Solved on Jan 21, 2024

Find the probability that a randomly selected thermometer reading is less than $-1.507$ Celsius, given the readings are normally distributed with $\mu = 0$ Celsius and $\sigma = 1.00$ Celsius.

STEP 1

Assumptions
1. The readings at freezing on a bundle of thermometers are normally distributed.
2. The mean of the distribution is $\mu = 0$ Celsius.
3. The standard deviation of the distribution is $\sigma = 1.00$ Celsius.
4. We are looking for the probability of a single reading being less than $-1.507$ Celsius.

STEP 2

To find the probability of a reading being less than $-1.507$ Celsius, we need to standardize this value using the Z-score formula. The Z-score is the number of standard deviations a data point is from the mean.
$Z = \frac{X - \mu}{\sigma}$
Where $X$ is the value from the normal distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation.

STEP 3

Now, plug in the values for $X$ , $\mu$ , and $\sigma$ to calculate the Z-score for $-1.507$ Celsius.
$Z = \frac{-1.507 - 0}{1.00}$

STEP 4

Calculate the Z-score.
$Z = \frac{-1.507}{1.00} = -1.507$

STEP 5

Now that we have the Z-score, we can find the probability of a reading being less than $-1.507$ Celsius by looking up this Z-score in the standard normal distribution table, which gives the probability that a standard normal variable is less than $Z$ .

STEP 6

If a standard normal distribution table is not available, we can use a calculator or software that provides the cumulative distribution function (CDF) for the standard normal distribution. The CDF will give us the probability that a standard normal variable is less than or equal to our Z-score.

STEP 7

Using the CDF of the standard normal distribution, we find the probability corresponding to the Z-score of $-1.507$ .

STEP 8

The probability of reading less than $-1.507$ Celsius is the value obtained from the CDF of the standard normal distribution at $Z = -1.507$ .
Let's denote this probability as $P(Z < -1.507)$ .

STEP 9

After finding the probability from the CDF or the standard normal distribution table, we have our final answer.
The probability of a thermometer reading less than $-1.507$ Celsius is $P(Z < -1.507)$ .

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Find the probability that a randomly selected thermometer reading is less than $-1.507$ Celsius, given the readings are normally distributed with $\mu = 0$ Celsius and $\sigma = 1.00$ Celsius.

STEP 1

STEP 2

STEP 3

Now, plug in the values for $X$ , $\mu$ , and $\sigma$ to calculate the Z-score for $-1.507$ Celsius.
$Z = \frac{-1.507 - 0}{1.00}$

STEP 4

Calculate the Z-score.
$Z = \frac{-1.507}{1.00} = -1.507$

STEP 5

Now that we have the Z-score, we can find the probability of a reading being less than $-1.507$ Celsius by looking up this Z-score in the standard normal distribution table, which gives the probability that a standard normal variable is less than $Z$ .

STEP 6

If a standard normal distribution table is not available, we can use a calculator or software that provides the cumulative distribution function (CDF) for the standard normal distribution. The CDF will give us the probability that a standard normal variable is less than or equal to our Z-score.

STEP 7

Using the CDF of the standard normal distribution, we find the probability corresponding to the Z-score of $-1.507$ .

STEP 8

The probability of reading less than $-1.507$ Celsius is the value obtained from the CDF of the standard normal distribution at $Z = -1.507$ .
Let's denote this probability as $P(Z < -1.507)$ .

STEP 9

After finding the probability from the CDF or the standard normal distribution table, we have our final answer.
The probability of a thermometer reading less than $-1.507$ Celsius is $P(Z < -1.507)$ .

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Find the probability that a randomly selected thermometer reading is less than −1.507-1.507−1.507 Celsius, given the readings are normally distributed with μ=0\mu = 0μ=0 Celsius and σ=1.00\sigma = 1.00σ=1.00 Celsius.

STEP 1

STEP 2

STEP 3

Now, plug in the values for XXX, μ\muμ, and σ\sigmaσ to calculate the Z-score for −1.507-1.507−1.507 Celsius.Z=−1.507−01.00Z = \frac{-1.507 - 0}{1.00}Z=1.00−1.507−0​

STEP 4

Calculate the Z-score.Z=−1.5071.00=−1.507Z = \frac{-1.507}{1.00} = -1.507Z=1.00−1.507​=−1.507

STEP 5

Now that we have the Z-score, we can find the probability of a reading being less than −1.507-1.507−1.507 Celsius by looking up this Z-score in the standard normal distribution table, which gives the probability that a standard normal variable is less than ZZZ.

STEP 6

If a standard normal distribution table is not available, we can use a calculator or software that provides the cumulative distribution function (CDF) for the standard normal distribution. The CDF will give us the probability that a standard normal variable is less than or equal to our Z-score.

STEP 7

Using the CDF of the standard normal distribution, we find the probability corresponding to the Z-score of −1.507-1.507−1.507.

STEP 8

The probability of reading less than −1.507-1.507−1.507 Celsius is the value obtained from the CDF of the standard normal distribution at Z=−1.507Z = -1.507Z=−1.507.Let's denote this probability as P(Z<−1.507)P(Z < -1.507)P(Z<−1.507).

STEP 9

After finding the probability from the CDF or the standard normal distribution table, we have our final answer.The probability of a thermometer reading less than −1.507-1.507−1.507 Celsius is P(Z<−1.507)P(Z < -1.507)P(Z<−1.507).

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the probability that a randomly selected thermometer reading is less than $-1.507$ Celsius, given the readings are normally distributed with $\mu = 0$ Celsius and $\sigma = 1.00$ Celsius.

Now, plug in the values for $X$ , $\mu$ , and $\sigma$ to calculate the Z-score for $-1.507$ Celsius.
$Z = \frac{-1.507 - 0}{1.00}$

Calculate the Z-score.
$Z = \frac{-1.507}{1.00} = -1.507$

Now that we have the Z-score, we can find the probability of a reading being less than $-1.507$ Celsius by looking up this Z-score in the standard normal distribution table, which gives the probability that a standard normal variable is less than $Z$ .

Using the CDF of the standard normal distribution, we find the probability corresponding to the Z-score of $-1.507$ .

The probability of reading less than $-1.507$ Celsius is the value obtained from the CDF of the standard normal distribution at $Z = -1.507$ .
Let's denote this probability as $P(Z < -1.507)$ .

After finding the probability from the CDF or the standard normal distribution table, we have our final answer.
The probability of a thermometer reading less than $-1.507$ Celsius is $P(Z < -1.507)$ .