The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at
the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some
give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive
numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the
frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and
tested. Find the temperature reading corresponding to the given information.
4) Find P40, the 40th percentile.
The temperature reading corresponding to the given information is -0.67°C. P40 is the 40th percentile, which is the temperature reading that corresponds to the 40th percentile of the normal distribution. This value is -0.67°C.
To find the temperature reading corresponding to the 40th percentile (P40), we need to use the standard normal distribution table or a calculator with a built-in function for finding percentiles.
Here's how to calculate P40:
1. Convert the given mean and standard deviation into a standard normal distribution by using the formula:
z = (x - μ) / σ
Where:
z = the z-score
x = the temperature reading (unknown)
μ = mean of the distribution (0°C)
σ = standard deviation of the distribution (1.00°C)
2. Once you have the z-score, refer to the standard normal distribution table or use a calculator to find the corresponding percentile. In this case, we are interested in finding the 40th percentile.
3. Locate the z-score in the table or use the calculator function to find the area under the standard normal curve to the left of the z-score. This area represents the percentile.
4. Using the table or calculator function, find the temperature reading (x) that corresponds to the area/percentile found in the previous step.
By following these steps, you will be able to find the temperature reading corresponding to the given information.
To find the temperature reading corresponding to the 40th percentile (P40), we need to find the z-score associated with the 40th percentile and then use that z-score to find the temperature reading.
The z-score represents the number of standard deviations a data point is from the mean. We can use the standard normal distribution table or a calculator to find the z-score corresponding to the 40th percentile.
From the table or calculator, find the z-score that corresponds to a cumulative probability of 0.40 (40th percentile). Let's assume that the z-score is denoted as z_40.
Once we have the z_40 value, we can use the formula for converting a z-score to a raw score (temperature reading):
Temperature reading = mean + (z-score * standard deviation)
Since the mean reading is 0°C and the standard deviation is 1.00°C, the formula becomes:
Temperature reading = 0 + (z_40 * 1.00)
Now you can substitute the value of z_40 that you obtained from the table or calculator, and calculate the temperature reading.