Step-by-Step Guidance for Introductory Statistics Worksheet

Q1a. Calculate $\sum x^2$ for the data set $x = \{7, 5, 3, -2, 9\}$.

Background

Topic: Summation Notation and Squaring Values

This question tests your ability to use summation notation and apply the squaring operation to each value in a data set.

Key Terms and Formulas

• $\sum x^2$ means "sum of the squares of each value in x".

• For a data set $x = \{x_1, x_2, ..., x_n\}$, $\sum x^2 = x_1^2 + x_2^2 + ... + x_n^2$.

Step-by-Step Guidance

1. List each value in the data set: $7, 5, 3, -2, 9$.

2. Square each value: $7^2, 5^2, 3^2, (-2)^2, 9^2$.

3. Write out the sum: $7^2 + 5^2 + 3^2 + (-2)^2 + 9^2$.

4. Add up the squared values to get the total sum.

Try solving on your own before revealing the answer!

Final Answer: $168$

$7^2 + 5^2 + 3^2 + (-2)^2 + 9^2 = 49 + 25 + 9 + 4 + 81 = 168$

We squared each value and summed them to get the result.

Q1b. Calculate $\sum f(x) \cdot x$ for $x = \{7, 5, 3, -2, 9\}$ and $f(x) = \sqrt{x}$.

Background

Topic: Function Application and Summation

This question tests your ability to apply a function to each value in a data set and then multiply by the original value, summing the results.

Key Terms and Formulas

• $f(x) = \sqrt{x}$ means "take the square root of x".

• $\sum f(x) \cdot x$ means "for each x, multiply $f(x)$ by $x$, then sum".

Step-by-Step Guidance

1. List each value in $x$: $7, 5, 3, -2, 9$.

2. Apply $f(x) = \sqrt{x}$ to each value. Note: $\sqrt{-2}$ is not a real number, so consider only real values.

3. Multiply each $f(x)$ by its corresponding $x$ value: $7 \cdot \sqrt{7}$, $5 \cdot \sqrt{5}$, $3 \cdot \sqrt{3}$, $-2 \cdot \sqrt{-2}$, $9 \cdot \sqrt{9}$.

4. Sum the results for all real values.

Try solving on your own before revealing the answer!

Final Answer: $7\sqrt{7} + 5\sqrt{5} + 3\sqrt{3} + 9\sqrt{9}$ (excluding $-2\sqrt{-2}$ as it is not real)

Numerically, $7\sqrt{7} \approx 18.52$, $5\sqrt{5} \approx 11.18$, $3\sqrt{3} \approx 5.20$, $9\sqrt{9} = 27$. The sum is approximately $61.9$.

We excluded $-2\sqrt{-2}$ since it is not a real number.

Q1c. Calculate $\frac{1}{n} \sum (x-2)^2$ for $x = \{7, 5, 3, -2, 9\}$.

Background

Topic: Mean of Squared Deviations

This question tests your ability to compute the average of squared differences from a constant (here, 2).

Key Terms and Formulas

• $n$ is the number of data points.

• For each $x$, calculate $(x-2)^2$.

• Sum these values and divide by $n$.

Step-by-Step Guidance

1. Count the number of data points: $n = 5$.

2. For each $x$, compute $x - 2$.

3. Square each result: $(x - 2)^2$.

4. Sum all squared values.

5. Divide the sum by $n$ to get the mean.

Try solving on your own before revealing the answer!

Final Answer: $29.2$

$(7-2)^2 = 25$, $(5-2)^2 = 9$, $(3-2)^2 = 1$, $(-2-2)^2 = 16$, $(9-2)^2 = 49$. Sum: $25+9+1+16+49=100$. Divide by $5$: $100/5 = 20$.

Correction: The sum is $100$, so the mean is $20$.

Q2a. Find the mean, median, mode, and midrange for the data set $\{-1, 2, 3, 3, 7, 12\}$.

Background

Topic: Measures of Central Tendency

This question tests your ability to calculate four common measures of center: mean, median, mode, and midrange.

Key Terms and Formulas

• Mean: $\text{mean} = \frac{\sum x}{n}$

• Median: Middle value when data is ordered

• Mode: Most frequent value

• Midrange: $\text{midrange} = \frac{\text{min} + \text{max}}{2}$

Step-by-Step Guidance

1. Order the data: $-1, 2, 3, 3, 7, 12$.

2. Calculate the mean: Add all values and divide by the number of values.

3. Find the median: Since there are 6 values, take the average of the 3rd and 4th values.

4. Identify the mode: Look for the value that appears most frequently.

5. Calculate the midrange: Add the smallest and largest values, then divide by 2.

Try solving on your own before revealing the answer!

Final Answer:

• Mean: $\frac{-1+2+3+3+7+12}{6} = 4.33$

• Median: $(3+3)/2 = 3$

• Mode: $3$

• Midrange: $\frac{-1+12}{2} = 5.5$

Each measure gives a different perspective on the center of the data.

Q2b. Find the mean, median, mode, and midrange for the data set $\{14, 22, 17, 2, 41, 19, 22, 21, 20, 6, 11, 31, 12, 10, 7\}$.

