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Assume that the readings at freezing on a bundle 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 2.4% of the thermometers are rejected because they have readings that are too high and another 2.4% 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. (Round answers to three decimal places when needed)

interval of acceptable thermometer readings = _________ ºC

User Antagony
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Final answer:

The cutoff values separating the rejected thermometers from the others are approximately -1.96°C and 1.96°C, using the z-scores of -1.96 and 1.96 for the lower and upper cutoff points, respectively, and considering the normal distribution's mean of 0°C and standard deviation of 1.00°C.

Step-by-step explanation:

To find the cutoff values that separate the rejected thermometers from the others, we will use the concept of normal distribution. Given that 2.4% of the thermometers are rejected for readings that are too low and another 2.4% for readings that are too high, we need to find the z-scores that correspond to these tail areas.

The area to the left of the lower cutoff point is 0.024, and the area to the right of the upper cutoff point (which is the same as the area to the left since the normal distribution is symmetric) is 1 - 0.024 = 0.976. Using a z-table or calculator, we can find the z-scores that correspond to these areas.

The z-score that corresponds to the area of 0.024 is approximately -1.96, and the z-score for the area of 0.976 is approximately 1.96. We can then convert these z-scores to actual temperature readings using the given mean (0°C) and standard deviation (1.00°C).

Lower cutoff: Mean + (z-score * Standard deviation) = 0 + (-1.96 * 1) = -1.96°C
Upper cutoff: Mean + (z-score * Standard deviation) = 0 + (1.96 * 1) = 1.96°C

The interval of acceptable thermometer readings is therefore from -1.96°C to 1.96°C.

User Attila
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