AP Statistics 1.7 Summary Statistics

[Maximum number: 4]

The manager of an automotive company is interested in comparing the gas mileages for cars manufactured in Country A and cars manufactured in Country B. The manager selected a random sample of 100 cars manufactured in Country A and a random sample of 100 cars manufactured in Country B. The gas mileages for each sample, in miles per gallon (mpg), are summarized in the boxplots.

Boxplots of Gas Mileage for Each Country

Scoring for Question 1	Score
Complete Response<br>Three parts essentially correct	4
Substantial Response<br>Two parts essentially correct and one part partially correct	3
Developing Response<br>Two parts essentially correct and no part partially correct OR<br>One part essentially correct and one or two parts partially correct OR<br>Three parts partially correct	2
Minimal Response<br>One part essentially correct and no parts partially correct OR<br>No part essentially correct and two parts partially correct	1

(a)

The manager will create a new boxplot with the combined data from the sample of cars manufactured in Country A and the sample of cars manufactured in Country B.

(i)

What is the range of the combined data set? Justify your answer.

Model Solution
The maximum is 40 mpg and the minimum is 14 mpg, so the range is $40-14=26\text{ mpg}$ . The boxplots show that all Country A mileages are below 40 mpg and all Country B mileages are above 14 mpg.

Scoring components
1. Correctly calculates 26 mpg as the range.
2. Justifies the range using 40 mpg as the maximum and 14 mpg as the minimum.

(ii)

What is a possible value of the median of the combined data set? Justify your answer by referencing the boxplots shown.

[Maximum number: 1]

The kelvin is a unit of measurement for temperature; 0 K is absolute zero, the temperature at which all thermal motion ceases. Conversion from Fahrenheit to Kelvin is given by $\mathrm{K}=\frac{5}{9} \times(\mathrm{F}-32)+273$ . The average daily temperature in Monrovia, Liberia, is 78.35°F with a standard deviation of 6.3°F. If a scientist converts Monrovia daily temperatures to the Kelvin scale, what will be the new mean and standard deviation?

A

Mean, 231.75 K; standard deviation, 3.5 K

B

Mean, 298.75 K; standard deviation, 3.5 K

C

Mean, 298.75 K; standard deviation, 258.72 K

D

Mean, 298.75 K; standard deviation, 276.5 K

A company sells a certain type of whistle. The price of the whistle varies from store to store. Julio, a statistician at the company, wants to estimate the mean price, in dollars ($), of this type of whistle at all stores that sell the whistle.

Julio wants to examine some characteristics of the distribution of the sample of whistle prices.

Julio called the managers of 20 randomly selected stores that sell the whistle and recorded the price of the whistle at each store. Following is a dotplot of Julio's data.

Price of the Whistle at 20 Stores

The summary statistics for Julio's data are shown in the following table.

Sample Size	Mean	Standard Deviation	Minimum	$\mathrm{Q}_{1}$	Median	$\mathrm{Q}_{3}$	Maximum
20	5.12	0.743	4.25	4.51	4.885	5.475	6.58

Summary Statistics for Julio's Data

Using the $1.5 \times$ IQR rule, determine whether there are any outliers in the sample of whistle prices. Justify your response.

Based on the $1.5 \times$ IQR rule, there are not whistle prices in this sample that would be considered outliers. A whistle price is an outlier using this method if it is more than 1.5 × IQR below the first quartile $\left(\mathrm{Q}_{1}\right)$ or more than $1.5 \times$ IQR above the third quartile $\left(\mathrm{Q}_{3}\right)$ . Because

\begin{aligned} & \mathrm{Q}_{1}-1.5 \times \mathrm{IQR} \\ & =4.51-1.5(5.475-4.51) \\ & =3.0625 \end{aligned}

and the minimum value (4.25) is greater than 3.0625, there are no outliers to the left. Because

\begin{aligned} & \mathrm{Q}_{3}+1.5 \times \mathrm{IQR} \\ & =5.475+1.5(5.475-4.51) \\ & =6.9225 \end{aligned}

and the maximum value (6.58) is less than 6.9225, there are no outliers to the right.

Scoring
Essentially correct (E) if the response satisfies the following four components:

In part (b-i) the response indicates the distribution is skewed to the right or approximately symmetric

In part (b-i) the response justifies the response to component 1 with appropriate reasoning based on the summary statistics

In part (b-ii) the response provides a justification for stating there are no outliers by correctly calculating the lower and upper outlier criteria with work shown

In part (b-ii) the response states whether there are outliers based on the calculated lower and upper outlier criteria

Partially correct (P) if the response satisfies three of the four components required for E.

OR
satisfies components 1 and 2 OR components 3 and 4 of components 1-4 required for E.

Incorrect (I) if the response does not meet the criteria for E or P.

