The medicinal thermometer commonly used in homes can be read ±0.1^∘ F, whereas those in the doct

step 1 So this question is a percent error calculation question. So they give you the error associated

The medicinal thermometer commonly used in homes can be read...

The medicinal thermometer commonly used in homes can be read $\pm 0.1^{\circ} \mathrm{F}$, whereas those in the doctor"s office may be accurate to $\pm 0.1^{\circ} \mathrm{C}$. In degrees Celsius, express the percent error expected from each of these thermometers in measuring a person's body temperature of $38.9^{\circ} \mathrm{C}$.

Key Concepts

Unit Conversion

This concept involves converting measurements from one unit system to another. For temperatures, the conversion between Fahrenheit and Celsius is done using the linear relation C = (F - 32)/1.8. This conversion is important not only for transforming temperature values but also for appropriately scaling any measurement uncertainty from one unit to another.

Measurement Uncertainty

Measurement uncertainty refers to the range within which the true value of a measurement is expected to lie, indicated by a ± value. It highlights the precision of the instrument employed and sets the stage for identifying how much the measured value might deviate from the actual value.

Relative Error (Percentage Error)

Relative error, often expressed as a percentage, is calculated by taking the absolute error (the uncertainty) and dividing it by the actual or measured value, then multiplying the result by 100. This concept allows one to assess the error's significance in relation to the size of the measurement, providing a normalized measure of accuracy.

Linear Error Propagation

In measurements involving linear transformations, such as converting temperature scales, the propagation of error is straightforward—the uncertainty in the output is the product of the input uncertainty and the absolute value of the slope of the conversion formula. This ensures that errors are correctly scaled when the measurement is transformed between units.

Transcript

00:01 So this question is a percent error calculation question.

00:05 So they give you the error associated with medicinal thermometers using homes, which is 0 .1 degrees fahrenheit.

00:11 And then they give you the error associated with thermometers using the doctor's office, which is 0 .1 degrees celsius.

00:19 And they want you to calculate the person error with respect to a person's body temperature who is 38 .9 degrees celsius.

00:27 So we're going to have to do two percent error calculations here, one for fahrenheit and one for celsius, or one for the home and then one for the doctor's office.

00:36 So first what we have to do is convert the body temperature of 38 .9 degrees celsius to fahrenheit so that we can do the calculation much easier.

00:45 So let's write out our degrees fahrenheit to, or degrees celsius degrees fahrenheit formula.

00:53 So degrees fahrenheit equals 9 over 5 degrees celsius, whatever number that is, plus 32.

01:01 And if you plug in 38 .9, you should get 102 .02 degrees fahrenheit.

01:11 Okay? so that's our value right here.

01:15 Now let's go over what percent error is.

01:18 So percent error, let me write out the equation.

01:20 Percent error is equal to the absolute value of the actual observed value, so va, minus the expected value ve over the expected value, again, ve.

01:43 Okay? are we going to do this times 100 % because we want to get a percentage here.

01:49 So basically, percent error is the percentage off of your expected value that your actual value was.

01:57 So how off is your actual value from your expected value? so if it's a high percent error, then you know your actual value is very different from your expected value.

02:06 Okay.

02:08 So we basically have all the information we need for this because they give you the error.

02:13 Okay, so they give you the error already, which is like 0 .1 degrees fahrenheit and then 0 .1 degrees celsius.

02:20 So that actually, this whole top part of this equation, actual minus expected, is just the error, is what they've given us already.

02:32 Because that is how much it's going to deviate, right? the medicinal thermometer in like homes is accurate to zero to plus minus 0 .1 degrees fahrenheit.

02:42 So that's how much it'll deviate from the actual value.

02:45 So we can just plug that in for our top part and we can find the percent error...

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