Question
Daniela recorded the low temperatures during the school day last week and this week. Her results are shown in the table below.
Low Temperatures during the School Day This Week and Last Week
Low Temperatures
This Week (Degrees Fahrenheit)
42
38
45
46
39
Low Temperatures
Last Week (Degrees Fahrenheit)
56
58
52
62
62
Daniela used the steps below to find a relationship between the difference in the mean temperatures and the mean absolute deviations of the data sets.
This Week Last Week
Step 1
Find the mean.
StartFraction 42 + 38 + 45 + 46 + 39 over 5 EndFraction = 42
Find the mean.
StartFraction 56 + 58 + 52 + 62 + 62 over 5 EndFraction = 58
Step 2
Find the mean absolute deviation.
StartFraction StartAbsoluteValue 42 minus 42 EndAbsoluteValue + StartAbsoluteValue 38 minus 42 EndAbsoluteValue + StartFraction StartAbsoluteValue 45 minus 42 EndAbsoluteValue + StartFraction StartAbsoluteValue 46 minus 42 EndAbsoluteValue + StartFraction StartAbsoluteValue 39 minus 42 EndAbsoluteValue over 5 EndFraction = 2.8
Find the mean absolute deviation.
StartFraction StartAbsoluteValue 56 minus 58 EndAbsoluteValue + StartAbsoluteValue 58 minus 58 EndAbsoluteValue + StartFraction StartAbsoluteValue 52 minus 58 EndAbsoluteValue + StartFraction StartAbsoluteValue 62 minus 58 EndAbsoluteValue + StartFraction StartAbsoluteValue 62 minus 58 EndAbsoluteValue over 5 EndFraction = 3.2
Step 3 Find the ratio of the differences of the means compared to the mean absolute deviation.
StartFraction 42 over 2.8 EndFraction = 15 Find the ratio of the differences of the means compared to the mean absolute deviation.
StartFraction 58 over 3.2 EndFraction = 18.125
Step 4 The difference in the means is about StartFraction 15 + 18 over 2 EndFraction = 16.5 times the mean absolute deviations.
In which step did Daniela make the first error?
Step 1
Step 2
Step 3
Step 4
Low Temperatures during the School Day This Week and Last Week
Low Temperatures
This Week (Degrees Fahrenheit)
42
38
45
46
39
Low Temperatures
Last Week (Degrees Fahrenheit)
56
58
52
62
62
Daniela used the steps below to find a relationship between the difference in the mean temperatures and the mean absolute deviations of the data sets.
This Week Last Week
Step 1
Find the mean.
StartFraction 42 + 38 + 45 + 46 + 39 over 5 EndFraction = 42
Find the mean.
StartFraction 56 + 58 + 52 + 62 + 62 over 5 EndFraction = 58
Step 2
Find the mean absolute deviation.
StartFraction StartAbsoluteValue 42 minus 42 EndAbsoluteValue + StartAbsoluteValue 38 minus 42 EndAbsoluteValue + StartFraction StartAbsoluteValue 45 minus 42 EndAbsoluteValue + StartFraction StartAbsoluteValue 46 minus 42 EndAbsoluteValue + StartFraction StartAbsoluteValue 39 minus 42 EndAbsoluteValue over 5 EndFraction = 2.8
Find the mean absolute deviation.
StartFraction StartAbsoluteValue 56 minus 58 EndAbsoluteValue + StartAbsoluteValue 58 minus 58 EndAbsoluteValue + StartFraction StartAbsoluteValue 52 minus 58 EndAbsoluteValue + StartFraction StartAbsoluteValue 62 minus 58 EndAbsoluteValue + StartFraction StartAbsoluteValue 62 minus 58 EndAbsoluteValue over 5 EndFraction = 3.2
Step 3 Find the ratio of the differences of the means compared to the mean absolute deviation.
StartFraction 42 over 2.8 EndFraction = 15 Find the ratio of the differences of the means compared to the mean absolute deviation.
StartFraction 58 over 3.2 EndFraction = 18.125
Step 4 The difference in the means is about StartFraction 15 + 18 over 2 EndFraction = 16.5 times the mean absolute deviations.
