Step-by-Step Guidance for STATC1000X Practice Test #1

Q1. What is the student’s mean score, given the scores and their percent contributions to the final grade?

Background

Topic: Weighted Mean

This question tests your ability to calculate a weighted mean, where each score contributes a different percentage to the final grade.

Key Terms and Formulas

• Weighted Mean: The mean where each value is multiplied by its weight (percentage), and the results are summed.

$\text{Weighted Mean} = \frac{\sum (x_i \times w_i)}{\sum w_i}$

• $x_i$ = individual score

• $w_i$ = weight (as a decimal, e.g., 20\% = 0.20)

Step-by-Step Guidance

1. List each score and its corresponding percent of the final grade. Convert each percent to a decimal (e.g., 20\% becomes 0.20).

2. Multiply each score by its decimal weight.

3. Add up all the weighted scores from step 2.

4. Add up all the weights (they should sum to 1 if the percentages are correct).

Try solving on your own before revealing the answer!

Final Answer: 93.1

Using the formula, the weighted mean is $100 \times 0.20 + 89 \times 0.30 + 100 \times 0.10 + 92 \times 0.40 = 93.1$.

This is the student's mean score, accounting for the different weights of each assessment.

Q2a. Find the mean, median, and mode of the LSAT scores: 174, 172, 169, 176, 169, 170, 175.

Background

Topic: Measures of Central Tendency

This question asks you to compute the mean, median, and mode for a small data set, and to consider if each measure is appropriate.

Key Terms and Formulas

• Mean: The arithmetic average.

• Median: The middle value when data are ordered.

• Mode: The value(s) that appear most frequently.

$\text{Mean} = \frac{\sum x_i}{n}$

Step-by-Step Guidance

1. Order the data from smallest to largest: 169, 169, 170, 172, 174, 175, 176.

2. To find the mean, add all the values and divide by the number of data points (7).

3. To find the median, identify the middle value in the ordered list.

4. To find the mode, look for the value(s) that appear most frequently.

Try solving on your own before revealing the answer!

Final Answer:

Mean: 172.1, Median: 172, Mode: 169

All three measures can be found and represent the center for this data set.

Q2b. Find the mean, median, and mode for the number of vehicle lanes at 16 Canadian border ports: 8, 6, 10, 3, 6, 11, 17, 2, 2, 6, 1, 10, 3, 19, 10, 5.

Background

Topic: Measures of Central Tendency

This question asks you to compute the mean, median, and mode for a larger data set, and to consider if each measure is appropriate.

Key Terms and Formulas

• Same as above: Mean, Median, Mode.

Step-by-Step Guidance

1. Order the data from smallest to largest.

2. Calculate the mean by summing all values and dividing by 16.

3. Find the median by averaging the 8th and 9th values in the ordered list.

4. Identify the mode(s) by finding the most frequent value(s).

Try solving on your own before revealing the answer!

Final Answer:

Mean: 8, Median: 7, Mode: 6

All three measures can be found, but the mean may be influenced by high values (like 17 and 19).

Q3. Use a stem-and-leaf plot to display the number of hours 24 nurses work per week.

Background

Topic: Data Visualization (Stem-and-Leaf Plot)

This question tests your ability to organize quantitative data into a stem-and-leaf plot, which helps visualize the distribution.

Key Terms and Concepts

• Stem-and-Leaf Plot: A way to display data where each value is split into a "stem" (all but the last digit) and a "leaf" (the last digit).

Step-by-Step Guidance

1. Order the data from smallest to largest.

2. Decide on the stems (e.g., 2 for 20s, 3 for 30s, 4 for 40s, 5 for 50s).

3. For each data value, write the leaf (last digit) next to the appropriate stem.

4. Arrange the leaves in ascending order for each stem.

Try constructing the plot on your own before revealing the answer!

Final Answer:

Stem | Leaf 2 | 4 3 | 0 2 2 3 5 5 6 6 6 6 8 8 9 4 | 0 0 0 0 0 0 0 0 0 0 8 5 | 0

This plot shows the distribution of hours worked per week.

Q4. Use a dot plot to display the life spans (in days) of 15 houseflies: 9, 9, 4, 11, 10, 5, 13, 9, 7, 11, 6, 8, 14, 10, 6.

Background

Topic: Data Visualization (Dot Plot)

This question tests your ability to create a dot plot, which is useful for small data sets to show frequency of each value.

Key Terms and Concepts

• Dot Plot: A simple graph where each data value is shown as a dot above its value on a number line.

Step-by-Step Guidance

1. Order the data from smallest to largest.

2. Draw a number line that includes all values from the minimum to the maximum.

3. For each data value, place a dot above the corresponding number on the line. Stack dots for repeated values.

Try drawing the dot plot before revealing the answer!

Final Answer:

4: • 5: • 6: •• 7: • 8: • 9: ••• 10: •• 11: •• 13: • 14: •

This dot plot visually represents the distribution of housefly life spans.

Q5. Use the frequency distribution to find (a) class width, (b) class midpoints, and (c) class boundaries for Toledo, OH, Average Normal Temperatures (℉).

Background

Topic: Frequency Distributions and Class Intervals

This question tests your understanding of how to interpret and analyze grouped data in a frequency distribution.

Key Terms and Formulas

• Class Width: The difference between the lower limits of two consecutive classes.

• Class Midpoint: The average of the lower and upper class limits.

• Class Boundaries: The values that separate classes without gaps.

$\text{Class Width} = \text{Lower limit of 2nd class} - \text{Lower limit of 1st class}$ $\text{Class Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$ $\text{Class Boundaries} = \text{Upper limit of one class} + 0.5$ and $\text{Lower limit of next class} - 0.5$

Step-by-Step Guidance

1. Find the class width by subtracting the lower limit of the first class from the lower limit of the second class.

2. Calculate the midpoint for each class by averaging its lower and upper limits.

3. Determine the class boundaries by finding the halfway point between the upper limit of one class and the lower limit of the next.

Try working through each part before revealing the answer!

Final Answer:

(a) Class width: 8 (b) Class midpoints: 28.5, 36.5, 44.5, 52.5, 60.5, 68.5, 76.5 (c) Class boundaries: 24.5, 32.5, 40.5, 48.5, 56.5, 64.5, 72.5, 80.5

These values help you interpret the frequency distribution accurately.

Q6. Identify the sampling technique used and discuss potential sources of bias (if any). Explain.

Background

Topic: Sampling Methods and Bias

This question tests your understanding of different sampling techniques and the concept of bias in data collection.

Key Terms and Concepts

• Sampling Technique: The method used to select a sample from a population (e.g., random, systematic, stratified, cluster, convenience).

• Bias: Systematic error introduced by the sampling method.

Step-by-Step Guidance

1. For each scenario, identify the sampling method (e.g., cluster, systematic, stratified, etc.).

2. Consider if the method could introduce bias (e.g., not all groups represented, convenience, etc.).

3. Explain your reasoning for both the technique and any potential bias.

Try identifying the techniques and biases before revealing the answer!

Final Answer:

a) Cluster sampling; possible bias if selected grids are not representative. b) Systematic sampling; possible bias if there's a pattern in arrivals. c) Stratified sampling; less bias if subplots are similar, but possible if subplots differ greatly.

Each method has strengths and weaknesses regarding bias and representativeness.