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 -2.443°C and -0.119°C.

 

To find the probability of obtaining a thermometer reading between \(-2.443^\circ C\) and \(-0.119^\circ C\), given that the readings are normally distributed with a mean \(\mu = 0^\circ C\) and a standard deviation \(\sigma = 1^\circ C\), we can use the standard normal distribution (z-distribution).

### Steps:

1. **Convert the values to z-scores**:
   The z-score for a value \(x\) in a normal distribution is calculated using the formula:
   \[
   z = \frac{x - \mu}{\sigma}
   \]
   - For \(x_1 = -2.443^\circ C\):
   \[
   z_1 = \frac{-2.443 - 0}{1} = -2.443
   \]
   - For \(x_2 = -0.119^\circ C\):
   \[
   z_2 = \frac{-0.119 - 0}{1} = -0.119
   \]

2. **Find the cumulative probabilities for each z-score**:
   Use standard normal distribution tables or a calculator to find the cumulative probabilities for each z-score:
   - \(P(z_1 = -2.443)\)
   - \(P(z_2 = -0.119)\)

3. **Subtract the cumulative probabilities**:
   The probability of obtaining a reading between \(-2.443^\circ C\) and \(-0.119^\circ C\) is:
   \[
   P(-2.443 < z < -0.119) = P(z_2) - P(z_1)
   \]

Let's compute the exact values.

I will calculate the cumulative probabilities for both z-scores.The probability of obtaining a thermometer reading between \(-2.443^\circ C\) and \(-0.119^\circ C\) is approximately \(0.445\), or 44.5%.

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions you might explore:
1. What is the probability of a reading being below \(-0.119^\circ C\)?
2. How do you find probabilities outside of a given range, such as above \(1^\circ C\)?
3. How does the standard deviation affect the shape of the normal distribution?
4. What is the probability of obtaining a reading above \(1^\circ C\)?
5. How can z-scores be used to compare different datasets?

**Tip**: When working with normal distributions, always convert raw scores into z-scores to standardize them for easy probability lookup.

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 -2.443°C and -0.119°C.

Discover how to find the probability of obtaining a thermometer reading between -2.443°C and -0.119°C using the standard normal distribution. This problem involves calculating z-scores and using probability tables. Suitable for students in Grades 10-12 studying statistics.

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