Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find the probability of obtaining a reading between 0.907°C and 2.994°C.

To find the probability of obtaining a reading between 0.907°C and 2.994°C, we will use the properties of the **normal distribution**.

The readings follow a normal distribution with:
- Mean (\(\mu\)) = 0°C
- Standard deviation (\(\sigma\)) = 1°C

The probability of obtaining a reading between 0.907°C and 2.994°C is equivalent to finding the area under the normal curve between these two values.

### Step 1: Convert the temperatures to **z-scores**.
The formula for the z-score is:

\[
z = \frac{x - \mu}{\sigma}
\]

- For \(x = 0.907^\circ C\):

\[
z_1 = \frac{0.907 - 0}{1} = 0.907
\]

- For \(x = 2.994^\circ C\):

\[
z_2 = \frac{2.994 - 0}{1} = 2.994
\]

### Step 2: Find the probabilities corresponding to these z-scores.
We can use a standard normal table or a calculator to find the probabilities for these z-scores.

- The probability corresponding to \(z_1 = 0.907\) is approximately **0.8176**.
- The probability corresponding to \(z_2 = 2.994\) is approximately **0.9986**.

### Step 3: Calculate the probability between the two z-scores.
The probability of obtaining a reading between 0.907°C and 2.994°C is the difference between the two probabilities:

\[
P(0.907 < X < 2.994) = P(z_2) - P(z_1) = 0.9986 - 0.8176 = 0.1810
\]

### Final Answer:
The probability of obtaining a reading between 0.907°C and 2.994°C is **0.1810** or **18.10%**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find probabilities for values below a certain z-score?
2. What is the probability of obtaining a reading less than 0°C in this scenario?
3. How do we calculate probabilities for non-standard normal distributions?
4. How does changing the mean and standard deviation affect the probability?
5. What is the probability of obtaining a reading above 3°C?

**Tip**: The total area under a normal distribution curve is always 1, representing a 100% probability for all possible outcomes.

Find the probability of obtaining a thermometer reading between 0.907°C and 2.994°C using the normal distribution. Learn to convert values to z-scores and calculate the area under the curve. Ideal for high school students studying statistics.

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