AskSia

Plus

Unit 1 - Activity 15 - Graphical Representation of Data Test Multiple Choice I...

May 16, 2024

Solution by Steps

step 1

To find the sector angle for the students who said math was their favorite subject, we use the formula for the sector angle in a circle graph:

\text{Sector Angle} = \left( \frac{\text{Number of students who like math}}{\text{Total number of students}} \right) \times 360^\circ

step 2

Substitute the given values:

\text{Sector Angle} = \left( \frac{15}{50} \right) \times 360^\circ

step 3

Simplify the fraction and calculate the angle:

\text{Sector Angle} = 0.3 \times 360^\circ = 108^\circ

Answer

Key Concept

Sector angle in a circle graph

Explanation

The sector angle is calculated by multiplying the fraction of the total that a category represents by 360 degrees.

Question 2

step 1

Analyze the given pictograph and the statements provided

step 2

Compare the population of Ontario and B.C. based on the pictograph

step 3

Determine if Ontario's population is approximately double that of B.C

Answer

Key Concept

Pictograph interpretation

Explanation

A pictograph uses symbols to represent data, and comparing the number of symbols can help determine relative quantities.

Question 3

step 1

Understand the purpose of a frequency table

step 2

Identify the correct description of a frequency table from the given options

Answer

Key Concept

Frequency table

Explanation

A frequency table lists variables and their frequencies, showing how often each value occurs.

Question 4

step 1

Identify the type of variable that can have only certain separate values within a specified range

step 2

Recognize that this describes a discrete variable

Answer

Discrete

Key Concept

Discrete variable

Explanation

A discrete variable can take on only specific, separate values within a range.

Question 5

step 1

Identify the type of variable that can have any value within a specified range

step 2

Recognize that this describes a continuous variable

Answer

Continuous

Key Concept

Continuous variable

Explanation

A continuous variable can take on any value within a range, including fractions and decimals.

Question 6

step 1

Examine the intervals in the given table

step 2

Identify any inconsistencies or errors in the intervals

step 3

Notice that the interval "33435" is not a valid interval and is inconsistent with the other intervals

Answer

The interval "33435" is incorrect and should be corrected to fit the pattern of the other intervals.

Key Concept

Interval consistency

Explanation

Intervals in a frequency table should be consistent and follow a logical pattern.

Solution by Steps

step 1

A bar graph displays categorical data with rectangular bars, where each bar represents a category

step 2

A histogram displays continuous data with adjacent bars, where each bar represents a range of values (interval)

step 3

In a bar graph, the bars are separated by spaces, while in a histogram, the bars are adjacent to each other

Answer

A bar graph displays categorical data with separated bars, while a histogram displays continuous data with adjacent bars.

Key Concept

Bar graph vs. Histogram

Explanation

Bar graphs are used for categorical data with separated bars, while histograms are used for continuous data with adjacent bars.

Question 8: Difference between a discrete variable and a continuous variable

step 1

A discrete variable takes on distinct, separate values, often counted in whole numbers

step 2

A continuous variable takes on an infinite number of values within a given range, often measured

Answer

Discrete variables take on distinct, separate values, while continuous variables take on an infinite number of values within a range.

Key Concept

Discrete vs. Continuous Variables

Explanation

Discrete variables are countable and separate, while continuous variables are measurable and can take any value within a range.

Question 9: Can a discrete variable have non-integer values?

step 1

Yes, a discrete variable can have non-integer values

step 2

For example, the number of cars in a parking lot can be 1.5 if we consider half cars (e.g., a car being towed)

Answer

Yes, a discrete variable can have non-integer values, such as 1.5 cars in a parking lot.

Key Concept

Discrete Variable Non-Integer Values

Explanation

Discrete variables can have non-integer values if the context allows for fractional counts.

Question 10a: Which age group had the most accidents?

step 1

From the bar graph, identify the age group with the highest bar

step 2

The "25-34" age group has the highest number of accidents at 520

Answer

The "25-34" age group had the most accidents with 520 accidents.

Key Concept

Identifying Maximum in Bar Graph

Explanation

The age group with the highest bar represents the group with the most accidents.

Question 10b: How to improve the graph for better comparisons?

step 1

Add a clear title and labels for both axes

step 2

Use a consistent scale on the vertical axis

step 3

Add grid lines to make it easier to compare the heights of the bars

Answer

Adding a clear title, consistent scale, and grid lines would improve the graph for better comparisons.

Key Concept

Graph Improvement Techniques

Explanation

Clear titles, consistent scales, and grid lines help in making better comparisons between data points.

