AP Statistics 1.6: Describing Distributions

Describe a quantitative distribution with shape, center, spread, and unusual features, then use those features to support a contextual claim.

[Maximum number: 4]

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.

(a)

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.

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

(i)

Describe the shape of the distribution of the sample of whistle prices. Justify your response using appropriate values from the summary statistics table.

(b)

It can often be difficult to determine whether the distribution of sample data is skewed by looking at a graph of the data and the summary statistics, particularly when the sample size is small. Thus, statisticians sometimes measure how skewed a data set is. One such measure is Pearson's coefficient of skewness, which is calculated using the following formula.

\text { Pearson's Coefficient of Skewness }=\frac{3(\bar{x}-m)}{s}

In the formula, $\bar{x}$ is the sample mean, m is the sample median, and s is the sample standard deviation.

(i)

(i) Calculate Pearson's coefficient of skewness for Julio's sample of 20 whistle prices. Show your work.

The following graph shows conclusions that can be made about the shape of the distribution of sample data based on Pearson's coefficient of skewness and sample size.

Model Solution
$\frac{3(5.12-4.885)}{0.743}\approx0.949.$

Scoring components
1. Calculates Pearson’s coefficient of skewness with work shown.

(ii)

Indicate the value of the Pearson's coefficient of skewness you calculated in part (c-i) for the appropriate sample size by marking it with an "X" on the preceding graph.

(c)

Consider your work in part (c).

Julio's inference procedure in part (a-i) needs one of the following requirements to be satisfied to verify the normality condition.

- The sample size is greater than or equal to 30.

- If the sample size is less than 30, the distribution of the sample data is not strongly skewed and does not have outliers.

(i)

What should you conclude about the shape of the distribution of the sample of whistle prices? Justify your response.

The following histograms summarize the teaching year for the teachers at two high schools, A and B.

Teaching year is recorded as an integer, with first-year teachers recorded as 1, second-year teachers recorded as 2, and so on. Both sets of data have a mean teaching year of 8.2, with data recorded from 200 teachers at High School A and 221 teachers at High School B. On the histograms, each interval represents possible integer values from the left endpoint up to but not including the right endpoint.

The standard deviation of the teaching year for the 221 teachers at High School B is 7.2. If one teacher is selected at random from High School B, what is the probability that the teaching year for the selected teacher will be within 1 standard deviation of the mean of 8.2 ? Justify your answer.

Part (c):
The interval mean plus or minus 1 standard deviation on either side of the mean is $8.2 \pm 7.2$ , or from 1.0 year to 15.4 years. Because teaching year is recorded as an integer, the interval includes teaching years 1 to 15. The number of teachers in that interval can be found by adding the heights of the five bars in the histogram for the intervals from 1 to 16, which includes 79+34+28+29+19=189. Therefore the probability is $\frac{189}{221} \approx 0.8552$ .

Scoring

Parts (a), (b), and (c) are scored as essentially correct (E), partially correct (P), or incorrect (I).
Part (a) is scored as follows:
Essentially correct (E) if the response satisfies the following three components:

States that the median is 6 for High School B and the median is 7 for High School A.

Provides a reasonable explanation of how the decision was made.

Provides the definition of the median or explicitly applies the definition of a median as a criterion in reaching their decision.

OR
Essentially correct (E) if the response satisfies the following three components:

States that the median is 6 for High School B and the median is 7 for High School A.

States that High School B shows a skewed distribution (or High School A shows a less skewed distribution).

Provides a reasonable explanation of how the more skewed distribution (High School B) would be the one with a larger separation between the mean and median.

Partially correct (P) if the response satisfies the first component and only one of the other two components required for E.

Incorrect (I) if the response does not meet the criteria for E or P.
Note: An incorrect statistical statement in the response will result in E being lowered to P, but not P being lowered to I. For example,

- If either distribution is described as left skewed, normal, or approximately normal;

- If the discussion would indicate a median different than 7 for High School A or a median different than 6 for High School B.

Part (b) is scored as follows:
Essentially correct (E) if the response satisfies the following two components:

The correct answer that the mean is 7.73.

Enough work to show that the answer was obtained as a weighted average of the two individual means.

Partially correct (P) if the response satisfies only one of the two components.
Incorrect (I) if the response does not satisfy the requirements for E or P.

Part (c) is scored as follows:
Essentially correct (E) if the response satisfies the following three components:

Calculates that the appropriate interval is 1 to 15.4 or 1 to 15 teaching years.

Correctly sums the counts of data values in the numerator based on the intervals provided.

Computes the probability using 221 as the denominator.

Partially correct (P) if the response satisfies only two of the three components;
OR
if the response reports the correct probability (0.8552) without supporting work.
Incorrect (I) if the response satisfies at most one of the three components.
Notes:

- If the response attempts to use the Empirical Rule or normal distribution to provide the desired probability, the response is scored I.

- If an incorrect count is shown in component 2, for instance by including the interval from 16 to 19, then component 3 is satisfied if that incorrect count is divided by 221 to find the reported probability.

- It is acceptable if the count is slightly off because of difficulty reading the exact heights of the bars in the histogram.

- If only one of component 2 or component 3 is missing, but the correct probability (0.8552) is reported, the response can be scored E.

- If the response recognizes that all values in the histogram bins up to 16 fall within one standard deviation of the mean and reports the interval as 1 to 16, component 1 is satisfied.

Complete Response
Three parts essentially correct

Substantial Response
Two parts essentially correct and one part partially correct

OR
Part (a) essentially correct and two parts partially correct

Developing Response
Two parts essentially correct and no parts partially correct

OR
Part (b) or part (c) essentially correct and one or two parts partially correct

OR
Three parts partially correct

Minimal Response
One part essentially correct

OR
No parts essentially correct and one or two parts partially correct

Intent of Question

The primary goals of this question were to assess a student's ability to (1) describe what constitutes a Type II error for a specific hypothesis test; (2) specify a rejection region in terms of values of the sample mean; (3) compute the power of a test for a specific value in the alternative hypothesis; (4) recognize the definition of power; and (5) understand the impact of increasing the sample size on the power of a test.

Solution

All question bank results loaded

Covered syllabus points

1.6 Descriptions for One Quantitative Variable Distributions question 1

Question (a)

Question (i)

Question (b)

Question (i)

Question (ii)

Question (c)

Question (i)

1.6 Descriptions for One Quantitative Variable Distributions question 2