Solved on Jan 21, 2024

Find the probability that a randomly selected thermometer reading is less than 1.507-1.507 Celsius, given the readings are normally distributed with μ=0\mu = 0 Celsius and σ=1.00\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 μ=0\mu = 0 Celsius.
3. The standard deviation of the distribution is σ=1.00\sigma = 1.00 Celsius.
4. We are looking for the probability of a single reading being less than 1.507-1.507 Celsius.

STEP 2

To find the probability of a reading being less than 1.507-1.507 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=XμσZ = \frac{X - \mu}{\sigma}
Where XX is the value from the normal distribution, μ\mu is the mean, and σ\sigma is the standard deviation.

STEP 3

Now, plug in the values for XX, μ\mu, and σ\sigma to calculate the Z-score for 1.507-1.507 Celsius.
Z=1.50701.00Z = \frac{-1.507 - 0}{1.00}

STEP 4

Calculate the Z-score.
Z=1.5071.00=1.507Z = \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 1.507-1.507 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 ZZ.

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.

STEP 8

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

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