Question

1. a set of data points is shown below. 8, 11, 12, 10, 9, 7, 5, 3, 9 which statement is true if a data point of 45 is added to the data set? a the mean and median will both increase. b the mean will increase and the median will decrease. c the mean will increase and the median will remain the same. d the mean and median will both decrease. questions 18 and 19 are gridded response items that require you to write your answers in the boxes provided on your answer sheet. write only one number or symbol in each box and fill in the circle in each column that matches what you have printed. fill in only one circle in each column. 18 what is the distance, in units, between the y - intercept of ( f(x)=x^{2}+7x - 18 ) and the y - intercept of the linear function that passes through the points shown in the table below? (\begin{array}{|c|c|c|c|c|}hline x&-5&10&25&60\hline g(x)&2&11&20&41\hlineend{array}) 19 what is the value of ( x ) in the equation shown below? ( 2(2x + 8)-4x = 10x + 4 )

Explanation:

Question 17 (Data Set and Measures)

First, let's analyze the original data set: \( 8, 11, 12, 10, 9, 7, 5, 3, 9 \)

Step 1: Find Original Mean

Sum of original data: \( 8 + 11 + 12 + 10 + 9 + 7 + 5 + 3 + 9 = 74 \)
Number of data points: \( 9 \)
Original mean: \( \frac{74}{9} \approx 8.22 \)

Step 2: Find Original Median

Sort the data: \( 3, 5, 7, 8, 9, 9, 10, 11, 12 \)
Median (middle value, 5th term): \( 9 \)

Step 3: Find Original Mode

Mode is the most frequent value: \( 9 \) (appears twice)

Step 4: Add 45 to the data set

New data set: \( 3, 5, 7, 8, 9, 9, 10, 11, 12, 45 \) (10 data points)

Step 5: Find New Mean

Sum of new data: \( 74 + 45 = 119 \)
New mean: \( \frac{119}{10} = 11.9 \) (increases)

Step 6: Find New Median

Sort new data: \( 3, 5, 7, 8, 9, 9, 10, 11, 12, 45 \)
Median (average of 5th and 6th terms): \( \frac{9 + 9}{2} = 9 \) (remains the same)

Step 7: Find New Mode

Mode is still \( 9 \) (appears twice, others once) (remains the same)

So the mean increases, and the median/mode remain the same. Wait, the options:

• A: Mean and median both increase? No, median stays.
• B: Mean increases, median decreases? No, median stays.
• C: Mean increases, median stays? Wait, original median was 9, new median is 9. Mode also stays. Wait, the option C says "The mean will increase and the median will remain the same." (Assuming mode is not mentioned, but median is. Let's recheck.)

Wait, original median: 9 (9th term, sorted: 3,5,7,8,9,9,10,11,12 – 5th term is 9). New data: 10 terms, 5th and 6th are 9 and 9, so median 9. Mean increases from ~8.22 to 11.9. So the correct option is C? Wait, the option C: "The mean will increase and the median will remain the same." (Yes, that matches.)

Question 18 (Linear Function and y-intercept)

We have a linear function \( g(x) \) passing through points \((-5, 2)\), \((10, 11)\), \((25, 20)\), \((60, 41)\). Let's find the slope first.

Step 1: Calculate Slope (\( m \))

Using two points, e.g., \((-5, 2)\) and \((10, 11)\):
\( m = \frac{11 - 2}{10 - (-5)} = \frac{9}{15} = \frac{3}{5} \)

Check with another pair: \((10, 11)\) and \((25, 20)\):
\( m = \frac{20 - 11}{25 - 10} = \frac{9}{15} = \frac{3}{5} \) (consistent)

Step 2: Find Equation of Line

Using point-slope form: \( y - y_1 = m(x - x_1) \) with \((-5, 2)\):
\( y - 2 = \frac{3}{5}(x + 5) \)
Simplify: \( y = \frac{3}{5}x + 3 + 2 = \frac{3}{5}x + 5 \)

So the y-intercept (when \( x = 0 \)) is \( 5 \).

