Question

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.
If 2.5% of the thermometers are rejected because they have readings that are too high and another 2.5% are rejected because they have readings that are too low, find the two readings that are cutoff values separating the rejected thermometers from the others.

Answer 1

Using the normal distribution, it is found that the two readings that are cutoff values separating the rejected thermometers from the others are -1.96ºC and 1.96ºC.

Normal Probability Distribution

The z-score of a measure X of a normally distributed variable with mean $\mu$ and standard deviation $\sigma$ is given by:

$Z = \frac{X - \mu}{\sigma}$

• The z-score measures how many standard deviations the measure is above or below the mean.

• Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.

The mean and the standard deviation are given, respectively, by:

$\mu = 0, \sigma = 1$.

The z-score that cuts off the bottom and top 2.5% of the distribution is $z = \pm 1.96$, hence:

$Z = \frac{X - \mu}{\sigma}$

$-1.96 = \frac{X - 0}{1}$

X = -1.96

$Z = \frac{X - \mu}{\sigma}$

$1.96 = \frac{X - 0}{1}$

X = 1.96

The two readings that are cutoff values separating the rejected thermometers from the others are -1.96ºC and 1.96ºC.

More can be learned about the normal distribution at brainly.com/question/27879230

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