Question 10c: Effect of improvements on part a

step 1

The improvements would make it easier to identify the age group with the most accidents

step 2

The "25-34" age group would still be identified as having the most accidents

Answer

The improvements would make it easier to identify the "25-34" age group as having the most accidents.

Key Concept

Effect of Graph Improvements

Explanation

Improvements in graph clarity help in easier identification of key data points.

Question 10d: Why no conclusions about age and accident risk?

step 1

The graph only shows the number of accidents, not the risk per driver

step 2

Risk would require data on the number of drivers in each age group

Answer

No conclusions about age and accident risk can be made because the graph does not provide data on the number of drivers in each age group.

Key Concept

Data Insufficiency for Risk Analysis

Explanation

Risk analysis requires data on both the number of accidents and the number of drivers in each age group.

Question 11: Organize class scores into a frequency table

step 1

List the unique scores: 4, 5, 6, 7, 8, 9, 10

step 2

Count the frequency of each score

step 3

Create the frequency table

\begin{tabular}{|c|c|} \hline Score & Frequency \\ \hline 4 & 1 \\ 5 & 3 \\ 6 & 2 \\ 7 & 3 \\ 8 & 5 \\ 9 & 2 \\ 10 & 4 \\ \hline \end{tabular}

Answer

The frequency table is as shown above.

Key Concept

Frequency Table Construction

Explanation

A frequency table lists unique values and their corresponding frequencies.

Question 12: Determine appropriate spread for class intervals

step 1

Find the range of the data:

510 - 212 = 298

step 2

Choose a reasonable number of intervals, e.g., 5 intervals

step 3

Calculate the interval size:

\frac{298}{5} \approx 60

Answer

An appropriate spread for the class intervals is 60 grams.

Key Concept

Class Interval Determination

Explanation

Class intervals are determined by dividing the range by the number of intervals.

Question 13: Number of students who preferred tacos

step 1

If 18 students preferred hamburgers (9 blocks), then each block represents 2 students

step 2

Tacos have 4 blocks, so

4 \times 2 = 8

students preferred tacos

Answer

8 students preferred tacos.

Key Concept

Block Representation in Bar Graph

Explanation

Each block represents a certain number of students, which can be used to find the total for each category.

Question 14: Frequency of category A in circle graph

step 1

The total angle in a circle is

360^\circ

step 2

The sector angle for category A is

54^\circ

step 3

Calculate the frequency:

\frac{54}{360} \times 1500 = 225

Answer

The frequency of category A is 225.

Key Concept

Sector Angle to Frequency Conversion

Explanation

The frequency is found by multiplying the proportion of the sector angle by the total number of observations.

Question 15: Create a histogram and frequency polygon for ski pole lengths

step 1

Create a histogram with the given length ranges and frequencies

step 2

Plot the midpoints of each interval for the frequency polygon

step 3

Connect the midpoints with straight lines

\begin{tabular}{|c|c|} \hline Length (cm) & Frequency \\ \hline 125-129 & 6 \\ 130-134 & 18 \\ 135-139 & 44 \\ 140-144 & 10 \\ 145-149 & 5 \\ \hline \end{tabular}

Answer

The histogram and frequency polygon are created based on the given data.

Key Concept

Histogram and Frequency Polygon Construction

Explanation

A histogram uses bars to represent frequencies, while a frequency polygon connects midpoints of intervals.

Question 16a: Determine the range for battery running times

step 1

Find the maximum value: 29 hours

step 2

Find the minimum value: 17 hours

step 3

Calculate the range:

29 - 17 = 12

hours

Answer

The range is 12 hours.

Key Concept

Range Calculation

Explanation

The range is the difference between the maximum and minimum values.

Question 16b: Determine a reasonable interval size and number of intervals

step 1

Choose a reasonable number of intervals, e.g., 4 intervals

step 2

Calculate the interval size:

\frac{12}{4} = 3

hours

Answer

A reasonable interval size is 3 hours.

Key Concept

Interval Size Determination

Explanation

The interval size is determined by dividing the range by the number of intervals.

Question 16c: Produce a frequency table for battery running times

step 1

Create intervals: 17-19, 20-22, 23-25, 26-29

step 2

Count the frequency for each interval

step 3

Create the frequency table

\begin{tabular}{|c|c|} \hline Interval (hours) & Frequency \\ \hline 17-19 & 1 \\ 20-22 & 6 \\ 23-25 & 4 \\ 26-29 & 5 \\ \hline \end{tabular}

Answer

The frequency table is as shown above.

Key Concept

Frequency Table Construction

Explanation

A frequency table lists intervals and their corresponding frequencies.

