A soft drink machine dispenses soft drink into cups.  Let the amount of liquid dispensed into a small drink (in ml) be given by the random variable X where X ~ N(150, 32).


What is the probability that a randomly selected small drink, has between 147 and 153 ml? (Please answer correct to 4dp).



We are given that the amount of liquid dispensed by the machine follows a normal distribution \(X \sim N(150, 32)\), where:

- The mean \(\mu = 150\) ml
- The variance \(\sigma^2 = 32\), so the standard deviation \(\sigma = \sqrt{32} = 5.6569\) ml.

We need to find the probability that the amount of liquid dispensed is between 147 and 153 ml:

\[
P(147 \leq X \leq 153)
\]

### Step 1: Convert to the standard normal distribution
We will standardize the normal variable \(X\) using the formula for the z-score:
\[
Z = \frac{X - \mu}{\sigma}
\]

For \(X = 147\):
\[
Z_1 = \frac{147 - 150}{5.6569} = \frac{-3}{5.6569} \approx -0.5303
\]

For \(X = 153\):
\[
Z_2 = \frac{153 - 150}{5.6569} = \frac{3}{5.6569} \approx 0.5303
\]

### Step 2: Find the probability using the standard normal distribution
Now we need to find the probability that \(Z\) is between \(-0.5303\) and \(0.5303\):

\[
P(-0.5303 \leq Z \leq 0.5303)
\]

Using the standard normal distribution tables or a calculator, we find the cumulative probabilities:

\[
P(Z \leq 0.5303) \approx 0.7019
\]
\[
P(Z \leq -0.5303) \approx 0.2981
\]

### Step 3: Compute the final probability
The probability that \(X\) is between 147 ml and 153 ml is:

\[
P(147 \leq X \leq 153) = P(Z \leq 0.5303) - P(Z \leq -0.5303) = 0.7019 - 0.2981 = 0.4038
\]

Thus, the probability that a randomly selected small drink has between 147 and 153 ml is **0.4038**.

Would you like more details on this process or have any further questions?

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Here are some related questions:
1. What is the probability that a small drink has less than 147 ml?
2. How would the probability change if the standard deviation was 4 ml instead?
3. What is the probability that a small drink has more than 153 ml?
4. How can you calculate the expected number of drinks between 147 and 153 ml in 1000 samples?
5. What percentage of drinks will contain more than 150 ml?

**Tip:** The z-score is a very useful tool for comparing data points from different normal distributions by standardizing them.

A soft drink machine dispenses soft drink into cups. Let the amount of liquid dispensed into a small drink (in ml) be given by the random variable X where X ~ N(150, 32). What is the probability that a randomly selected small drink has between 147 and 153 ml? (Please answer correct to 4dp).

Learn how to calculate the probability that a randomly selected small drink contains between 147 and 153 ml using a normal distribution with a mean of 150 ml and a standard deviation of 5.6569 ml. This step-by-step guide uses the Z-score formula and standard normal distribution table. Suitable for Grades 10-12.

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