Question

challenge the table shows the diameters of 11 bolts. diameters (inches): 0.43, 0.48, 0.44, 0.47; 0.45, 0.51, 0.46, 0.46; 0.48, 0.49, 0.50. you want to find the diameter that marks the middle of the lower half of the data set. which data display would be best to represent the data? which diameter marks the middle of the lower half of the data set? use pencil and paper. what if four more measurements exactly the same as the one you found are included with the data set? what effect does this have on the middle of the lower half of the data set? ... which data display would be best to represent the data? a would be best to represent the data.

Explanation:

To solve the problem of finding the diameter that marks the middle of the lower half of the data set (the first quartile, \(Q_1\)) and determining the best data display, we follow these steps:

Step 1: Organize the Data

First, we list all the diameters from the table and sort them in ascending order. The diameters are:
\(0.43, 0.44, 0.45, 0.46, 0.46, 0.47, 0.48, 0.48, 0.49, 0.50, 0.51\)

Step 2: Identify the Lower Half

The data set has 11 values. The lower half of the data (for finding \(Q_1\)) includes the first 5 values (since \(11\) is odd, we exclude the median, which is the 6th value: \(0.47\)).

Lower half data: \(0.43, 0.44, 0.45, 0.46, 0.46\)

Step 3: Find the Middle of the Lower Half

The middle of the lower half (the median of the lower half) is the 3rd value in this sorted sub - set.

For the sub - set \(0.43, 0.44, 0.45, 0.46, 0.46\), the median (middle value) is \(0.45\). So the diameter that marks the middle of the lower half of the data set is \(0.45\).

Step 4: Best Data Display

A box - and - whisker plot is the best data display for showing the quartiles (including the middle of the lower half, \(Q_1\)) of a data set. It clearly visualizes the minimum, \(Q_1\), median, \(Q_3\), and maximum values, making it ideal for identifying the middle of the lower half.

Step 5: Effect of Adding Four More Measurements

If we add four more measurements, we first need to re - sort the new data set (now with \(11 + 4=15\) values). Then we re - identify the lower half (the first 7 values, since \(15\) is odd, we exclude the median, the 8th value). The median of this new lower half (new \(Q_1\)) will depend on the values of the four added measurements. If the added measurements are less than or equal to the original \(Q_1(0.45)\), the new \(Q_1\) may decrease or stay the same; if they are greater, the new \(Q_1\) may increase or stay the same, but it will change the position and potentially the value of the middle of the lower half.

Final Answers

• The diameter marking the middle of the lower half: \(0.45\)
• Best data display: Box - and - whisker plot
• Effect of adding four measurements: The middle of the lower half ( \(Q_1\)) will likely change (its value and position in the data set) depending on the values of the added measurements.

Answer:

To solve the problem of finding the diameter that marks the middle of the lower half of the data set (the first quartile, \(Q_1\)) and determining the best data display, we follow these steps:

Step 1: Organize the Data

First, we list all the diameters from the table and sort them in ascending order. The diameters are:
\(0.43, 0.44, 0.45, 0.46, 0.46, 0.47, 0.48, 0.48, 0.49, 0.50, 0.51\)

Step 2: Identify the Lower Half

The data set has 11 values. The lower half of the data (for finding \(Q_1\)) includes the first 5 values (since \(11\) is odd, we exclude the median, which is the 6th value: \(0.47\)).

Lower half data: \(0.43, 0.44, 0.45, 0.46, 0.46\)

Step 3: Find the Middle of the Lower Half

The middle of the lower half (the median of the lower half) is the 3rd value in this sorted sub - set.

For the sub - set \(0.43, 0.44, 0.45, 0.46, 0.46\), the median (middle value) is \(0.45\). So the diameter that marks the middle of the lower half of the data set is \(0.45\).

Step 4: Best Data Display

A box - and - whisker plot is the best data display for showing the quartiles (including the middle of the lower half, \(Q_1\)) of a data set. It clearly visualizes the minimum, \(Q_1\), median, \(Q_3\), and maximum values, making it ideal for identifying the middle of the lower half.

Step 5: Effect of Adding Four More Measurements

If we add four more measurements, we first need to re - sort the new data set (now with \(11 + 4=15\) values). Then we re - identify the lower half (the first 7 values, since \(15\) is odd, we exclude the median, the 8th value). The median of this new lower half (new \(Q_1\)) will depend on the values of the four added measurements. If the added measurements are less than or equal to the original \(Q_1(0.45)\), the new \(Q_1\) may decrease or stay the same; if they are greater, the new \(Q_1\) may increase or stay the same, but it will change the position and potentially the value of the middle of the lower half.

Final Answers

• The diameter marking the middle of the lower half: \(0.45\)
• Best data display: Box - and - whisker plot
• Effect of adding four measurements: The middle of the lower half ( \(Q_1\)) will likely change (its value and position in the data set) depending on the values of the added measurements.