In which step did Daniela make the first error?
Step 1
Step 2
Step 3
Step 4
Answers
GPT-4o mini
To determine where Daniela made the first error, we can evaluate each step she conducted:
### Step 1: Find the Mean
- **This Week:**
\[
\text{Mean} = \frac{42 + 38 + 45 + 46 + 39}{5} = \frac{210}{5} = 42
\]
This calculation is correct.
- **Last Week:**
\[
\text{Mean} = \frac{56 + 58 + 52 + 62 + 62}{5} = \frac{290}{5} = 58
\]
This calculation is also correct.
### Step 2: Find the Mean Absolute Deviation (MAD)
- **This Week:**
\[
\text{MAD} = \frac{|42 - 42| + |38 - 42| + |45 - 42| + |46 - 42| + |39 - 42|}{5}
\]
\[
= \frac{0 + 4 + 3 + 4 + 3}{5} = \frac{14}{5} = 2.8
\]
This calculation is correct.
- **Last Week:**
\[
\text{MAD} = \frac{|56 - 58| + |58 - 58| + |52 - 58| + |62 - 58| + |62 - 58|}{5}
\]
\[
= \frac{2 + 0 + 6 + 4 + 4}{5} = \frac{16}{5} = 3.2
\]
This calculation is also correct.
### Step 3: Find the Ratio of the Differences of the Means to the MAD
- **This Week:**
\[
\text{Ratio} = \frac{42}{2.8} = 15
\]
This calculation is correct.
- **Last Week:**
\[
\text{Ratio} = \frac{58}{3.2} = 18.125
\]
This calculation is also correct.
### Step 4: Find the Average of the Ratios
In this step, she calculates:
\[
\text{Average Ratio} = \frac{15 + 18.125}{2} = \frac{33.125}{2} = 16.5625
\]
However, she rounded this incorrectly (from 16.5625 to 16.5).
### Conclusion:
Daniela did not make any errors in the computations of means or mean absolute deviations. The rounding in Step 4 is technically not an error in calculation, but rather a rounding decision.
Therefore, **the answer is Step 4**, where she made an incorrect simplification in rounding the average of the ratios. However, if considering strict computation errors, there were none.
### Step 1: Find the Mean
- **This Week:**
\[
\text{Mean} = \frac{42 + 38 + 45 + 46 + 39}{5} = \frac{210}{5} = 42
\]
This calculation is correct.
- **Last Week:**
\[
\text{Mean} = \frac{56 + 58 + 52 + 62 + 62}{5} = \frac{290}{5} = 58
\]
This calculation is also correct.
### Step 2: Find the Mean Absolute Deviation (MAD)
- **This Week:**
\[
\text{MAD} = \frac{|42 - 42| + |38 - 42| + |45 - 42| + |46 - 42| + |39 - 42|}{5}
\]
\[
= \frac{0 + 4 + 3 + 4 + 3}{5} = \frac{14}{5} = 2.8
\]
This calculation is correct.
- **Last Week:**
\[
\text{MAD} = \frac{|56 - 58| + |58 - 58| + |52 - 58| + |62 - 58| + |62 - 58|}{5}
\]
\[
= \frac{2 + 0 + 6 + 4 + 4}{5} = \frac{16}{5} = 3.2
\]
This calculation is also correct.
### Step 3: Find the Ratio of the Differences of the Means to the MAD
- **This Week:**
\[
\text{Ratio} = \frac{42}{2.8} = 15
\]
This calculation is correct.
- **Last Week:**
\[
\text{Ratio} = \frac{58}{3.2} = 18.125
\]
This calculation is also correct.
### Step 4: Find the Average of the Ratios
In this step, she calculates:
\[
\text{Average Ratio} = \frac{15 + 18.125}{2} = \frac{33.125}{2} = 16.5625
\]
However, she rounded this incorrectly (from 16.5625 to 16.5).
### Conclusion:
Daniela did not make any errors in the computations of means or mean absolute deviations. The rounding in Step 4 is technically not an error in calculation, but rather a rounding decision.
Therefore, **the answer is Step 4**, where she made an incorrect simplification in rounding the average of the ratios. However, if considering strict computation errors, there were none.