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Chapter 4: Problem 23

If the temperature at which a certain compound melts is a random variable with mean value \(120^{\circ} \mathrm{C}\) and standard deviation \(2^{\circ} \mathrm{C}\), what are the mean temperature and standard deviation measured in \({ }^{\circ} \mathrm{F}\) ? [Hint: \(\left.{ }^{\circ} \mathrm{F}=1.8^{\circ} \mathrm{C}+32 .\right]\)

Short Answer

Expert verified
Mean: \(248^{\circ} \text{F}\), Standard Deviation: \(3.6^{\circ} \text{F}\).

Step by step solution

01

Convert Mean Temperature from Celsius to Fahrenheit

The formula to convert Celsius to Fahrenheit is given by \( ^{\circ} \text{F} = 1.8 \times ^{\circ} \text{C} + 32 \). To find the mean temperature in Fahrenheit, we substitute the mean value from Celsius into this formula:\[\mu_F = 1.8 \times 120 + 32\]
02

Calculate Converted Mean Temperature

Solve the equation from the previous step:\[\mu_F = 1.8 \times 120 + 32 = 216 + 32 = 248\]Thus, the mean temperature in Fahrenheit is \(248^{\circ} \text{F}\).
03

Convert Standard Deviation from Celsius to Fahrenheit

In converting standard deviation, only the scaling factor is used because standard deviation is a measure of spread, not an absolute value. So, we multiply the Celsius standard deviation by 1.8:\[\sigma_F = 1.8 \times 2 = 3.6\]
04

Present the Results

The mean temperature in Fahrenheit is \(248^{\circ} \text{F}\) and the standard deviation is \(3.6^{\circ} \text{F}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Temperature Conversion
When converting mean temperatures from Celsius to Fahrenheit, we follow a straightforward mathematical formula. This conversion is necessary because different regions use different temperature scales, and it's crucial to have consistency in experiments and data analysis. The formula to convert Celsius (\(^\circ \mathrm{C}\)) to Fahrenheit (\(^\circ \mathrm{F}\)) is:
  • \[^\circ \mathrm{F} = 1.8 \times ^\circ \mathrm{C} + 32\]
To apply this, consider a mean temperature of \(120^{\circ} \mathrm{C}\). We multiply by 1.8 and then add 32:
  • \[1.8 \times 120 + 32 = 216 + 32\]
  • This yields \(248^{\circ} \mathrm{F}\)
Therefore, when expressing the mean in Fahrenheit, you observe a notable increase caused by the conversion formula. Understanding this mathematical relationship is key when working with different units or analyzing global data sets.
Standard Deviation Conversion
Converting standard deviation from Celsius to Fahrenheit involves a slight variance from mean conversion. Instead of applying the full formula used for mean, we focus on just the scaling factor, 1.8. This is because standard deviation represents data spread rather than an exact temperature. For example, given a standard deviation of \(2^{\circ} \mathrm{C}\), we multiply by 1.8:
  • \[\sigma_F = 1.8 \times 2 = 3.6\]
It's crucial to note that when converting standard deviation, the additive constant (32 in the mean conversion formula) is not used. The reason is simple: standard deviation doesn't derive from specific degrees but measures variation or dispersion in temperatures. Thus, the conversion results in \(3.6^{\circ} \mathrm{F}\) for the standard deviation, reflecting just the scale adjustment when shifting from Celsius to Fahrenheit. This ensures consistent interpretation of data variability regardless of the temperature scale used.
Fahrenheit and Celsius
The Fahrenheit and Celsius scales are two primary ways of measuring temperature in daily life or scientific research. Celsius, commonly used around the world, is part of the metric system and is based on the freezing and boiling points of water:
  • 0掳C is the freezing point of water.
  • 100掳C is the boiling point at standard atmospheric pressure.
On the other hand, Fahrenheit is more prevalent in the United States. In this scale:
  • 32掳F marks the freezing point of water.
  • 212掳F denotes the boiling point of water under the same conditions.
The distinct differences between these scales lead to the necessity of conversions for accurate temperature communication across regions that use different systems. Understanding both scales, how they compare, and how to switch between them is vital for anyone involved in scientific work or global discussions on climate and weather. This is because temperature plays a significant role in experiments, forecasts, and analyses of environmental phenomena.

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