Assume that the readings on the thermometers are normally distributed with a mean of 0 degrees and standard deviation of 1.00 degrees C. A thermometer is randomly selected and tested. Draw a sketch and find the temperature reading corresponding to p subscript 99, the percentile. This is the temperature reading separating the bottom 99% from the top 1%
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To find the temperature reading corresponding to p99, the 99th percentile, we can use the standard normal distribution table or a calculator with a normal distribution function. Let's go through the steps to obtain the temperature reading.
Step 1: Draw a Sketch of the Standard Normal Distribution
Start by drawing a graph that represents a standard normal distribution. The x-axis represents the temperature readings, and the y-axis represents the probability density.
Step 2: Identify the Area under the Curve
Looking at the problem statement, we are interested in finding the temperature reading that separates the bottom 99% (p ≤ 0.99) from the top 1% (p > 0.99). This means we need to find the z-score that corresponds to an area of 0.99 under the curve.
Step 3: Convert Percentile to Z-Score
Using a standard normal distribution table or a calculator, find the z-score that corresponds to a cumulative probability (area under the curve) of 0.99. In this case, the area to the left of the z-score is 0.99.
Step 4: Calculate the Temperature Reading
The z-score we obtained in the previous step corresponds to the number of standard deviations away from the mean. Since the mean is 0 degrees and the standard deviation is 1.00 degrees C, we can multiply the z-score by 1.00 to obtain the temperature reading corresponding to the desired percentile.
For example, if the z-score is 2.33, we can calculate the temperature reading as:
Temperature Reading = Mean + (Z-Score × Standard Deviation)
Temperature Reading = 0 + (2.33 × 1.00)
Temperature Reading = 2.33 degrees Celsius
Therefore, the temperature reading corresponding to p99, the 99th percentile, is 2.33 degrees Celsius.