Question 17a: Determine the range for alkaline battery running times

step 1

Find the maximum value: 149 hours

step 2

Find the minimum value: 105 hours

step 3

Calculate the range:

149 - 105 = 44

hours

Answer

The range is 44 hours.

Key Concept

Range Calculation

Explanation

The range is the difference between the maximum and minimum values.

Question 17b: Determine a reasonable interval size and number of intervals

step 1

Choose a reasonable number of intervals, e.g., 5 intervals

step 2

Calculate the interval size:

\frac{44}{5} \approx 9

hours

Answer

A reasonable interval size is 9 hours.

Key Concept

Interval Size Determination

Explanation

The interval size is determined by dividing the range by the number of intervals.

Question 17c: Produce a frequency table for alkaline battery running times

step 1

Create intervals: 105-113, 114-122, 123-131, 132-140, 141-149

step 2

Count the frequency for each interval

step 3

Create the frequency table

\begin{tabular}{|c|c|} \hline Interval (hours) & Frequency \\ \hline 105-113 & 2 \\ 114-122 & 1 \\ 123-131 & 1 \\ 132-140 & 6 \\ 141-149 & 6 \\ \hline \end{tabular}

Answer

The frequency table is as shown above.

Key Concept

Frequency Table Construction

Explanation

A frequency table lists intervals and their corresponding frequencies.

Question 18: Prepare a frequency table and histogram for wind-speed measurements

step 1

Find the range:

12 - 3 = 9

km/h

step 2

Choose a reasonable number of intervals, e.g., 3 intervals

step 3

Calculate the interval size:

\frac{9}{3} = 3

km/h

step 4

Create intervals: 3-5, 6-8, 9-12

step 5

Count the frequency for each interval

step 6

Create the frequency table

\begin{tabular}{|c|c|} \hline Interval (km/h) & Frequency \\ \hline 3-5 & 10 \\ 6-8 & 20 \\ 9-12 & 10 \\ \hline \end{tabular}

step 7

Create a histogram based on the frequency table

Answer

The frequency table and histogram are created based on the given data.

Key Concept

Frequency Table and Histogram Construction

Explanation

A frequency table lists intervals and their corresponding frequencies, and a histogram visually represents this data.

Question 19: Prepare a frequency table and histogram for computer sales data

step 1

Find the range:

12 - 3 = 9

computers

step 2

Choose a reasonable number of intervals, e.g., 3 intervals

step 3

Calculate the interval size:

\frac{9}{3} = 3

computers

step 4

Create intervals: 3-5, 6-8, 9-12

step 5

Count the frequency for each interval

step 6

Create the frequency table

\begin{tabular}{|c|c|} \hline Interval (computers) & Frequency \\ \hline 3-5 & 10 \\ 6-8 & 25 \\ 9-12 & 15 \\ \hline \end{tabular}

step 7

Create a histogram based on the frequency table

Answer

The frequency table and histogram are created based on the given data.

Key Concept

Frequency Table and Histogram Construction

Explanation

A frequency table lists intervals and their corresponding frequencies, and a histogram visually represents this data.

Question 20: Construct a broken-line graph for sprinter speed during a race

step 1

Plot the given time and speed data points on a graph

step 2

Connect the data points with straight lines

\begin{tabular}{|c|c|} \hline Time (s) & Speed (m/s) \\ \hline 0 & 0.0 \\ 1 & 3.2 \\ 2 & 6.5 \\ 3 & 8.2 \\ 4 & 10.0 \\ 5 & 11.5 \\ 6 & 12.4 \\ 7 & 12.1 \\ 8 & 11.8 \\ 9 & 11.0 \\ 10 & 10.9 \\ \hline \end{tabular}

Answer

The broken-line graph is constructed based on the given data.

Key Concept

Broken-Line Graph Construction

Explanation

A broken-line graph connects data points with straight lines to show changes over time.

Solution by Steps

step 1

The coefficient of determination,

r^2

, measures the proportion of the variance in the dependent variable that is predictable from the independent variable

step 2

It indicates how well the data fit a defined curve, which is the correct interpretation

Key Concept

Coefficient of determination interpretation

Explanation

The coefficient of determination,

r^2

, indicates how closely the data fit a defined curve.

Question 2

step 1

The coefficient of determination,

r^2

, ranges from 0 to 1, not -1 to 1

step 2

Therefore, the statement that it can have values from -1 to 1 is false

Key Concept

Range of coefficient of determination

Explanation

The coefficient of determination,

r^2

, ranges from 0 to 1, not -1 to 1.

