AP Statistics: Descriptive Statistics, Distributions, and Data Analysis

Q1. About 75 students in the sample were absent...

Background

Topic: Interpreting Summary Statistics and Quartiles

This question tests your understanding of how to interpret summary statistics (mean, median, quartiles) and use them to estimate the number of observations in different parts of a data distribution.

Key Terms and Formulas:

• Mean: The average value of the data set.

• Median: The middle value when the data is ordered.

• Quartiles (Q1, Q3): Values that divide the data into quarters.

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

Step-by-Step Guidance

1. Recall that the first quartile (Q1) is the value below which 25% of the data fall, and the third quartile (Q3) is the value below which 75% of the data fall.

2. Use the sample size (300 students) to estimate how many students fall below or above certain quartiles or the mean/median.

3. Consider what each answer choice is asking: for example, how many students were absent fewer than 3.5 days (the median), more than 4.1 days (the mean), more than 6 days (Q3), or between 2 and 6 days (between Q1 and Q3).

4. Set up calculations for each option using the quartile positions (e.g., 25% of 300, 50% of 300, etc.).

Try solving on your own before revealing the answer!

Final Answer: D) between 2 and 6 days

Between Q1 (2 days) and Q3 (6 days) is the interquartile range, which contains the middle 50% of the data. 50% of 300 students is 150 students, so about 75 students would be in each quartile. Thus, about 75 students were absent between 2 and 6 days.

Q2. Which of the following could be the 8th value in the data set?

Background

Topic: Effects of Adding a Value on Standard Deviation and Range

This question tests your understanding of how adding a new value to a data set affects the standard deviation and range.

Key Terms and Formulas:

• Standard Deviation ($s$): A measure of spread in the data.

• Range: The difference between the maximum and minimum values.

Step-by-Step Guidance

1. Recall that the range stays the same only if the new value is not less than the minimum or greater than the maximum of the original set.

2. Check the original minimum (3) and maximum (23) values.

3. Consider which options would keep the range unchanged (i.e., the new value must be between 3 and 23, inclusive).

4. Think about which value would increase the standard deviation. Adding a value far from the mean (11) will increase the standard deviation.

5. Set up which of the options fit both criteria: range unchanged and standard deviation increased.

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Final Answer: C) 21

Adding 21 keeps the range the same (since 23 is still the maximum), and because 21 is far from the mean (11), it increases the standard deviation.

Q3. Which of the following could be the interquartile range of the number of hours worked?

Background

Topic: Interpreting Histograms and Calculating the Interquartile Range (IQR)

This question tests your ability to estimate the IQR from a histogram or bar graph.

Key Terms and Formulas:

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

• Quartiles: Q1 is the 25th percentile, Q3 is the 75th percentile.

Step-by-Step Guidance

1. Count the total number of employees (41) and determine the positions of Q1 and Q3 (the 11th and 31st values when ordered).

2. Use the histogram to estimate the values at these positions by adding up frequencies until you reach the 11th and 31st employees.

3. Subtract the estimated Q1 value from the estimated Q3 value to get the IQR.

4. Compare your result to the answer choices to see which is most reasonable.

Try solving on your own before revealing the answer!

Final Answer: B) 19

By estimating the quartile positions from the histogram, the IQR is closest to 19 hours.

Q4. What would be the weight of a raspberry with the same standardized score as a strawberry that weighs 17.5 grams?

Background

Topic: Standardized Scores (z-scores)

This question tests your ability to use z-scores to compare values from different distributions.

Key Terms and Formulas:

• z-score: $z = \frac{x - \mu}{\sigma}$

• $x$ = observed value, $\mu$ = mean, $\sigma$ = standard deviation

Step-by-Step Guidance

1. Calculate the z-score for the strawberry: $z = \frac{17.5 - 15}{1.67}$

2. Set up the equation for the raspberry: $z = \frac{x - 4.5}{0.5}$

3. Set the two z-scores equal to each other, since the question asks for the raspberry with the same standardized score.

4. Solve for $x$ in the raspberry equation.

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Final Answer: A) 5.25 grams

The raspberry with the same z-score as the 17.5g strawberry weighs 5.25 grams.

Q5. For which of the following will the value of the statistic for hours spent babysitting be equal to the value of the corresponding statistic for amount of money earned from babysitting?

Background

Topic: Effects of Linear Transformations on Statistics

This question tests your understanding of how multiplying data by a constant affects measures like mean, standard deviation, quartiles, and z-scores.

