 .  Assume that the readings at freezing on a bundle 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 less than -0.859°C.

To solve this problem, we need to find the probability that a normally distributed thermometer reading is less than \(-0.859^\circ\text{C}\). We are given that the distribution has a mean \(\mu = 0^\circ\text{C}\) and a standard deviation \(\sigma = 1^\circ\text{C}\).

The formula for calculating the probability in a normal distribution is based on finding the corresponding **z-score**, which tells us how many standard deviations the value is away from the mean.

The z-score is calculated by the formula:

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

Where:
- \(x\) is the value for which we want to find the probability (in this case, \(-0.859^\circ\text{C}\)),
- \(\mu\) is the mean (0°C),
- \(\sigma\) is the standard deviation (1°C).

### Step 1: Calculate the z-score
Substitute the values into the formula:

\[
z = \frac{-0.859 - 0}{1} = -0.859
\]

### Step 2: Find the probability
Now, we need to find the probability that a standard normal variable \(Z\) is less than \(-0.859\). This is denoted as:

\[
P(Z < -0.859)
\]

Using a standard normal distribution table (or a calculator), we find the cumulative probability corresponding to \(z = -0.859\).

From the table, \(P(Z < -0.859) \approx 0.1956\).

### Final Answer:
The probability of obtaining a reading less than \(-0.859^\circ\text{C}\) is approximately **0.1956** or **19.56%**.

---

Would you like more details on how to read the z-score table or further clarification on this topic?

Here are five related questions you might explore:
1. What is the probability of obtaining a reading greater than 1.5°C?
2. How would the result change if the standard deviation was 0.5°C instead of 1°C?
3. What is the probability of a reading being between -1°C and 1°C?
4. How do we find the z-score if we know the probability and need the value?
5. What does a z-score of 0 represent in this context?

**Tip:** For faster calculations, a cumulative distribution function (CDF) calculator can be used to find probabilities for z-scores directly.

Assume that the readings at freezing on a bundle 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 less than -0.859°C.

Learn how to calculate the probability of obtaining a reading less than -0.859°C from a normally distributed thermometer with a mean of 0°C and a standard deviation of 1°C. This detailed guide walks you through the steps of calculating the z-score and finding the probability using the standard normal distribution. Perfect for students in Grades 11-12 or introductory college statistics.

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