Question 19 (Solve for \( x \))

Equation: \( 2(2x + 8) - 4x = 10x + 4 \)

Step 1: Expand Left Side

\( 4x + 16 - 4x = 10x + 4 \)

Step 2: Simplify Left Side

\( 16 = 10x + 4 \)

Step 3: Solve for \( x \)

Subtract 4: \( 12 = 10x \)
Divide by 10: \( x = \frac{12}{10} = \frac{6}{5} = 1.2 \)

Final Answers:

1. C. The mean will increase and the median will remain the same.
2. The y-intercept is \( \boldsymbol{5} \).
3. The value of \( x \) is \( \boldsymbol{\frac{6}{5}} \) (or \( 1.2 \)).

Answer:

Question 17 (Data Set and Measures)

First, let's analyze the original data set: \( 8, 11, 12, 10, 9, 7, 5, 3, 9 \)

Step 1: Find Original Mean

Sum of original data: \( 8 + 11 + 12 + 10 + 9 + 7 + 5 + 3 + 9 = 74 \)
Number of data points: \( 9 \)
Original mean: \( \frac{74}{9} \approx 8.22 \)

Step 2: Find Original Median

Sort the data: \( 3, 5, 7, 8, 9, 9, 10, 11, 12 \)
Median (middle value, 5th term): \( 9 \)

Step 3: Find Original Mode

Mode is the most frequent value: \( 9 \) (appears twice)

Step 4: Add 45 to the data set

New data set: \( 3, 5, 7, 8, 9, 9, 10, 11, 12, 45 \) (10 data points)

Step 5: Find New Mean

Sum of new data: \( 74 + 45 = 119 \)
New mean: \( \frac{119}{10} = 11.9 \) (increases)

Step 6: Find New Median

Sort new data: \( 3, 5, 7, 8, 9, 9, 10, 11, 12, 45 \)
Median (average of 5th and 6th terms): \( \frac{9 + 9}{2} = 9 \) (remains the same)

Step 7: Find New Mode

Mode is still \( 9 \) (appears twice, others once) (remains the same)

So the mean increases, and the median/mode remain the same. Wait, the options:

• A: Mean and median both increase? No, median stays.
• B: Mean increases, median decreases? No, median stays.
• C: Mean increases, median stays? Wait, original median was 9, new median is 9. Mode also stays. Wait, the option C says "The mean will increase and the median will remain the same." (Assuming mode is not mentioned, but median is. Let's recheck.)

Wait, original median: 9 (9th term, sorted: 3,5,7,8,9,9,10,11,12 – 5th term is 9). New data: 10 terms, 5th and 6th are 9 and 9, so median 9. Mean increases from ~8.22 to 11.9. So the correct option is C? Wait, the option C: "The mean will increase and the median will remain the same." (Yes, that matches.)

Question 18 (Linear Function and y-intercept)

We have a linear function \( g(x) \) passing through points \((-5, 2)\), \((10, 11)\), \((25, 20)\), \((60, 41)\). Let's find the slope first.

Step 1: Calculate Slope (\( m \))

Using two points, e.g., \((-5, 2)\) and \((10, 11)\):
\( m = \frac{11 - 2}{10 - (-5)} = \frac{9}{15} = \frac{3}{5} \)

Check with another pair: \((10, 11)\) and \((25, 20)\):
\( m = \frac{20 - 11}{25 - 10} = \frac{9}{15} = \frac{3}{5} \) (consistent)

Step 2: Find Equation of Line

Using point-slope form: \( y - y_1 = m(x - x_1) \) with \((-5, 2)\):
\( y - 2 = \frac{3}{5}(x + 5) \)
Simplify: \( y = \frac{3}{5}x + 3 + 2 = \frac{3}{5}x + 5 \)

So the y-intercept (when \( x = 0 \)) is \( 5 \).

Question 19 (Solve for \( x \))

Equation: \( 2(2x + 8) - 4x = 10x + 4 \)

Step 1: Expand Left Side

\( 4x + 16 - 4x = 10x + 4 \)

Step 2: Simplify Left Side

\( 16 = 10x + 4 \)

Step 3: Solve for \( x \)

Subtract 4: \( 12 = 10x \)
Divide by 10: \( x = \frac{12}{10} = \frac{6}{5} = 1.2 \)

Final Answers:

1. C. The mean will increase and the median will remain the same.
2. The y-intercept is \( \boldsymbol{5} \).
3. The value of \( x \) is \( \boldsymbol{\frac{6}{5}} \) (or \( 1.2 \)).