Key Terms and Formulas:

• Linear Transformation: $y = a + bx$

• Mean, standard deviation, quartiles, and z-scores are affected differently by multiplication.

Step-by-Step Guidance

1. Recall that multiplying each value by a constant multiplies the mean, standard deviation, and quartiles by that constant, but does not change the z-scores.

2. Consider each statistic in the answer choices and how it is affected by multiplying by 12 (the hourly rate).

3. Determine which statistic remains unchanged between the two variables (hours and money earned).

Try solving on your own before revealing the answer!

Final Answer: D) The standardized score of the maximum

The z-score (standardized score) does not change when all values are multiplied by a constant.

Q6. The dotplot shows the number of goals scored by the United States Women's National Team in each of their 24 soccer matches in a recent season.

Background

Topic: Percentiles, Mean, Median, Standard Deviation, and Effects of Outliers

This question tests your ability to interpret dotplots, calculate percentiles, mean, median, and understand the effect of removing an outlier on standard deviation.

Key Terms and Formulas:

• Percentile: The percentage of data values below a given value.

• Mean: $\bar{x} = \frac{\sum x_i}{n}$

• Median: The middle value in an ordered data set.

• Standard Deviation: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Step-by-Step Guidance

1. For percentile: Count how many matches had fewer than 4 goals, divide by 24, and multiply by 100 to get the percentile.

2. For mean and median: Add up all the goals and divide by 24 for the mean; order the data and find the middle value(s) for the median.

3. For standard deviation interpretation: Explain what the value means in the context of goals scored.

4. For the effect of removing 13: Consider how removing an outlier affects the spread (standard deviation) of the data.

Try solving on your own before revealing the answer!

Final Answer:

• a. 4 goals is at the 33rd percentile (8 out of 24 matches had fewer than 4 goals).

• b. Mean ≈ 4.83, Median = 4

• c. The standard deviation of 2.55 means that the number of goals scored typically varies by about 2.55 from the mean.

• d. Removing 13 (an outlier) would decrease the standard deviation, making the data less spread out.

Q7. All 82 students in Mrs. Gallas’ classes were asked how many concerts they had attended in the past year. The results are in the table.

Background

Topic: Five-Number Summary, Outlier Detection, Boxplots, and Transformations

This question tests your ability to find the five-number summary, check for outliers using the 1.5 × IQR rule, create a boxplot, and apply transformations to data.

Key Terms and Formulas:

• Five-number summary: Minimum, Q1, Median, Q3, Maximum

• Interquartile Range (IQR): $IQR = Q_3 - Q_1$

• Outlier rule: Outlier if $x < Q_1 - 1.5 \times IQR$ or $x > Q_3 + 1.5 \times IQR$

Step-by-Step Guidance

1. List the data in order and find the minimum, Q1, median, Q3, and maximum.

2. Calculate the IQR and use the 1.5 × IQR rule to check for outliers.

3. Draw a boxplot using the five-number summary.

4. To find the median and IQR of the amount spent, multiply each value by $122.50$ and apply the same summary statistics.

Try solving on your own before revealing the answer!

Final Answer:

• a. Five-number summary: Min = 0, Q1 = 1, Median = 1, Q3 = 3, Max = 7

• b. No outliers (using 1.5 × IQR rule)

• c. Boxplot: Drawn using the five-number summary

• d. Median amount spent = $122.50, IQR = $245.00

Q8. The distribution of the number of songs on each of the original albums by Beyoncé and Taylor Swift are shown below.

Background

Topic: Comparing Distributions Using Summary Statistics

This question tests your ability to compare two distributions using measures such as mean, median, standard deviation, and quartiles.

Key Terms and Formulas:

• Mean, Median, Standard Deviation, Quartiles

• Comparing center, spread, and shape of distributions

Step-by-Step Guidance

1. Count the number of albums for each artist with at least 15 songs using the summary statistics or dotplots.

2. Compare the means, medians, standard deviations, and quartiles to describe similarities and differences in the distributions.

3. Discuss which artist has more variability and which has a higher center.

Try solving on your own before revealing the answer!

Final Answer:

• a. Beyoncé: 3 albums, Taylor Swift: 5 albums have at least 15 songs.

• b. Taylor Swift's albums have a slightly higher mean and more variability; Beyoncé's albums are more tightly clustered.