Background

Topic: Measures of Central Tendency

This question tests your ability to calculate mean, median, mode, and midrange for a larger data set.

Key Terms and Formulas

• Mean: $\text{mean} = \frac{\sum x}{n}$

• Median: Middle value when data is ordered

• Mode: Most frequent value

• Midrange: $\text{midrange} = \frac{\text{min} + \text{max}}{2}$

Step-by-Step Guidance

1. Order the data from smallest to largest.

2. Count the number of values ($n = 15$).

3. Calculate the mean: Add all values and divide by $n$.

4. Find the median: The 8th value in the ordered list.

5. Identify the mode: Look for the value(s) that appear most frequently.

6. Calculate the midrange: Add the smallest and largest values, then divide by 2.

Try solving on your own before revealing the answer!

Final Answer:

• Mean: $18.2$

• Median: $17$

• Mode: $22$

• Midrange: $\frac{2+41}{2} = 21.5$

Ordering the data and applying each formula gives these results.

Q3a. Categorize the data collected for "How many days per week do you eat dessert after dinner?" Is it qualitative, discrete (quantitative), or continuous (quantitative)?

Background

Topic: Data Types

This question tests your understanding of different types of data: qualitative, discrete quantitative, and continuous quantitative.

Key Terms

• Qualitative: Describes categories or qualities (not numbers).

• Discrete Quantitative: Countable numbers (like days per week).

• Continuous Quantitative: Measured values that can take any value within a range.

Step-by-Step Guidance

1. Consider the question: "How many days per week do you eat dessert after dinner?"

2. Think about the possible answers: 0, 1, 2, ..., 7 (whole numbers).

3. Determine if the data is categorical or numerical.

4. If numerical, decide if it is discrete (countable) or continuous (measurable).

Try solving on your own before revealing the answer!

Final Answer: Discrete Quantitative

The data is countable (number of days), so it is discrete quantitative.

Q3b. Calculate the mean, median, mode, and midrange for the class data on "days per week eating dessert".

Background

Topic: Measures of Central Tendency

This question tests your ability to calculate mean, median, mode, and midrange for a real-world data set.

Key Terms and Formulas

• Mean: $\text{mean} = \frac{\sum x}{n}$

• Median: Middle value when data is ordered

• Mode: Most frequent value

• Midrange: $\text{midrange} = \frac{\text{min} + \text{max}}{2}$

Step-by-Step Guidance

1. Collect the data from the class (list all values).

2. Order the data from smallest to largest.

3. Calculate the mean: Add all values and divide by the number of data points.

4. Find the median: Identify the middle value(s).

5. Identify the mode: Find the value(s) that appear most frequently.

6. Calculate the midrange: Add the smallest and largest values, then divide by 2.

Try solving on your own before revealing the answer!

Final Answer:

The mean, median, mode, and midrange depend on the actual class data collected. Use the formulas above to compute each measure once you have the data.

Q3c. Construct the relative frequency distribution for the class data on "days per week eating dessert".

Background

Topic: Frequency and Relative Frequency Distributions

This question tests your ability to organize data into a frequency table and calculate relative frequencies.

Key Terms and Formulas

• Frequency: Number of times each value occurs.

• Relative Frequency: $\text{relative frequency} = \frac{\text{frequency}}{\text{total number of data points}}$

Step-by-Step Guidance

1. List all possible values (e.g., 0 to 7 days).

2. Count how many times each value appears in the data.

3. Calculate the relative frequency for each value using the formula above.

4. Organize the results in a table with columns for value, frequency, and relative frequency.

Try solving on your own before revealing the answer!

Final Answer:

The relative frequency distribution depends on the class data. For each value, divide its frequency by the total number of responses to get the relative frequency.

Q3d. Draw a histogram for the relative frequency distribution.

Background

Topic: Histograms

This question tests your ability to visually represent a frequency distribution using a histogram.

Key Terms

• Histogram: A bar graph showing frequency or relative frequency for each value or interval.

Step-by-Step Guidance

1. Use the relative frequency distribution table from the previous step.

2. On the horizontal axis, label the values (e.g., 0 to 7 days).

3. On the vertical axis, label the relative frequency (from 0 to 1).

4. Draw bars for each value, with height equal to its relative frequency.

Try solving on your own before revealing the answer!

Final Answer:

The histogram will show bars for each value (days per week), with heights corresponding to their relative frequencies. Use your table to draw the bars accurately.

Q3e. Which measure of center seems most appropriate to describe the center of the data, and why?

Background

Topic: Choosing Measures of Center

This question tests your ability to interpret data and select the most meaningful measure of center based on the distribution.

Key Terms

• Mean: Sensitive to outliers.

• Median: Resistant to outliers/skewed data.

• Mode: Most common value.

Step-by-Step Guidance

1. Examine the shape of the data distribution (from your histogram).

2. Consider whether the data is symmetric, skewed, or has outliers.

3. Recall that the mean is affected by extreme values, while the median is more robust.

4. Think about which measure best represents the "typical" value for this data.

Try solving on your own before revealing the answer!

Final Answer:

If the data is skewed or has outliers, the median is usually the most appropriate measure of center. If the data is symmetric and has no outliers, the mean may be suitable. The mode can be useful if one value is much more common than others.