{\sigma}$, where $X$ is the value of interest, $\mu$ is the mean, and $\sigma$ is the standard deviation

step 3

Substituting the given values into the Z-score formula gives us $Z = \frac{1 - 1}{0.05} = \frac{0}{0.05} = 0$

step 4

However, we need the Z-score for $X < 1$, so we calculate $Z = \frac{1 - 1}{0.05} = \frac{0}{0.05} = 0$. This is the Z-score for the exact mean, so we need to look at the Z-score for a value just less than 1

step 5

Since the Z-score for $X = 1$ is 0, and we want $P(X < 1)$, we look up the Z-score for a value just less than 0 in the standard normal distribution table

step 6

The Z-score for values less than 0 in the standard normal distribution table corresponds to a probability of 0.5

step 7

Therefore, $P(X < 1) = 0.5$

Answer

$P(X < 1) = 0.5$

Key Concept

Standardizing a Normal Distribution

Explanation

To find probabilities related to normal distributions, we standardize the variable to a Z-score and use the standard normal distribution table to find the corresponding probability

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