The mean of the daily low temperatures for a 6-day period in a certain city was recorded as being 17.0ºF. It was later determined that the low temperature for 1 of these 6 days was recorded incorrectly. If that day’s low temperature was 3.0ºF higher than originally recorded, what is the difference between the incorrectly recorded mean and the correct mean?



To solve this problem, we'll calculate both the incorrectly recorded mean and the correct mean, then find the difference between the two.

### Step 1: Understanding the Problem
We know the following:
- The incorrect mean of the 6-day period is 17.0°F.
- The low temperature for one of the days was recorded 3.0°F higher than it should have been.

### Step 2: Incorrectly Recorded Mean
The mean of the temperatures for the 6 days is 17.0°F. This mean is calculated by dividing the sum of the temperatures for the 6 days by 6. Let the sum of the recorded temperatures be \( S_{\text{incorrect}} \). 

\[
\frac{S_{\text{incorrect}}}{6} = 17.0
\]
Multiplying both sides by 6 to find the sum of the temperatures:
\[
S_{\text{incorrect}} = 17.0 \times 6 = 102.0
\]
So, the sum of the recorded temperatures is 102.0°F.

### Step 3: Corrected Mean
We are told that one day's temperature was recorded 3.0°F higher than it should have been. Let's denote the incorrect temperature for this day as \( T_{\text{incorrect}} \), and the correct temperature for that day as \( T_{\text{correct}} \).

Since \( T_{\text{incorrect}} = T_{\text{correct}} + 3.0 \), the correct temperature can be written as:
\[
T_{\text{correct}} = T_{\text{incorrect}} - 3.0
\]
Now, let's calculate the correct sum of temperatures, \( S_{\text{correct}} \), by subtracting 3.0°F from the incorrect sum:
\[
S_{\text{correct}} = S_{\text{incorrect}} - 3.0 = 102.0 - 3.0 = 99.0
\]

### Step 4: Correct Mean
Now, we calculate the correct mean by dividing the correct sum by 6:
\[
\text{Correct Mean} = \frac{S_{\text{correct}}}{6} = \frac{99.0}{6} = 16.5
\]

### Step 5: Difference Between the Incorrect and Correct Means
The difference between the incorrectly recorded mean and the correct mean is:
\[
\text{Difference} = 17.0 - 16.5 = 0.5
\]

### Final Answer:
The difference between the incorrectly recorded mean and the correct mean is **0.5°F**.

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Would you like more details or have any questions? Here are some related questions to explore:

1. How would the answer change if the temperature error was larger than 3.0°F?
2. What is the impact of increasing or decreasing the number of days in this problem?
3. How would you calculate the mean if there were more than one incorrect reading?
4. How can this method be used for correcting the mean in other statistical problems?
5. What other types of average calculations might be affected by similar errors?

**Tip:** Always remember that small errors in data can have a significant impact on the calculated averages, especially when the number of data points is small.

The mean of the daily low temperatures for a 6-day period in a certain city was recorded as being 17.0ºF. It was later determined that the low temperature for 1 of these 6 days was recorded incorrectly. If that day’s low temperature was 3.0ºF higher than originally recorded, what is the difference between the incorrectly recorded mean and the correct mean?

Learn how to correct the mean of a 6-day temperature data set when one day’s temperature was recorded 3.0°F too high. This problem explains the process of recalculating the correct mean and the difference caused by the error. Suitable for students in Grades 7-9.

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