Answer :
To solve this question, we need to find the value of [tex]\( x \)[/tex] that makes a particular number appear most frequently (the mode) in the given data set.
Let's analyze the data set: [tex]\( 3, 5, 6, 7, 5, 4, 7, 5, 6, (x+1), 8, 7 \)[/tex].
1. Count the frequency of each number:
- 3 appears 1 time
- 4 appears 1 time
- 5 appears 3 times
- 6 appears 2 times
- 7 appears 3 times
- 8 appears 1 time
Currently, the numbers 5 and 7 are the modes, each appearing 3 times.
2. Determine the impact of [tex]\( x+1 \)[/tex]:
The number [tex]\( x+1 \)[/tex] is the unknown value. Depending on what [tex]\( x+1 \)[/tex] is, it may change which number is the mode. To find [tex]\( x \)[/tex], we need to see how it can affect the current mode frequencies.
3. Checking possible values for [tex]\( x+1 \)[/tex]:
- If [tex]\( x+1 = 5 \)[/tex], then the new frequency of 5 becomes 4 (since 5 appears 3 times already).
- If [tex]\( x+1 = 7 \)[/tex], then the new frequency of 7 becomes 4 (same reasoning as above).
- If [tex]\( x+1 = 3, 4, 6, \)[/tex] or [tex]\( 8 \)[/tex], none of these values will surpass the frequency of 4 achieved by setting [tex]\( x+1 \)[/tex] to 5 or 7.
If [tex]\( x+1 = 5 \)[/tex] or [tex]\( x+1 = 7 \)[/tex], that value will clearly become the sole mode.
4. Choose a suitable value for [tex]\( x \)[/tex]:
To calculate [tex]\( x \)[/tex] when [tex]\( x+1 = 5 \)[/tex]:
[tex]\[
x + 1 = 5 \implies x = 4
\][/tex]
To calculate [tex]\( x \)[/tex] when [tex]\( x+1 = 7 \)[/tex]:
[tex]\[
x + 1 = 7 \implies x = 6
\][/tex]
5. Conclusion:
The values of [tex]\( x \)[/tex] that can make a single number the mode are [tex]\( x = 4 \)[/tex] (for 5 to be the mode) or [tex]\( x = 6 \)[/tex] (for 7 to be the mode). Either of these values is a correct answer. You can choose [tex]\( x = 4 \)[/tex] to make 5 the mode.
Let's analyze the data set: [tex]\( 3, 5, 6, 7, 5, 4, 7, 5, 6, (x+1), 8, 7 \)[/tex].
1. Count the frequency of each number:
- 3 appears 1 time
- 4 appears 1 time
- 5 appears 3 times
- 6 appears 2 times
- 7 appears 3 times
- 8 appears 1 time
Currently, the numbers 5 and 7 are the modes, each appearing 3 times.
2. Determine the impact of [tex]\( x+1 \)[/tex]:
The number [tex]\( x+1 \)[/tex] is the unknown value. Depending on what [tex]\( x+1 \)[/tex] is, it may change which number is the mode. To find [tex]\( x \)[/tex], we need to see how it can affect the current mode frequencies.
3. Checking possible values for [tex]\( x+1 \)[/tex]:
- If [tex]\( x+1 = 5 \)[/tex], then the new frequency of 5 becomes 4 (since 5 appears 3 times already).
- If [tex]\( x+1 = 7 \)[/tex], then the new frequency of 7 becomes 4 (same reasoning as above).
- If [tex]\( x+1 = 3, 4, 6, \)[/tex] or [tex]\( 8 \)[/tex], none of these values will surpass the frequency of 4 achieved by setting [tex]\( x+1 \)[/tex] to 5 or 7.
If [tex]\( x+1 = 5 \)[/tex] or [tex]\( x+1 = 7 \)[/tex], that value will clearly become the sole mode.
4. Choose a suitable value for [tex]\( x \)[/tex]:
To calculate [tex]\( x \)[/tex] when [tex]\( x+1 = 5 \)[/tex]:
[tex]\[
x + 1 = 5 \implies x = 4
\][/tex]
To calculate [tex]\( x \)[/tex] when [tex]\( x+1 = 7 \)[/tex]:
[tex]\[
x + 1 = 7 \implies x = 6
\][/tex]
5. Conclusion:
The values of [tex]\( x \)[/tex] that can make a single number the mode are [tex]\( x = 4 \)[/tex] (for 5 to be the mode) or [tex]\( x = 6 \)[/tex] (for 7 to be the mode). Either of these values is a correct answer. You can choose [tex]\( x = 4 \)[/tex] to make 5 the mode.