Additional Notes:

- If the response states skewed right, to satisfy component 2 the response should refer to the median being less than the mean. If the response states approximately symmetric, the response should refer to the median and mean being close together.

- A response that indicates skewed left does not satisfy components 1 or 2.

- If the response says the distribution is skewed to the right, examples of responses that satisfy component 2 include:

the mean is higher than the median,

the ratio between the mean and median is greater than 1,

the difference between the maximum and median is greater than the difference between the median and minimum, and

the difference between the maximum and third quartile is greater than the difference between the first quartile and minimum.

- For responses that score P, the number of components satisfied should be considered if holistic scoring is required.

The manager of an automotive company is interested in comparing the gas mileages for cars manufactured in Country A and cars manufactured in Country B. The manager selected a random sample of 100 cars manufactured in Country A and a random sample of 100 cars manufactured in Country B. The gas mileages for each sample, in miles per gallon (mpg), are summarized in the boxplots.

Boxplots of Gas Mileage for Each Country

The manager will create a new boxplot with the combined data from the sample of cars manufactured in Country A and the sample of cars manufactured in Country B.

What is a possible value of the median of the combined data set? Justify your answer by referencing the boxplots shown.

The manager of an automotive company is interested in comparing the gas mileages for cars manufactured in Country A and cars manufactured in Country B. The manager selected a random sample of 100 cars manufactured in Country A and a random sample of 100 cars manufactured in Country B. The gas mileages for each sample, in miles per gallon (mpg), are summarized in the boxplots.

Boxplots of Gas Mileage for Each Country

The manager will create a new boxplot with the combined data from the sample of cars manufactured in Country A and the sample of cars manufactured in Country B.

What is the range of the combined data set? Justify your answer.

Model Solution
The maximum is 40 mpg and the minimum is 14 mpg, so the range is $40-14=26\text{ mpg}$ . The boxplots show that all Country A mileages are below 40 mpg and all Country B mileages are above 14 mpg.

Scoring components
1. Correctly calculates 26 mpg as the range.
2. Justifies the range using 40 mpg as the maximum and 14 mpg as the minimum.

Studies have shown that foods rich in compounds known as flavonoids help lower blood pressure. Researchers conducted a study to investigate whether there was a greater reduction in blood pressure for people who consumed dark chocolate, which contains flavonoids, than people who consumed white chocolate, which does not contain flavonoids. Twenty-five healthy adults agreed to participate in the study and add 3.5 ounces of chocolate to their daily diets. Of the 25 participants, 13 were randomly assigned to the dark chocolate group and the rest were assigned to the white chocolate group. All participants had their blood pressure recorded, in millimeters of mercury (mmHg), before adding chocolate to their daily diets and again 30 days after adding chocolate to their daily diets.
The reduction in blood pressure (before minus after) for each of the participants in the two groups is shown in the dotplots below.

The researchers found the mean reduction in blood pressure for those who consumed dark chocolate is $\bar{x}_{\text {dark }}=6.08 \mathrm{mmHg}$ and the mean reduction in blood pressure for those who consumed white chocolate is $\bar{x}_{\text {white }}=0.42 \mathrm{mmHg}$ .
Determine and compare the medians of the reduction in blood pressure for the two groups.

The researchers found the mean reduction in blood pressure for those who consumed dark chocolate is x_dark =6.08 mmHg and the mean reduction in blood pressure for those who consumed white chocolate is x_white =0.42 mmHg.

Determine and compare the medians of the reduction in blood pressure for the two groups.

Emma is moving to a large city and is investigating typical monthly rental prices of available one-bedroom apartments. She obtained a random sample of rental prices for 50 one-bedroom apartments taken from a Web site where people voluntarily list available apartments.

The distribution of the 50 rental prices of the available apartments is shown in the following histogram.

Because Emma does not have the resources to develop the theoretical sampling distribution, she estimates the sampling distribution of the sample median using a process called bootstrapping. In the bootstrapping process, a computer program performs the following steps.

- Take a random sample, with replacement, of size 50 from the original sample.

- Calculate and record the median of the sample.

- Repeat the process to obtain a total of 15,000 medians.

Emma ran the bootstrap process, and the following frequency table is the bootstrap distribution showing her results of generating 15,000 medians.

The bootstrap distribution provides an approximation of the sampling distribution of the sample median. A confidence interval for the median can be constructed using a percentage of the values in the middle of the bootstrap distribution.

Emma wants to estimate the typical rental price of a one-bedroom apartment in the city. Based on the distribution shown, what is a disadvantage of using the mean rather than the median as an estimate of the typical rental price?

AP Statistics 1.7 Summary Statistics for One Quantitative Variable Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question (a)

Question (i)

Question (ii)

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7