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%.

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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.