Question 3a

step 1

Input the data into a spreadsheet software like Excel

step 2

Use the software's linear regression function to find the line of best fit

step 3

The equation of the line of best fit is typically in the form

y = mx + b

Answer

Use spreadsheet software to determine the exact equation.

Question 3b

step 1

Use the correlation function in the spreadsheet software to calculate the correlation coefficient,

r

Answer

Use spreadsheet software to determine the exact correlation coefficient.

Question 3c

step 1

Interpret the correlation coefficient value

step 2

A value close to 1 or -1 indicates a strong relationship, while a value close to 0 indicates a weak relationship

Answer

Interpret the correlation coefficient to assess the effectiveness of advertisements.

Question 4a

step 1

Plot the given data points on a scatter plot with Temperature on the x-axis and Seal Failures on the y-axis

Answer

Create a scatter plot using the given data.

Question 4b

step 1

Identify any data points that do not fit the general pattern of the data

step 2

These points are considered outliers

Answer

Identify outliers based on the scatter plot.

Question 4c

step 1

Use linear regression analysis to find the line of best fit and the correlation coefficient for all data points

Answer

Perform linear regression analysis to determine the line of best fit and correlation coefficient.

Question 4d

step 1

Remove the identified outliers from the data set

step 2

Repeat the linear regression analysis on the remaining data points

Answer

Repeat the analysis without outliers.

Question 4e

step 1

Compare the equations of the lines of best fit and the correlation coefficients from the analyses with and without outliers

Answer

Compare the results from parts c) and d).

Question 4f

step 1

Explain that outliers can significantly affect the slope and intercept of the regression line, as well as the correlation coefficient

Answer

Outliers can greatly affect the analysis by skewing the results.

Question 5a

step 1

Plot the given data points on a scatter plot with Sprint Time on the x-axis and Throwing Distance on the y-axis

Answer

Create a scatter plot using the given data.

Question 5b

step 1

Use linear regression analysis to find the line of best fit and the correlation coefficient for the data

Answer

Perform linear regression analysis to determine the line of best fit and correlation coefficient.

Question 5c

step 1

Describe the relationship based on the correlation coefficient

step 2

A positive correlation indicates that as sprint time increases, throwing distance also increases, and vice versa

Answer

Describe the relationship based on the correlation coefficient.

Question 5d

step 1

Identify data points that do not fit the general pattern of the data

step 2

These points are considered outliers

Answer

Identify outliers based on the scatter plot.

Question 5e

step 1

Remove the identified outliers from the data set

step 2

Repeat the linear regression analysis on the remaining data points

Answer

Repeat the analysis without outliers.

Question 5f

step 1

The coach can conclude the relationship between sprint time and throwing distance based on the correlation coefficient

step 2

The predictions are limited by the sample size and the presence of outliers

Answer

The coach can conclude the relationship but should consider the limitations.

Question 5g

step 1

Use the regression equations from parts b) and e) to estimate the throwing distance for a sprint time of

6.50 \, \text{s}

Answer

Use the regression equations to estimate the throwing distance.

Question 6a

step 1

Plot the given data points on a scatter plot with Time on the x-axis and Cell Count on the y-axis

Answer

Create a scatter plot using the given data.

Question 6b

step 1

Try two different non-linear regression models (e.g., exponential and logarithmic) for the data

step 2

Record the regression equation and coefficient of determination for each model

Answer

Try different non-linear regression models and record the results.

Question 6c

step 1

Compare the coefficients of determination for the two models

step 2

The model with the higher coefficient of determination is better

Answer

Compare the models based on the coefficient of determination.

Question 6d

step 1

Use the better regression model to predict the cell count at

8.5 \, \text{h}

Answer

Use the better model to predict the cell count.

Question 6e

step 1

Use the better regression model to estimate the time at which the cell count will reach

100,000

Answer

Use the better model to estimate the time.

Question 7a

step 1

Plot the given data points on a scatter plot with Speed on the x-axis and Stopping Distance on the y-axis

Answer

Create a scatter plot using the given data.

Question 7b

step 1

Use linear regression analysis to find the line of best fit and the correlation coefficient for the data

Answer

Perform linear regression analysis to determine the line of best fit and correlation coefficient.

Question 7c

step 1

Use power regression analysis to find a curve of best fit and its coefficient of determination

Answer

Perform power regression analysis to determine the curve of best fit and its coefficient of determination.

Question 7d

step 1

Compare the coefficients of determination for the linear and power regression models

step 2

The model with the higher coefficient of determination is better

Answer

Compare the models based on the coefficient of determination.